#### Sample records for fractional differential equation

1. Differential operator multiplication method for fractional differential equations

Tang, Shaoqiang; Ying, Yuping; Lian, Yanping; Lin, Stephen; Yang, Yibo; Wagner, Gregory J.; Liu, Wing Kam

2016-11-01

Fractional derivatives play a very important role in modeling physical phenomena involving long-range correlation effects. However, they raise challenges of computational cost and memory storage requirements when solved using current well developed numerical methods. In this paper, the differential operator multiplication method is proposed to address the issues by considering a reaction-advection-diffusion equation with a fractional derivative in time. The linear fractional differential equation is transformed into an integer order differential equation by the proposed method, which can fundamentally fix the aforementioned issues for select fractional differential equations. In such a transform, special attention should be paid to the initial conditions for the resulting differential equation of higher integer order. Through numerical experiments, we verify the proposed method for both fractional ordinary differential equations and partial differential equations.

2. Numerical approaches to fractional calculus and fractional ordinary differential equation

Li, Changpin; Chen, An; Ye, Junjie

2011-05-01

Nowadays, fractional calculus are used to model various different phenomena in nature, but due to the non-local property of the fractional derivative, it still remains a lot of improvements in the present numerical approaches. In this paper, some new numerical approaches based on piecewise interpolation for fractional calculus, and some new improved approaches based on the Simpson method for the fractional differential equations are proposed. We use higher order piecewise interpolation polynomial to approximate the fractional integral and fractional derivatives, and use the Simpson method to design a higher order algorithm for the fractional differential equations. Error analyses and stability analyses are also given, and the numerical results show that these constructed numerical approaches are efficient.

3. A generalized fractional sub-equation method for fractional differential equations with variable coefficients

Tang, Bo; He, Yinnian; Wei, Leilei; Zhang, Xindong

2012-08-01

In this Letter, a generalized fractional sub-equation method is proposed for solving fractional differential equations with variable coefficients. Being concise and straightforward, this method is applied to the space-time fractional Gardner equation with variable coefficients. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics.

4. Modeling some real phenomena by fractional differential equations

Almeida, Ricardo; Bastos, Nuno R. O.; Monteiro, M. Teresa T.

2016-11-01

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional differential equations may model more efficiently certain problems than ordinary differential equations. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data, as well as other related parameters.

5. Efficient modified Chebyshev differentiation matrices for fractional differential equations

Dabiri, Arman; Butcher, Eric A.

2017-09-01

This paper compares several fractional operational matrices for solving a system of linear fractional differential equations (FDEs) of commensurate or incommensurate order. For this purpose, three fractional collocation differentiation matrices (FCDMs) based on finite differences are first proposed and compared with Podlubny's matrix previously used in the literature, after which two new efficient FCDMs based on Chebyshev collocation are proposed. It is shown via an error analysis that the use of the well-known property of fractional differentiation of polynomial bases applied to these methods results in a limitation in the size of the obtained Chebyshev-based FCDMs. To compensate for this limitation, a new fast spectrally accurate FCDM for fractional differentiation which does not require the use of the gamma function is proposed. Then, the Schur-Pade and Schur decomposition methods are implemented to enhance and improve numerical stability. Therefore, this method overcomes the previous limitation regarding the size limitation. In several illustrative examples, the convergence and computation time of the proposed FCDMs are compared and their advantages and disadvantages are outlined.

6. Exp-function method for solving fractional partial differential equations.

PubMed

Zheng, Bin

2013-01-01

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

7. Periodicity and positivity of a class of fractional differential equations.

PubMed

Ibrahim, Rabha W; Ahmad, M Z; Mohammed, M Jasim

2016-01-01

Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.

8. On the singular perturbations for fractional differential equation.

PubMed

Atangana, Abdon

2014-01-01

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

9. Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

PubMed Central

Ahmad, M. Z.; Hasan, M. K.; Abbasbandy, S.

2013-01-01

We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided. PMID:24082853

10. Solving fuzzy fractional differential equations using Zadeh's extension principle.

PubMed

Ahmad, M Z; Hasan, M K; Abbasbandy, S

2013-01-01

We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided.

11. Existence of a coupled system of fractional differential equations

SciTech Connect

Ibrahim, Rabha W.; Siri, Zailan

2015-10-22

We manage the existence and uniqueness of a fractional coupled system containing Schrödinger equations. Such a system appears in quantum mechanics. We confirm that the fractional system under consideration admits a global solution in appropriate functional spaces. The solution is shown to be unique. The method is based on analytic technique of the fixed point theory. The fractional differential operator is considered from the virtue of the Riemann-Liouville differential operator.

12. Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.

PubMed

Baranwal, Vipul K; Pandey, Ram K; Singh, Om P

2014-01-01

We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

13. Analytical schemes for a new class of fractional differential equations

Agrawal, O. P.

2007-05-01

Fractional differential equations (FDEs) considered so far contain mostly left (or forward) fractional derivatives. In this paper, we present analytical solutions for a class of FDEs which contain both the left and the right (or the forward and the backward) fractional derivatives. The methods presented use properties of fractional integral operators (which, in many cases, lead to Volterra-type integral equations), an operational approach and a successive approximation method to obtain the solutions. The methods are demonstrated using some examples. The FDEs considered may come from fractional variational calculus (FVC) or from other physical principles. In the case of fractional variational problems (FVPs), the transversality conditions are used to identify appropriate boundary conditions and to solve the problems. It is hoped that this study will lead to further investigations in the field and more elegant solutions would be found.

14. Jacobi wavelet operational matrix of fractional integration for solving fractional integro-differential equation

Rong, Loh Jian; Chang, Phang

2016-02-01

In this paper, we first define generalized shifted Jacobi polynomial on interval and then use it to define Jacobi wavelet. Then, the operational matrix of fractional integration for Jacobi wavelet is being derived to solve fractional differential equation and fractional integro-differential equation. This method can be seen as a generalization of other orthogonal wavelet operational methods, e.g. Legendre wavelets, Chebyshev wavelets of 1st kind, Chebyshev wavelets of 2nd kind, etc. which are special cases of the Jacobi wavelets. We apply our method to a special type of fractional integro-differential equation of Fredholm type.

15. A procedure to construct exact solutions of nonlinear fractional differential equations.

PubMed

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

16. A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

PubMed Central

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972

17. Renormalization of tracer turbulence leading to fractional differential equations.

PubMed

Sánchez, R; Carreras, B A; Newman, D E; Lynch, V E; van Milligen, B Ph

2006-07-01

For many years quasilinear renormalization has been applied to numerous problems in turbulent transport. This scheme relies on the localization hypothesis to derive a linear transport equation from a simplified stochastic description of the underlying microscopic dynamics. However, use of the localization hypothesis narrows the range of transport behaviors that can be captured by the renormalized equations. In this paper, we construct a renormalization procedure that manages to avoid the localization hypothesis completely and produces renormalized transport equations, expressed in terms of fractional differential operators, that exhibit much more of the transport phenomenology observed in nature. This technique provides a first step toward establishing a rigorous link between the microscopic physics of turbulence and the fractional transport models proposed phenomenologically for a wide variety of turbulent systems such as neutral fluids or plasmas.

18. A Solution to the Fundamental Linear Fractional Order Differential Equation

NASA Technical Reports Server (NTRS)

Hartley, Tom T.; Lorenzo, Carl F.

1998-01-01

This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

19. Some existence results on nonlinear fractional differential equations.

PubMed

Baleanu, Dumitru; Rezapour, Shahram; Mohammadi, Hakimeh

2013-05-13

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η

20. Matrix approach to discrete fractional calculus II: Partial fractional differential equations

Podlubny, Igor; Chechkin, Aleksei; Skovranek, Tomas; Chen, YangQuan; Vinagre Jara, Blas M.

2009-05-01

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359-386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.

1. Exact solutions and maximal dimension of invariant subspaces of time fractional coupled nonlinear partial differential equations

2017-01-01

We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.

2. Oscillation of a class of fractional differential equations with damping term.

PubMed

Qin, Huizeng; Zheng, Bin

2013-01-01

We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

3. On invariant analysis of some time fractional nonlinear systems of partial differential equations. I

Singla, Komal; Gupta, R. K.

2016-10-01

An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.

4. Fuzzy fractional functional differential equations under Caputo gH-differentiability

Hoa, Ngo Van

2015-05-01

In this paper the fuzzy fractional functional differential equations (FFFDEs) under the Caputo generalized Hukuhara differentiability are introduced. We study the existence and uniqueness results of solutions for FFFDEs under some suitable conditions. Also the solution to fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives by a modified Adams-Bashforth-Moulton method (MABMM) is presented. The method is illustrated by solving some examples.

5. The (G'/G)-expansion method for the nonlinear time fractional differential equations

Unsal, Omer; Guner, Ozkan; Bekir, Ahmet; Cevikel, Adem C.

2017-01-01

In this paper, we obtain exact solutions of two time fractional differential equations using Jumarie's modified Riemann-Liouville derivative which is encountered in mathematical physics and applied mathematics; namely (3 + 1)-dimensional time fractional KdV-ZK equation and time fractional ADR equation by using fractional complex transform and (G/'G )-expansion method. It is shown that the considered transform and method are very useful in solving nonlinear fractional differential equations.

6. A study of impulsive multiterm fractional differential equations with single and multiple base points and applications.

PubMed

2014-01-01

We discuss the existence and uniqueness of solutions for initial value problems of nonlinear singular multiterm impulsive Caputo type fractional differential equations on the half line. Our study includes the cases for a single base point fractional differential equation as well as multiple base points fractional differential equation. The asymptotic behavior of solutions for the problems is also investigated. We demonstrate the utility of our work by applying the main results to fractional-order logistic models.

7. A Study of Impulsive Multiterm Fractional Differential Equations with Single and Multiple Base Points and Applications

PubMed Central

2014-01-01

We discuss the existence and uniqueness of solutions for initial value problems of nonlinear singular multiterm impulsive Caputo type fractional differential equations on the half line. Our study includes the cases for a single base point fractional differential equation as well as multiple base points fractional differential equation. The asymptotic behavior of solutions for the problems is also investigated. We demonstrate the utility of our work by applying the main results to fractional-order logistic models. PMID:24578623

8. New Solutions of Three Nonlinear Space- and Time-Fractional Partial Differential Equations in Mathematical Physics

Yao, Ruo-Xia; Wang, Wei; Chen, Ting-Hua

2014-11-01

Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.

9. An efficient technique for higher order fractional differential equation.

PubMed

2016-01-01

In this study, we establish exact solutions of fractional Kawahara equation by using the idea of [Formula: see text]-expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.

10. Exponential rational function method for space-time fractional differential equations

Aksoy, Esin; Kaplan, Melike; Bekir, Ahmet

2016-04-01

In this paper, exponential rational function method is applied to obtain analytical solutions of the space-time fractional Fokas equation, the space-time fractional Zakharov Kuznetsov Benjamin Bona Mahony, and the space-time fractional coupled Burgers' equations. As a result, some exact solutions for them are successfully established. These solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.

11. An approximation method for fractional integro-differential equations

Emiroglu, Ibrahim

2015-12-01

In this work, an approximation method is proposed for fractional order linear Fredholm type integrodifferential equations with boundary conditions. The Sinc collocation method is applied to the examples and its efficiency and strength is also discussed by some special examples. The results of the proposed method are compared to the available analytic solutions.

12. On the convergence of difference schemes for fractional differential equations with Robin boundary conditions

Bazzaev, A. K.; Shkhanukov-Lafishev, M. Kh.

2017-01-01

Locally one-dimensional difference schemes for partial differential equations with fractional order derivatives with respect to time and space in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.

13. Soliton solution and other solutions to a nonlinear fractional differential equation

Guner, Ozkan; Unsal, Omer; Bekir, Ahmet; Kadem, Abdelouahab

2017-01-01

In this paper, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the ansatz method and the functional variable method are used to construct exact solutions for (3+1)-dimensional time fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation. This fractional equation is turned into another nonlinear ordinary differential equation by fractional complex transform then these methods are applied to solve it. As a result, some new exact solutions obtained.

14. Practical stability with respect to initial time difference for Caputo fractional differential equations

Agarwal, Ravi; O'Regan, D.; Hristova, S.; Cicek, M.

2017-01-01

Practical stability with initial data difference for nonlinear Caputo fractional differential equations is studied. This type of stability generalizes known concepts of stability in the literature. It enables us to compare the behavior of two solutions when both initial values and initial intervals are different. In this paper the concept of practical stability with initial time difference is generalized to Caputo fractional differential equations. A definition of the derivative of Lyapunov like function along the given nonlinear Caputo fractional differential equation is given. Comparison results using this definition and scalar fractional differential equations are proved. Sufficient conditions for several types of practical stability with initial time difference for nonlinear Caputo fractional differential equations are obtained via Lyapunov functions. Some examples are given to illustrate the results.

15. Numerical solutions and solitary wave solutions of fractional KDV equations using modified fractional reduced differential transform method

Saha Ray, S.

2013-12-01

In this paper, the modified fractional reduced differential transform method (MFRDTM) has been proposed and it is implemented for solving fractional KdV (Korteweg-de Vries) equations. The fractional derivatives are described in the Caputo sense. In this paper, the reduced differential transform method is modified to be easily employed to solve wide kinds of nonlinear fractional differential equations. In this new approach, the nonlinear term is replaced by its Adomian polynomials. Thus the nonlinear initial-value problem can be easily solved with less computational effort. In order to show the power and effectiveness of the present modified method and to illustrate the pertinent features of the solutions, several fractional KdV equations with different types of nonlinearities are considered. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional KdV equations.

16. Constructing conservation laws for fractional-order integro-differential equations

Lukashchuk, S. Yu.

2015-08-01

In a class of functions depending on linear integro-differential fractional-order variables, we prove an analogue of the fundamental operator identity relating the infinitesimal operator of a point transformation group, the Euler-Lagrange differential operator, and Noether operators. Using this identity, we prove fractional-differential analogues of the Noether theorem and its generalizations applicable to equations with fractional-order integrals and derivatives of various types that are Euler-Lagrange equations. In explicit form, we give fractional-differential generalizations of Noether operators that gives an efficient way to construct conservation laws, which we illustrate with three examples.

17. Laplace transform method for linear sequential Riemann Liouville and Caputo fractional differential equations

Vatsala, Aghalaya S.; Sowmya, M.

2017-01-01

Study of nonlinear sequential fractional differential equations of Riemann-Lioville type and Caputo type initial value problem are very useful in applications. In order to develop any iterative methods to solve the nonlinear problems, we need to solve the corresponding linear problem. In this work, we develop Laplace transform method to solve the linear sequential Riemann-Liouville fractional differential equations as well as linear sequential Caputo fractional differential equations of order nq which is sequential of order q. Also, nq is chosen such that (n-1) < nq < n. All our results yield the integer results as a special case when q tends to 1.

18. Exact solutions of some fractional differential equations by various expansion methods

Topsakal, Muammer; Guner, Ozkan; Bekir, Ahmet; Unsal, Omer

2016-10-01

In this paper, we construct the exact solutions of some nonlinear spacetime fractional differential equations involving modified Riemann-Liouville derivative in mathematical physics and applied mathematics; namely the fractional modified Benjamin-Bona- Mahony (mBBM) and Kawahara equations by using G'/G and (G'/G, 1/G)-expansion methods.

19. A new analytical approach to solve some of the fractional-order partial differential equations

2017-03-01

The aim of the present paper is to present an analytical method for the time fractional biological population model, time fractional Burgers, time fractional Cahn-Hilliard, space-time fractional Whitham-Broer-Kaup, space-time fractional Fokas equations by using the generalized tanh-coth method. The fractional derivative is described in the sense of the modified Riemann-Liouville derivatives. The method gives an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. We have obtained the exact solutions for the aforementioned nonlinear fractional equations. A generalized fractional complex transform is appropriately used to convert these fractional equations to ordinary differential equations which subsequently resulted into number of exact solutions.

20. A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

NASA Technical Reports Server (NTRS)

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)

2002-01-01

We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.

1. Bright and dark soliton solutions for some nonlinear fractional differential equations

Ozkan, Guner; Ahmet, Bekir

2016-03-01

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona-Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann-Liouville sense.

2. Exact solutions for the fractional differential equations by using the first integral method

2015-03-01

In this paper, we apply the first integral method to study the solutions of the nonlinear fractional modified Benjamin-Bona-Mahony equation, the nonlinear fractional modified Zakharov-Kuznetsov equation and the nonlinear fractional Whitham-Broer-Kaup-Like systems. This method is based on the ring theory of commutative algebra. The results obtained by the proposed method show that the approach is effective and general. This approach can also be applied to other nonlinear fractional differential equations, which are arising in the theory of solitons and other areas.

3. Couple of the variational iteration method and fractional-order Legendre functions method for fractional differential equations.

PubMed

Yin, Fukang; Song, Junqiang; Leng, Hongze; Lu, Fengshun

2014-01-01

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid "noise terms" is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique.

4. A hybrid algorithm for Caputo fractional differential equations

Salgado, G. H. O.; Aguirre, L. A.

2016-04-01

This paper is concerned with the numerical solution of fractional initial value problems (FIVP) in sense of Caputo's definition for dynamical systems. Unlike for integer-order derivatives that have a single definition, there is more than one definition of non integer-order derivatives and the solution of an FIVP is definition-dependent. In this paper, the chief differences of the main definitions of fractional derivatives are revisited and a numerical algorithm to solve an FIVP for Caputo derivative is proposed. The main advantages of the algorithm are twofold: it can be initialized with integer-order derivatives, and it is faster than the corresponding standard algorithm. The performance of the proposed algorithm is illustrated with examples which suggest that it requires about half the computation time to achieve the same accuracy than the standard algorithm.

5. The Generation of a Series of Multiwing Chaotic Attractors Using Integer and Fractional Order Differential Equation Systems

Xu, Fei

In this article, we present a systematic approach to design chaos generators using integer order and fractional order differential equation systems. A series of multiwing chaotic attractors and grid multiwing chaotic attractors are obtained using linear integer order differential equation systems with switching controls. The existence of chaotic attractors in the corresponding fractional order differential equation systems is also investigated. We show that, using the nonlinear fractional order differential equation system, or linear fractional order differential equation systems with switching controls, a series of multiwing chaotic attractors can be obtained.

6. Boundedness of the solutions for certain classes of fractional differential equations with application to adaptive systems.

PubMed

Aguila-Camacho, Norelys; Duarte-Mermoud, Manuel A

2016-01-01

This paper presents the analysis of three classes of fractional differential equations appearing in the field of fractional adaptive systems, for the case when the fractional order is in the interval α ∈(0,1] and the Caputo definition for fractional derivatives is used. The boundedness of the solutions is proved for all three cases, and the convergence to zero of the mean value of one of the variables is also proved. Applications of the obtained results to fractional adaptive schemes in the context of identification and control problems are presented at the end of the paper, including numerical simulations which support the analytical results.

7. Multiple positive solutions to a coupled systems of nonlinear fractional differential equations.

PubMed

Shah, Kamal; Khan, Rahmat Ali

2016-01-01

In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov's fixed point theorem for the considered problem. Further, we also established sufficient conditions for existence of multiplicity results for positive solutions. Also, we developed some conditions under which the considered coupled system of fractional order differential equations has no positive solution. Appropriate examples are also provided which demonstrate our results.

8. New operational matrices for solving fractional differential equations on the half-line.

PubMed

Bhrawy, Ali H; Taha, Taha M; Alzahrani, Ebraheem O; Alzahrani, Ebrahim O; Baleanu, Dumitru; Alzahrani, Abdulrahim A

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

9. New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

PubMed Central

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. PMID:25996369

10. Existence results for a system of coupled hybrid fractional differential equations.

PubMed

Ahmad, Bashir; Ntouyas, Sotiris K; Alsaedi, Ahmed

2014-01-01

This paper studies the existence of solutions for a system of coupled hybrid fractional differential equations with Dirichlet boundary conditions. We make use of the standard tools of the fixed point theory to establish the main results. The existence and uniqueness result is elaborated with the aid of an example.

11. Toward the existence and uniqueness of solutions for fractional integro-differential equations under uncertainty

Ahmadian, A.; Ismail, F.; Senu, N.; Salahshour, S.; Suleiman, M.

2016-06-01

The main contribution of the current paper is to obtain new results on the existence and uniqueness of the solution of fractional integro-differential equations under uncertainty with nonlocal conditions. For this purpose, we have used two basic tools, the contraction mapping principle and Krasnoselskii's fixed-point theorem. Indeed, we have considered the original problem involving fuzzy Caputo differentiability, together with fuzzy nonlinear condition.

12. The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

PubMed

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

13. High-order fractional partial differential equation transform for molecular surface construction

PubMed Central

Hu, Langhua; Chen, Duan; Wei, Guo-Wei

2013-01-01

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model

14. High-order fractional partial differential equation transform for molecular surface construction.

PubMed

Hu, Langhua; Chen, Duan; Wei, Guo-Wei

2013-01-01

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model

15. Application of the principal fractional meta-trigonometric functions for the solution of linear commensurate-order time-invariant fractional differential equations.

PubMed

Lorenzo, C F; Hartley, T T; Malti, R

2013-05-13

A new and simplified method for the solution of linear constant coefficient fractional differential equations of any commensurate order is presented. The solutions are based on the R-function and on specialized Laplace transform pairs derived from the principal fractional meta-trigonometric functions. The new method simplifies the solution of such fractional differential equations and presents the solutions in the form of real functions as opposed to fractional complex exponential functions, and thus is directly applicable to real-world physics.

16. A new mixed element method for a class of time-fractional partial differential equations.

PubMed

Liu, Yang; Li, Hong; Gao, Wei; He, Siriguleng; Fang, Zhichao

2014-01-01

A kind of new mixed element method for time-fractional partial differential equations is studied. The Caputo-fractional derivative of time direction is approximated by two-step difference method and the spatial direction is discretized by a new mixed element method, whose gradient belongs to the simple (L (2)(Ω)(2)) space replacing the complex H(div; Ω) space. Some a priori error estimates in L (2)-norm for the scalar unknown u and in (L (2))(2)-norm for its gradient σ. Moreover, we also discuss a priori error estimates in H (1)-norm for the scalar unknown u.

17. A New Mixed Element Method for a Class of Time-Fractional Partial Differential Equations

PubMed Central

Li, Hong; Gao, Wei; He, Siriguleng; Fang, Zhichao

2014-01-01

A kind of new mixed element method for time-fractional partial differential equations is studied. The Caputo-fractional derivative of time direction is approximated by two-step difference method and the spatial direction is discretized by a new mixed element method, whose gradient belongs to the simple (L2(Ω)2) space replacing the complex H(div; Ω) space. Some a priori error estimates in L2-norm for the scalar unknown u and in (L2)2-norm for its gradient σ. Moreover, we also discuss a priori error estimates in H1-norm for the scalar unknown u. PMID:24737957

18. Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations.

PubMed

Sánchez, R; Carreras, B A; van Milligen, B Ph

2005-01-01

The fluid limit of a recently introduced family of nonintegrable (nonlinear) continuous-time random walks is derived in terms of fractional differential equations. In this limit, it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales. For this reason, the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems, ranging from the study of self-organized criticality to the modeling of turbulent transport in fluids and plasmas.

19. An improved non-classical method for the solution of fractional differential equations

Birk, Carolin; Song, Chongmin

2010-10-01

A procedure to construct temporally local schemes for the computation of fractional derivatives is proposed. The frequency-domain counterpart (i ω) α of the fractional differential operator of order α is expressed as an improper integral of a rational function in i ω. After applying a quadrature rule, the improper integral is approximated by a series of partial fractions. Each term of the partial fractions corresponds to an exponential kernel in the time domain. The convolution integral in a fractional derivative can be evaluated recursively leading to a local scheme. As the arguments of the exponential functions are always real and negative, the scheme is stable. The present procedure provides a convenient way to evaluate the quality of a given algorithm by examining its accuracy in fitting the function (i ω) α . It is revealed that the non-classical solution methods for fractional differential equations proposed by Yuan and Agrawal (ASME J Vib Acoust 124:321-324, 2002) and by Diethelm (Numer Algorithms 47:361-390, 2008) can also be interpreted as applying specific quadrature rules to evaluate the improper integral numerically. Over a wider range of frequencies, Diethelm’s algorithm provides a more accurate fitting than the YA algorithm. Therefore, it leads to better performance. Further exploiting this advantage of the proposed derivation, a novel quadrature rule leading to an even better performance than Diethelm’s algorithm is proposed. Significant gains in accuracy are achieved at the extreme ends of the frequency range. This results in significant improvements in accuracy for late time responses. Several numerical examples, including fractional differential equations of degree α = 0.3 and α = 1.5, demonstrate the accuracy and efficiency of the proposed method.

20. From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

SciTech Connect

Angstmann, C.N.; Donnelly, I.C.; Henry, B.I.; Jacobs, B.A.; Langlands, T.A.M.; Nichols, J.A.

2016-02-15

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.

1. On mixed derivatives type high dimensional multi-term fractional partial differential equations approximate solutions

2017-01-01

In this research article, we derive and analyze an efficient spectral method based on the operational matrices of three dimensional orthogonal Jacobi polynomials to solve numerically the mixed partial derivatives type multi-terms high dimensions generalized class of fractional order partial differential equations. We transform the considered fractional order problem to an easily solvable algebraic equations with the aid of the operational matrices. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some test problems are considered to confirm the accuracy and validity of the proposed numerical method. The convergence of the method is ensured by comparing our Matlab software simulations based obtained results with the exact solutions in the literature, yielding negligible errors. Moreover, comparative results discussed in the literature are extended and improved in this study.

2. Stability, boundedness, and lagrange stability of fractional differential equations with initial time difference.

PubMed

Çiçek, Muhammed; Yakar, Coşkun; Oğur, Bülent

2014-01-01

Differential inequalities, comparison results, and sufficient conditions on initial time difference stability, boundedness, and Lagrange stability for fractional differential systems have been evaluated.

3. Fractional chemotaxis diffusion equations.

PubMed

Langlands, T A M; Henry, B I

2010-05-01

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.

4. A model to determine the petroleum pressure in a well using fractional differential equations

Brito Martinez, Beatriz; Brambila Paz, Fernando; Fuentes Ruiz, Carlos

2016-11-01

A noninvasive method was used to determine the pressure of petroleum leaving a well. The mathematical model is based on nonlinear fractional differential equations. This model comes from the fractal dimension of the porous medium. The problem is solved in three stages. In the first stage the fractal dimension of the porous medium is determined. We show that microwaves reflected and transmitted through soil have a fractal dimension which is correlated with the fractal dimension of the porous medium. The fractal signature of microwave scattering correlates with certain physical and mechanical properties of soils (porosity, permeability, conductivity, etc.). In the second stage we use three partial fractional equations as a mathematical model to study the diffusion inside the porous medium. In this model sub-diffusive phenomenon occurs if fractal derivative is between zero and one and supra-diffusive occurs if the derivative is greater than 1 and less than 2. Finally in the third stage the mathematical model is used to determinate the petroleum pressure output in a Mexican oil field, which contains three partial fractional equations with triple porosity and permeability.

5. A fractional differential equation for a MEMS viscometer used in the oil industry

Fitt, A. D.; Goodwin, A. R. H.; Ronaldson, K. A.; Wakeham, W. A.

2009-07-01

A mathematical model is developed for a micro-electro-mechanical system (MEMS) instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration. It is shown that, in one mode of operation, the displacement of the device satisfies a fractional differential equation (FDE). The theory of FDEs is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme. It is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised. It is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring, where the resulting FDE is nonlinear.

6. Fractional vector calculus and fractional Maxwell's equations

SciTech Connect

Tarasov, Vasily E.

2008-11-15

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered.

7. Legendre wavelet operational matrix of fractional derivative through wavelet-polynomial transformation and its applications on non-linear system of fractional order differential equations

Isah, Abdulnasir; Chang, Phang

2016-06-01

In this article we propose the wavelet operational method based on shifted Legendre polynomial to obtain the numerical solutions of non-linear systems of fractional order differential equations (NSFDEs). The operational matrix of fractional derivative derived through wavelet-polynomial transformation are used together with the collocation method to turn the NSFDEs to a system of non-linear algebraic equations. Illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

8. Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives

2008-08-01

The existence and uniqueness theorems for functional right-left delay and left-right advanced fractional functional differential equations with bounded delay and advance, respectively, are proved. The continuity with respect to the initial function for these equations is also proved under some Lipschitz kind conditions. The Q-operator is used to transform the delay-type equation to an advanced one and vice versa. An example is given to clarify the results.

9. A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh

Jia, Jinhong; Wang, Hong

2015-10-01

Numerical methods for fractional differential equations generate full stiffness matrices, which were traditionally solved via Gaussian type direct solvers that require O (N3) of computational work and O (N2) of memory to store where N is the number of spatial grid points in the discretization. We develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite volume schemes defined on a locally refined composite mesh for fractional differential equations to resolve boundary layers of the solutions. Numerical results are presented to show the utility of the method.

10. Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup

Gou, Haide; Li, Baolin

2017-01-01

In this paper, we study local and global existence of mild solution for an impulsive fractional functional integro differential equation with non-compact semi-group in Banach spaces. We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems. The theorems proved in this paper improve and extend some related conclusions on this topic. Finally, two applications are given to illustrate that our results are valuable.

11. Haar based numerical solution of Fredholm-Volterra fractional integro-differential equation with nonlocal boundary conditions

Setia, Amit; Prakash, Bijil; Vatsala, Aghalaya S.

2017-01-01

In this paper, a numerical method is proposed to solve the Fredholm-Volterra fractional integro-differential equation with nonlocal boundary conditions by using Haar wavelets. A collocation based Galerkin's method is applied by using Haar wavelets as basis functions over the interval [0, 1). It converts the Fredholm-Volterra fractional integro-differential equation into a system of m linear equations. On incorporating q nonlocal boundary conditions, it leads to further q equations. All together it will give a system of (m + q) linear equations in (m + q) variables which can be solved. A variety of test examples are considered to illustrate the proposed method. The actual error is also measured with respect to a norm and the results are validated through error bounds.

12. Green function of the double-fractional Fokker-Planck equation: path integral and stochastic differential equations.

PubMed

Kleinert, H; Zatloukal, V

2013-11-01

The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

13. Fractional differential equations based modeling of microbial survival and growth curves: model development and experimental validation.

PubMed

Kaur, A; Takhar, P S; Smith, D M; Mann, J E; Brashears, M M

2008-10-01

A fractional differential equations (FDEs)-based theory involving 1- and 2-term equations was developed to predict the nonlinear survival and growth curves of foodborne pathogens. It is interesting to note that the solution of 1-term FDE leads to the Weibull model. Nonlinear regression (Gauss-Newton method) was performed to calculate the parameters of the 1-term and 2-term FDEs. The experimental inactivation data of Salmonella cocktail in ground turkey breast, ground turkey thigh, and pork shoulder; and cocktail of Salmonella, E. coli, and Listeria monocytogenes in ground beef exposed at isothermal cooking conditions of 50 to 66 degrees C were used for validation. To evaluate the performance of 2-term FDE in predicting the growth curves-growth of Salmonella typhimurium, Salmonella Enteritidis, and background flora in ground pork and boneless pork chops; and E. coli O157:H7 in ground beef in the temperature range of 22.2 to 4.4 degrees C were chosen. A program was written in Matlab to predict the model parameters and survival and growth curves. Two-term FDE was more successful in describing the complex shapes of microbial survival and growth curves as compared to the linear and Weibull models. Predicted curves of 2-term FDE had higher magnitudes of R(2) (0.89 to 0.99) and lower magnitudes of root mean square error (0.0182 to 0.5461) for all experimental cases in comparison to the linear and Weibull models. This model was capable of predicting the tails in survival curves, which was not possible using Weibull and linear models. The developed model can be used for other foodborne pathogens in a variety of food products to study the destruction and growth behavior.

14. Existence and uniqueness theorems for impulsive fractional differential equations with the two-point and integral boundary conditions.

PubMed

Mardanov, M J; Mahmudov, N I; Sharifov, Y A

2014-01-01

We study a boundary value problem for the system of nonlinear impulsive fractional differential equations of order α (0 < α ≤ 1) involving the two-point and integral boundary conditions. Some new results on existence and uniqueness of a solution are established by using fixed point theorems. Some illustrative examples are also presented. We extend previous results even in the integer case α = 1.

15. A general fractional differential equation associated with an integral operator with the H-function in the kernel

Srivastava, H. M.; Harjule, P.; Jain, R.

2015-01-01

In this paper, we introduce and investigate a fractional integral operator which contains Fox's H-function in its kernel. We find solutions to some fractional differential equations by using this operator. The results derived in this paper generalize the results obtained in earlier works by Kilbas et al. [7] and Srivastava and Tomovski [23]. A number of corollaries and consequences of the main results are also considered. Using some of these corollaries, graphical illustrations are presented and it is found that the graphs given here are quite comparable to the physical phenomena of decay processes.

16. Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order

2017-03-01

In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x = x(x , y , z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.

17. Some operational tools for solving fractional and higher integer order differential equations: A survey on their mutual relations

Kiryakova, Virginia S.

2012-11-01

The Laplace Transform (LT) serves as a basis of the Operational Calculus (OC), widely explored by engineers and applied scientists in solving mathematical models for their practical needs. This transform is closely related to the exponential and trigonometric functions (exp, cos, sin) and to the classical differentiation and integration operators, reducing them to simple algebraic operations. Thus, the classical LT and the OC give useful tool to handle differential equations and systems with constant coefficients. Several generalizations of the LT have been introduced to allow solving, in a similar way, of differential equations with variable coefficients and of higher integer orders, as well as of fractional (arbitrary non-integer) orders. Note that fractional order mathematical models are recently widely used to describe better various systems and phenomena of the real world. This paper surveys briefly some of our results on classes of such integral transforms, that can be obtained from the LT by means of "transmutations" which are operators of the generalized fractional calculus (GFC). On the list of these Laplace-type integral transforms, we consider the Borel-Dzrbashjan, Meijer, Krätzel, Obrechkoff, generalized Obrechkoff (multi-index Borel-Dzrbashjan) transforms, etc. All of them are G- and H-integral transforms of convolutional type, having as kernels Meijer's G- or Fox's H-functions. Besides, some special functions (also being G- and H-functions), among them - the generalized Bessel-type and Mittag-Leffler (M-L) type functions, are generating Gel'fond-Leontiev (G-L) operators of generalized differentiation and integration, which happen to be also operators of GFC. Our integral transforms have operational properties analogous to those of the LT - they do algebrize the G-L generalized integrations and differentiations, and thus can serve for solving wide classes of differential equations with variable coefficients of arbitrary, including non-integer order

18. On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations.

PubMed

Edelman, Mark

2015-07-01

In this paper, we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the system's variable is a total (a convolution) of the generating functions of all previous values of the variable with weights, which are powers of the time passed. In discrete cases, these systems can be described by difference equations in which a fractional difference on the left hand side is equal to a total (also a convolution) of the generating functions of all previous values of the system's variable with the fractional Eulerian number weights on the right hand side. In the continuous limit, the considered systems can be described by the Grünvald-Letnikov fractional differential equations, which are equivalent to the Volterra integral equations of the second kind. New properties of the fractional Eulerian numbers and possible applications of the results are discussed.

19. Relativistic equations with fractional and pseudodifferential operators

SciTech Connect

Babusci, D.; Dattoli, G.; Quattromini, M.

2011-06-15

In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with noninteger exponents. We apply the method to equations resembling generalizations of the heat equations and discuss the possibility of extending the procedure to the relativistic Schroedinger and Dirac equations.

20. Multiple positive solutions to nonlinear boundary value problems of a system for fractional differential equations.

PubMed

Zhai, Chengbo; Hao, Mengru

2014-01-01

By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by -D(0+)(ν1)y1(t) = λ1a1(t)f(y1(t), y2(t)), - D(0+)(ν2)y2(t) = λ2a2(t)g(y1(t), y2(t)), where D(0+)(ν) is the standard Riemann-Liouville fractional derivative, ν1, ν2 ∈ (n - 1, n] for n > 3 and n ∈ N, subject to the boundary conditions y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = 0 = [D(0+ (α)y2(t)] t=1, for 1 ≤ α ≤ n - 2, or y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = ϕ1(y1), [D(0+)(α)y2(t)] t=1 = ϕ2(y2), for 1 ≤ α ≤ n - 2, ϕ1, ϕ2 ∈ C([0,1], R). Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.

1. Parametrically defined differential equations

Polyanin, A. D.; Zhurov, A. I.

2017-01-01

The paper deals with nonlinear ordinary differential equations defined parametrically by two relations. It proposes techniques to reduce such equations, of the first or second order, to standard systems of ordinary differential equations. It obtains the general solution to some classes of nonlinear parametrically defined ODEs dependent on arbitrary functions. It outlines procedures for the numerical solution of the Cauchy problem for parametrically defined differential equations.

2. Solving Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1987-01-01

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

3. Nonlinear differential equations

SciTech Connect

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

4. Symmetry classification of time-fractional diffusion equation

Naeem, I.; Khan, M. D.

2017-01-01

In this article, a new approach is proposed to construct the symmetry groups for a class of fractional differential equations which are expressed in the modified Riemann-Liouville fractional derivative. We perform a complete group classification of a nonlinear fractional diffusion equation which arises in fractals, acoustics, control theory, signal processing and many other applications. Introducing the suitable transformations, the fractional derivatives are converted to integer order derivatives and in consequence the nonlinear fractional diffusion equation transforms to a partial differential equation (PDE). Then the Lie symmetries are computed for resulting PDE and using inverse transformations, we derive the symmetries for fractional diffusion equation. All cases are discussed in detail and results for symmetry properties are compared for different values of α. This study provides a new way of computing symmetries for a class of fractional differential equations.

5. On fractional Langevin equation involving two fractional orders

Baghani, Omid

2017-01-01

In numerical analysis, it is frequently needed to examine how far a numerical solution is from the exact one. To investigate this issue quantitatively, we need a tool to measure the difference between them and obviously this task is accomplished by the aid of an appropriate norm on a certain space of functions. For example, Sobolev spaces are indispensable part of theoretical analysis of partial differential equations and boundary integral equations, as well as are necessary for the analysis of some numerical methods for the solving of such equations. But most of articles that appear in this field usually use ‖.‖∞ in the space of C[a, b] which is very restrictive. In this paper, we introduce a new norm that is convenient for the fractional and singular differential equations. Using this norm, the existence and uniqueness of initial value problems for nonlinear Langevin equation with two different fractional orders are studied. In fact, the obtained results could be used for the classical cases. Finally, by two examples we show that we cannot always speak about the existence and uniqueness of solutions just by using the previous methods.

6. SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER

DOEpatents

Collier, D.M.; Meeks, L.A.; Palmer, J.P.

1960-05-10

A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.

7. Stochastic differential equations

SciTech Connect

Sobczyk, K. )

1990-01-01

This book provides a unified treatment of both regular (or random) and Ito stochastic differential equations. It focuses on solution methods, including some developed only recently. Applications are discussed, in particular an insight is given into both the mathematical structure, and the most efficient solution methods (analytical as well as numerical). Starting from basic notions and results of the theory of stochastic processes and stochastic calculus (including Ito's stochastic integral), many principal mathematical problems and results related to stochastic differential equations are expounded here for the first time. Applications treated include those relating to road vehicles, earthquake excitations and offshore structures.

8. Do Differential Equations Swing?

ERIC Educational Resources Information Center

Maruszewski, Richard F., Jr.

2006-01-01

One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…

9. Modelling by Differential Equations

ERIC Educational Resources Information Center

Chaachoua, Hamid; Saglam, Ayse

2006-01-01

This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…

10. Fractional Schrödinger equation.

PubMed

2002-11-01

Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrödinger equations.

11. A fractional Dirac equation and its solution

Muslih, Sami I.; Agrawal, Om P.; Baleanu, Dumitru

2010-02-01

This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.

12. Power-law spatial dispersion from fractional Liouville equation

SciTech Connect

Tarasov, Vasily E.

2013-10-15

A microscopic model in the framework of fractional kinetics to describe spatial dispersion of power-law type is suggested. The Liouville equation with the Caputo fractional derivatives is used to obtain the power-law dependence of the absolute permittivity on the wave vector. The fractional differential equations for electrostatic potential in the media with power-law spatial dispersion are derived. The particular solutions of these equations for the electric potential of point charge in this media are considered.

13. A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations

Ke, Rihuan; Ng, Michael K.; Sun, Hai-Wei

2015-12-01

In this paper, we study the block lower triangular Toeplitz-like with tri-diagonal blocks system which arises from the time-fractional partial differential equation. Existing fast numerical solver (e.g., fast approximate inversion method) cannot handle such linear system as the main diagonal blocks are different. The main contribution of this paper is to propose a fast direct method for solving this linear system, and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system. Our idea is based on the divide-and-conquer strategy and together with the fast Fourier transforms for calculating Toeplitz matrix-vector multiplication. The complexity needs O (MNlog2 ⁡ M) arithmetic operations, where M is the number of blocks (the number of time steps) in the system and N is the size (number of spatial grid points) of each block. Numerical examples from the finite difference discretization of time-fractional partial differential equations are also given to demonstrate the efficiency of the proposed method.

14. Perturbed nonlinear differential equations

NASA Technical Reports Server (NTRS)

Proctor, T. G.

1974-01-01

For perturbed nonlinear systems, a norm, other than the supremum norm, is introduced on some spaces of continuous functions. This makes possible the study of new types of behavior. A study is presented on a perturbed nonlinear differential equation defined on a half line, and the existence of a family of solutions with special boundedness properties is established. The ideas developed are applied to the study of integral manifolds, and examples are given.

15. A Fractional Variational Approach to the Fractional Basset-Type Equation

Baleanu, Dumitru; Garra, Roberto; Petras, Ivo

2013-08-01

In this paper we discuss an application of fractional variational calculus to the Basset-type fractional equations. It is well known that the unsteady motion of a sphere immersed in a Stokes fluid is described by an integro-differential equation involving derivative of real order. Here we study the inverse problem, i.e. we consider the problem from a Lagrangian point of view in the framework of fractional variational calculus. In this way we find an application of fractional variational methods to a classical physical model, finding a Basset-type fractional equation starting from a Lagrangian depending on derivatives of fractional order.

16. Lie symmetries and conservation laws for the time fractional Derrida-Lebowitz-Speer-Spohn equation

Rui, Wenjuan; Zhang, Xiangzhi

2016-05-01

This paper investigates the invariance properties of the time fractional Derrida-Lebowitz-Speer-Spohn (FDLSS) equation with Riemann-Liouville derivative. By using the Lie group analysis method of fractional differential equations, we derive Lie symmetries for the FDLSS equation. In a particular case of scaling transformations, we transform the FDLSS equation into a nonlinear ordinary fractional differential equation. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

17. Perturbed nonlinear differential equations

NASA Technical Reports Server (NTRS)

Proctor, T. G.

1972-01-01

The existence of a solution defined for all t and possessing a type of boundedness property is established for the perturbed nonlinear system y = f(t,y) + F(t,y). The unperturbed system x = f(t,x) has a dichotomy in which some solutions exist and are well behaved as t increases to infinity, and some solution exists and are well behaved as t decreases to minus infinity. A similar study is made for a perturbed nonlinear differential equation defined on a half line, R+, and the existence of a family of solutions with special boundedness properties is established. The ideas are applied to integral manifolds.

18. Fokker Planck equation with fractional coordinate derivatives

Tarasov, Vasily E.; Zaslavsky, George M.

2008-11-01

Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1<α<2.

19. Fast parareal iterations for fractional diffusion equations

Wu, Shu-Lin; Zhou, Tao

2017-01-01

Numerical methods for fractional PDEs is a hot topic recently. This work is concerned with the parareal algorithm for system of ODEs u‧ (t) + Au (t) = f that arising from semi-discretizations of time-dependent fractional diffusion equations with nonsymmetric Riemann-Liouville fractional derivatives. The spatial semi-discretization of this kind of fractional derivatives often results in a coefficient matrix A with spectrum σ (A)

20. Solving Nonlinear Coupled Differential Equations

NASA Technical Reports Server (NTRS)

Mitchell, L.; David, J.

1986-01-01

Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.

1. A high-speed algorithm for computation of fractional differentiation and fractional integration.

PubMed

2013-05-13

A high-speed algorithm for computing fractional differentiations and fractional integrations in fractional differential equations is proposed. In this algorithm, the stored data are not the function to be differentiated or integrated but the weighted integrals of the function. The intervals of integration for the memory can be increased without loss of accuracy as the computing time-step n increases. The computing cost varies as n log n, as opposed to n(2) of standard algorithms.

2. Approximate solutions to fractional subdiffusion equations

Hristov, J.

2011-03-01

The work presents integral solutions of the fractional subdiffusion equation by an integral method, as an alternative approach to the solutions employing hypergeometric functions. The integral solution suggests a preliminary defined profile with unknown coefficients and the concept of penetration (boundary layer). The prescribed profile satisfies the boundary conditions imposed by the boundary layer that allows its coefficients to be expressed through its depth as unique parameter. The integral approach to the fractional subdiffusion equation suggests a replacement of the real distribution function by the approximate profile. The solution was performed with Riemann-Liouville time-fractional derivative since the integral approach avoids the definition of the initial value of the time-derivative required by the Laplace transformed equations and leading to a transition to Caputo derivatives. The method is demonstrated by solutions to two simple fractional subdiffusion equations (Dirichlet problems): 1) Time-Fractional Diffusion Equation, and 2) Time-Fractional Drift Equation, both of them having fundamental solutions expressed through the M-Wright function. The solutions demonstrate some basic issues of the suggested integral approach, among them: a) Choice of the profile, b) Integration problem emerging when the distribution (profile) is replaced by a prescribed one with unknown coefficients; c) Optimization of the profile in view to minimize the average error of approximations; d) Numerical results allowing comparisons to the known solutions expressed to the M-Wright function and error estimations.

3. PREFACE: Fractional Differentiation and its Applications (FDA08) Fractional Differentiation and its Applications (FDA08)

Baleanu, Dumitru; Tenreiro Machado, J. A.

2009-10-01

The international workshop, Fractional Differentiation and its Applications (FDA08), held at Cankaya University, Ankara, Turkey on 5-7 November 2008, was the third in an ongoing series of conferences dedicated to exploring applications of fractional calculus in science, engineering, economics and finance. Fractional calculus, which deals with derivatives and integrals of any order, is now recognized as playing an important role in modeling multi-scale problems that span a wide range of time or length scales. Fractional calculus provides a natural link to the intermediate-order dynamics that often reflects the complexity of micro- and nanostructures through fractional-order differential equations. Unlike the more established techniques of mathematical physics, the methods of fractional differentiation are still under development; while it is true that the ideas of fractional calculus are as old as the classical integer-order differential operators, modern work is proceeding by both expanding the capabilities of this mathematical tool and by widening its range of applications. Hence, the interested reader will find papers here that focus on the underlying mathematics of fractional calculus, that extend fractional-order operators into new domains, and that apply well established methods to experimental and theoretical problems. The organizing committee invited presentations from experts representing the international community of scholars in fractional calculus and welcomed contributions from the growing number of researchers who are applying fractional differentiation to complex technical problems. The selection of papers in this topical issue of Physica Scripta reflects the success of the FDA08 workshop, with the emergence of a variety of novel areas of application. With these ideas in mind, the guest editors would like to honor the many distinguished scientists that have promoted the development of fractional calculus and, in particular, Professor George M

4. Linear determining equations for differential constraints

SciTech Connect

Kaptsov, O V

1998-12-31

A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie operators. As applications of this approach equations of an ideal incompressible fluid and non-linear heat equations are discussed.

5. Pendulum Motion and Differential Equations

ERIC Educational Resources Information Center

Reid, Thomas F.; King, Stephen C.

2009-01-01

A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…

6. New iterative method for fractional gas dynamics and coupled Burger's equations.

PubMed

Al-Luhaibi, Mohamed S

2015-01-01

This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger's equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterative method is easy to implement and accurate when applied to time-fractional partial differential equations.

7. Fractional field equations for highly improbable events

Kleinert, H.

2013-06-01

Free and weakly interacting particles perform approximately Gaussian random walks with collisions. They follow a second-quantized nonlinear Schrödinger equation, or relativistic versions of it. By contrast, the fields of strongly interacting particles extremize more involved effective actions obeying fractional wave equations with anomalous dimensions. Their particle orbits perform universal Lévy walks with heavy tails, in which rare events are much more frequent than in Gaussian random walks. Such rare events are observed in exceptionally strong windgusts, monster or rogue waves, earthquakes, and financial crashes. While earthquakes may destroy entire cities, the latter have the potential of devastating entire economies.

8. Fractional Solutions of Bessel Equation with N-Method

PubMed Central

Bas, Erdal; Yilmazer, Resat; Panakhov, Etibar

2013-01-01

This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the N-fractional calculus operator Nν method, we derive the fractional solutions of the equation. PMID:24023534

9. Fast permutation preconditioning for fractional diffusion equations.

PubMed

Wang, Sheng-Feng; Huang, Ting-Zhu; Gu, Xian-Ming; Luo, Wei-Hua

2016-01-01

In this paper, an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is used to discretize the fractional diffusion equations with constant diffusion coefficients. The coefficient matrix possesses the Toeplitz structure and the fast Toeplitz matrix-vector product can be utilized to reduce the computational complexity from [Formula: see text] to [Formula: see text], where N is the number of grid points. Two preconditioned iterative methods, named bi-conjugate gradient method for Toeplitz matrix and bi-conjugate residual method for Toeplitz matrix, are proposed to solve the relevant discretized systems. Finally, numerical experiments are reported to show the effectiveness of our preconditioners.

10. A unifying fractional wave equation for compressional and shear waves.

PubMed

Holm, Sverre; Sinkus, Ralph

2010-01-01

This study has been motivated by the observed difference in the range of the power-law attenuation exponent for compressional and shear waves. Usually compressional attenuation increases with frequency to a power between 1 and 2, while shear wave attenuation often is described with powers less than 1. Another motivation is the apparent lack of partial differential equations with desirable properties such as causality that describe such wave propagation. Starting with a constitutive equation which is a generalized Hooke's law with a loss term containing a fractional derivative, one can derive a causal fractional wave equation previously given by Caputo [Geophys J. R. Astron. Soc. 13, 529-539 (1967)] and Wismer [J. Acoust. Soc. Am. 120, 3493-3502 (2006)]. In the low omegatau (low-frequency) case, this equation has an attenuation with a power-law in the range from 1 to 2. This is consistent with, e.g., attenuation in tissue. In the often neglected high omegatau (high-frequency) case, it describes attenuation with a power-law between 0 and 1, consistent with what is observed in, e.g., dynamic elastography. Thus a unifying wave equation derived properly from constitutive equations can describe both cases.

11. Generalized Ordinary Differential Equation Models.

PubMed

Miao, Hongyu; Wu, Hulin; Xue, Hongqi

2014-10-01

Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method.

12. Fractional Differential and Integral Inequalities with Applications

DTIC Science & Technology

2016-02-14

nanotechnology . Consider the equation (3.1) ( )(t) = f(, () ) + (, () ), (0) = 0, where 0 < < 1, (, () ), (, ...boundary conditions. References [1] D. Baleanu, Z. B. Guvencs̈ , J.A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications...Rivero, J. Trujillo and M. Pilar Velasco, “On Deterministic Fractional Models,” New Trends in Nanotechnology and Fractional Calculus Applications, edited

13. A fractional diffusion equation model for cancer tumor

Iyiola, Olaniyi Samuel; Zaman, F. D.

2014-10-01

In this article, we consider cancer tumor models and investigate the need for fractional order derivative as compared to the classical first order derivative in time. Three different cases of the net killing rate are taken into account including the case where net killing rate of the cancer cells is dependent on the concentration of the cells. At first, we use a relatively new analytical technique called q-Homotopy Analysis Method on the resulting time-fractional partial differential equations to obtain analytical solution in form of convergent series with easily computable components. Our numerical analysis enables us to give some recommendations on the appropriate order (fractional) of derivative in time to be used in modeling cancer tumor.

14. Symmetry algebras of linear differential equations

Shapovalov, A. V.; Shirokov, I. V.

1992-07-01

The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.

15. Systems of Nonlinear Hyperbolic Partial Differential Equations

DTIC Science & Technology

1997-12-01

McKinney) Travelling wave solutions of the modified Korteweg - deVries -Burgers Equation . J. Differential Equations , 116 (1995), 448-467. 4. (with D.G...SUBTITLE Systems of Nonlinear Hyperbolic Partial Differential Equations 6. AUTHOR’S) Michael Shearer PERFORMING ORGANIZATION NAMES(S) AND...DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 words) This project concerns properties of wave propagation in partial differential equations that are nonlinear

16. Fractional-order difference equations for physical lattices and some applications

SciTech Connect

Tarasov, Vasily E.

2015-10-15

Fractional-order operators for physical lattice models based on the Grünwald-Letnikov fractional differences are suggested. We use an approach based on the models of lattices with long-range particle interactions. The fractional-order operators of differentiation and integration on physical lattices are represented by kernels of lattice long-range interactions. In continuum limit, these discrete operators of non-integer orders give the fractional-order derivatives and integrals with respect to coordinates of the Grünwald-Letnikov types. As examples of the fractional-order difference equations for physical lattices, we give difference analogs of the fractional nonlocal Navier-Stokes equations and the fractional nonlocal Maxwell equations for lattices with long-range interactions. Continuum limits of these fractional-order difference equations are also suggested.

17. On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.

PubMed

Yang, Xiao-Jun; Tenreiro Machado, J A; Baleanu, Dumitru; Cattani, Carlo

2016-08-01

This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

18. Nonlinear acoustic wave equations with fractional loss operators.

PubMed

Prieur, Fabrice; Holm, Sverre

2011-09-01

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations.

19. Solution of time fractional Black-Scholes European option pricing equation arising in financial market

Ravi Kanth, A. S. V.; Aruna, K.

2016-12-01

In this paper, we present fractional differential transform method (FDTM) and modified fractional differential transform method (MFDTM) for the solution of time fractional Black-Scholes European option pricing equation. The method finds the solution without any discretization, transformation, or restrictive assumptions with the use of appropriate initial or boundary conditions. The efficiency and exactitude of the proposed methods are tested by means of three examples.

20. Least-Squares Spectral Method for the solution of a fractional advection-dispersion equation

Carella, Alfredo Raúl; Dorao, Carlos Alberto

2013-01-01

Fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields. This article describes a Least Squares Spectral Method for solving advection-dispersion equations using Caputo or Riemann-Liouville fractional derivatives. A Gauss-Lobatto-Jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition. Exponential convergence rate of the operator is verified when increasing the order of the approximation. Solutions are calculated for fractional-time and fractional-space differential equations. Comparisons with finite difference schemes are included. A significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate.

1. Generalized space-time fractional diffusion equation with composite fractional time derivative

Tomovski, Živorad; Sandev, Trifce; Metzler, Ralf; Dubbeldam, Johan

2012-04-01

We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grünwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of the form δ(x)ṡ tΓ/(1-β) (β>0).

2. Lie algebras and linear differential equations.

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

3. MACSYMA's symbolic ordinary differential equation solver

NASA Technical Reports Server (NTRS)

Golden, J. P.

1977-01-01

The MACSYMA's symbolic ordinary differential equation solver ODE2 is described. The code for this routine is delineated, which is of interest because it is written in top-level MACSYMA language, and may serve as a good example of programming in that language. Other symbolic ordinary differential equation solvers are mentioned.

4. Solving Differential Equations Using Modified Picard Iteration

ERIC Educational Resources Information Center

Robin, W. A.

2010-01-01

Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

5. Symbolic Solution of Linear Differential Equations

NASA Technical Reports Server (NTRS)

Feinberg, R. B.; Grooms, R. G.

1981-01-01

An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.

6. Stochastic differential equation model to Prendiville processes

Granita, Bahar, Arifah

2015-10-01

The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.

7. Stochastic differential equation model to Prendiville processes

SciTech Connect

Granita; Bahar, Arifah

2015-10-22

The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.

8. Sparse dynamics for partial differential equations

PubMed Central

Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D.; Osher, Stanley

2013-01-01

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms. PMID:23533273

9. Discrete Surface Modelling Using Partial Differential Equations.

PubMed

Xu, Guoliang; Pan, Qing; Bajaj, Chandrajit L

2006-02-01

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.

10. Ordinary differential equation for local accumulation time.

PubMed

Berezhkovskii, Alexander M

2011-08-21

Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential equation. Using this equation one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential equation, as was done in previous studies. We derive this ordinary differential equation together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this equation.

11. Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions

Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C.

2016-08-01

Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

12. Fourier spectral method for higher order space fractional reaction-diffusion equations

2016-11-01

Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose a fast and accurate method for numerical solutions of space fractional reaction-diffusion equations. The proposed method is based on an exponential integrator scheme in time and the Fourier spectral method in space. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency, and a completely straightforward extension to high spatial dimensions. Although, in general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives, we introduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme justified by a number of computational experiments, this includes two and three dimensional partial differential equations. Numerical experiments are provided to validate the effectiveness of the proposed approach.

13. Sobolev type equations of time-fractional order with periodical boundary conditions

Plekhanova, Marina

2016-08-01

The existence of a unique local solution for a class of time-fractional Sobolev type partial differential equations endowed by the Cauchy initial conditions and periodical with respect to every spatial variable boundary conditions on a parallelepiped is proved. General results are applied to study of the unique solvability for the initial boundary value problem to Benjamin-Bona-Mahony-Burgers and Allair partial differential equations.

14. Connecting Related Rates and Differential Equations

ERIC Educational Resources Information Center

Brandt, Keith

2012-01-01

This article points out a simple connection between related rates and differential equations. The connection can be used for in-class examples or homework exercises, and it is accessible to students who are familiar with separation of variables.

15. Fractional Schrödinger equation in optics.

PubMed

Longhi, Stefano

2015-03-15

In quantum mechanics, the space-fractional Schrödinger equation provides a natural extension of the standard Schrödinger equation when the Brownian trajectories in Feynman path integrals are replaced by Levy flights. Here an optical realization of the fractional Schrödinger equation, based on transverse light dynamics in aspherical optical cavities, is proposed. As an example, a laser implementation of the fractional quantum harmonic oscillator is presented in which dual Airy beams can be selectively generated under off-axis longitudinal pumping.

16. Control problems for semilinear neutral differential equations in Hilbert spaces.

PubMed

Jeong, Jin-Mun; Cho, Seong Ho

2014-01-01

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied.

17. Control Problems for Semilinear Neutral Differential Equations in Hilbert Spaces

PubMed Central

Jeong, Jin-Mun; Cho, Seong Ho

2014-01-01

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied. PMID:24772022

18. Silicon isotope fractionation during magmatic differentiation

Savage, Paul S.; Georg, R. Bastian; Williams, Helen M.; Burton, Kevin W.; Halliday, Alex N.

2011-10-01

The Si isotopic composition of Earth's mantle is thought to be homogeneous (δ 30Si = -0.29 ± 0.08‰, 2 s.d.) and not greatly affected by partial melting and recycling. Previous analyses of evolved igneous material indicate that such rocks are isotopically heavy relative to the mantle. To understand this variation, it is necessary to investigate the degree of Si isotopic fractionation that takes place during magmatic differentiation. Here we report Si isotopic compositions of lavas from Hekla volcano, Iceland, which has formed in a region devoid of old, geochemically diverse crust. We show that Si isotopic composition varies linearly as a function of silica content, with more differentiated rocks possessing heavier isotopic compositions. Data for samples from the Afar Rift Zone, as well as various igneous USGS standards are collinear with the Hekla trend, providing evidence of a fundamental relationship between magmatic differentiation and Si isotopes. The effect of fractionation has been tested by studying cumulates from the Skaergaard Complex, which show that olivine and pyroxene are isotopically light, and plagioclase heavy, relative to the Si isotopic composition of the Earth's mantle. Therefore, Si isotopes can be utilised to model the competing effects of mafic and felsic mineral fractionation in evolving silicate liquids and cumulates. At an average SiO 2 content of ˜60 wt.%, the predicted δ 30Si value of the continental crust that should result from magmatic fractionation alone is -0.23 ± 0.05‰ (2 s.e.), barely heavier than the mantle. This is, at most, a maximum estimate, as this does not take into account weathered material whose formation drives the products toward lighter δ 30Si values. Mass balance calculations suggest that removal of continental crust of this composition from the upper mantle will not affect the Si isotopic composition of the mantle.

19. Normal Forms for Nonautonomous Differential Equations

Siegmund, Stefan

2002-01-01

We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λj-∑ni=1 ℓiλi≠0 for eigenvalues is generalized to the new nonresonance condition λj∩∑ni=1 ℓiλi=∅ for spectral intervals.

20. Program for solution of ordinary differential equations

NASA Technical Reports Server (NTRS)

Sloate, H.

1973-01-01

A program for the solution of linear and nonlinear first order ordinary differential equations is described and user instructions are included. The program contains a new integration algorithm for the solution of initial value problems which is particularly efficient for the solution of differential equations with a wide range of eigenvalues. The program in its present form handles up to ten state variables, but expansion to handle up to fifty state variables is being investigated.

1. A class of neutral functional differential equations.

NASA Technical Reports Server (NTRS)

Melvin, W. R.

1972-01-01

Formulation and study of the initial value problem for neutral functional differential equations. The existence, uniqueness, and continuation of solutions to this problem are investigated, and an analysis is made of the dependence of the solutions on the initial conditions and parameters, resulting in the derivation of a continuous dependence theorem in which the fundamental mathematical principles underlying the continuous dependence problem for a very general system of nonlinear neutral functional differential equations are separated out.

2. Leapfrog/Finite Element Method for Fractional Diffusion Equation

PubMed Central

Zhao, Zhengang; Zheng, Yunying

2014-01-01

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove an L 2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis. PMID:24955431

3. A Unified Introduction to Ordinary Differential Equations

ERIC Educational Resources Information Center

Lutzer, Carl V.

2006-01-01

This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)

4. Nonlinear scalar field equations involving the fractional Laplacian

Byeon, Jaeyoung; Kwon, Ohsang; Seok, Jinmyoung

2017-04-01

In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.

5. Numerical solution of the one-dimensional fractional convection diffusion equations based on Chebyshev operational matrix.

PubMed

Xie, Jiaquan; Huang, Qingxue; Yang, Xia

2016-01-01

In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent.

6. Analytical and numerical validation for solving the fractional Klein-Gordon equation using the fractional complex transform and variational iteration methods

2016-09-01

In this paper, we implement the fractional complex transform method to convert the nonlinear fractional Klein-Gordon equation (FKGE) to an ordinary differential equation. We use the variational iteration method (VIM) to solve the resulting ODE. The fractional derivatives are presented in terms of the Caputo sense. Some numerical examples are presented to validate the proposed techniques. Finally, a comparison with the numerical solution using Runge-Kutta of order four is given.

7. Spectral analysis and structure preserving preconditioners for fractional diffusion equations

Donatelli, Marco; Mazza, Mariarosa; Serra-Capizzano, Stefano

2016-02-01

Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grünwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.

8. Lipschitz regularity of solutions for mixed integro-differential equations

Barles, Guy; Chasseigne, Emmanuel; Ciomaga, Adina; Imbert, Cyril

We establish new Hölder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hölder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.

9. Differential geometry techniques for sets of nonlinear partial differential equations

NASA Technical Reports Server (NTRS)

Estabrook, Frank B.

1990-01-01

An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.

10. Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law

Chen, Li-Qun; Zhao, Wei-Jia; Zu, Jean W.

2004-12-01

This paper deals with the transverse vibration of an initially stressed moving viscoelastic string obeying a fractional differentiation constitutive law. The governing equation is derived from Newtonian second law of motion, and reduced to a set of non-linear differential-integral equations based on Galerkin's truncation. A numerical approach is proposed to solve numerically the differential-integral equation through developing an approximate expression of the fractional derivatives involved. Some numerical examples are presented to highlight the effects of viscoelastic parameters and frequencies of parametric excitations on the transient responses of the axially moving string.

11. An enriched finite element method to fractional advection-diffusion equation

Luan, Shengzhi; Lian, Yanping; Ying, Yuping; Tang, Shaoqiang; Wagner, Gregory J.; Liu, Wing Kam

2017-03-01

In this paper, an enriched finite element method with fractional basis [ 1,x^{α }] for spatial fractional partial differential equations is proposed to obtain more stable and accurate numerical solutions. For pure fractional diffusion equation without advection, the enriched Galerkin finite element method formulation is demonstrated to simulate the exact solution successfully without any numerical oscillation, which is advantageous compared to the traditional Galerkin finite element method with integer basis [ 1,x] . For fractional advection-diffusion equation, the oscillatory behavior becomes complex due to the introduction of the advection term which can be characterized by a fractional element Peclet number. For the purpose of addressing the more complex numerical oscillation, an enriched Petrov-Galerkin finite element method is developed by using a dimensionless fractional stabilization parameter, which is formulated through a minimization of the residual of the nodal solution. The effectiveness and accuracy of the enriched finite element method are demonstrated by a series of numerical examples of fractional diffusion equation and fractional advection-diffusion equation, including both one-dimensional and two-dimensional, steady-state and time-dependent cases.

12. Multiscale functions, scale dynamics, and applications to partial differential equations

Cresson, Jacky; Pierret, Frédéric

2016-05-01

Modeling phenomena from experimental data always begins with a choice of hypothesis on the observed dynamics such as determinism, randomness, and differentiability. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following: "With a finite set of data concerning a phenomenon, can we recover its underlying nature? From this problem, we introduce in this paper the definition of multi-scale functions, scale calculus, and scale dynamics based on the time scale calculus [see Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (Springer Science & Business Media, 2001)] which is used to introduce the notion of scale equations. These definitions will be illustrated on the multi-scale Okamoto's functions. Scale equations are analysed using scale regimes and the notion of asymptotic model for a scale equation under a particular scale regime. The introduced formalism explains why a single scale equation can produce distinct continuous models even if the equation is scale invariant. Typical examples of such equations are given by the scale Euler-Lagrange equation. We illustrate our results using the scale Newton's equation which gives rise to a non-linear diffusion equation or a non-linear Schrödinger equation as asymptotic continuous models depending on the particular fractional scale regime which is considered.

13. On the solutions of fractional order of evolution equations

Morales-Delgado, V. F.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.

2017-01-01

In this paper we present a discussion of generalized Cauchy problems in a diffusion wave process, we consider bi-fractional-order evolution equations in the Riemann-Liouville, Liouville-Caputo, and Caputo-Fabrizio sense. Through Fourier transforms and Laplace transform we derive closed-form solutions to the Cauchy problems mentioned above. Similarly, we establish fundamental solutions. Finally, we give an application of the above results to the determination of decompositions of Dirac type for bi-fractional-order equations and write a formula for the moments for the fractional vibration of a beam equation. This type of decomposition allows us to speak of internal degrees of freedom in the vibration of a beam equation.

14. New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods

S Saha, Ray

2016-04-01

In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

15. Differential equation models for sharp threshold dynamics.

PubMed

Schramm, Harrison C; Dimitrov, Nedialko B

2014-01-01

We develop an extension to differential equation models of dynamical systems to allow us to analyze probabilistic threshold dynamics that fundamentally and globally change system behavior. We apply our novel modeling approach to two cases of interest: a model of infectious disease modified for malware where a detection event drastically changes dynamics by introducing a new class in competition with the original infection; and the Lanchester model of armed conflict, where the loss of a key capability drastically changes the effectiveness of one of the sides. We derive and demonstrate a step-by-step, repeatable method for applying our novel modeling approach to an arbitrary system, and we compare the resulting differential equations to simulations of the system's random progression. Our work leads to a simple and easily implemented method for analyzing probabilistic threshold dynamics using differential equations.

16. Stochastic Differential Equation of Earthquakes Series

Mariani, Maria C.; Tweneboah, Osei K.; Gonzalez-Huizar, Hector; Serpa, Laura

2016-07-01

This work is devoted to modeling earthquake time series. We propose a stochastic differential equation based on the superposition of independent Ornstein-Uhlenbeck processes driven by a Γ (α, β ) process. Superposition of independent Γ (α, β ) Ornstein-Uhlenbeck processes offer analytic flexibility and provides a class of continuous time processes capable of exhibiting long memory behavior. The stochastic differential equation is applied to the study of earthquakes by fitting the superposed Γ (α, β ) Ornstein-Uhlenbeck model to earthquake sequences in South America containing very large events (Mw ≥ 8). We obtained very good fit of the observed magnitudes of the earthquakes with the stochastic differential equations, which supports the use of this methodology for the study of earthquakes sequence.

17. Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations

SciTech Connect

Petzold, L; Cao, Y; Li, S; Serban, R

2005-08-09

Sensitivity analysis generates essential information for model development, design optimization, parameter estimation, optimal control, model reduction and experimental design. In this paper we describe the forward and adjoint methods for sensitivity analysis, and outline some of our recent work on theory, algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) and time-dependent partial differential equation (PDE) systems.

18. A differential equation for approximate wall distance

Fares, E.; Schröder, W.

2002-07-01

A partial differential equation to compute the distance from a surface is derived and solved numerically. The benefit of such a formulation especially in combination with turbulence models is shown. The details of the formulation as well as several examples demonstrating the influence of its parameters are presented. The proposed formulation has computational advantages and can be favourably incorporated into one- and two-equation turbulence models like e.g. the Spalart-Allmaras, the Secundov or Menter's SST model. Copyright

19. Algorithms For Integrating Nonlinear Differential Equations

NASA Technical Reports Server (NTRS)

Freed, A. D.; Walker, K. P.

1994-01-01

Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

20. An efficient method for systems of variable coefficient coupled Burgers' equation with time-fractional derivative.

PubMed

2013-01-01

A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. Results indicate that the introduced method is promising for solving other types of systems of nonlinear fractional-order partial differential equations.

1. Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime

Liang, Jin-Rong; Wang, Jun; Lǔ, Long-Jin; Gu, Hui; Qiu, Wei-Yuan; Ren, Fu-Yao

2012-01-01

In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X α, H ( t)= X H ( S α ( t)), 0< α, H<1, driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional Fokker-Planck equation and Black-Scholes formula. We obtain the fractional Fokker-Planck equation governing the dynamics of the probability density function of the composite-diffusive fractional Brownian motion and find the Black-Scholes differential equation driven by the stock asset X α, H ( t) and the corresponding Black-Scholes formula for the fair prices of European option.

2. Reply to "Comment on 'Fractional quantum mechanics' and 'Fractional Schrödinger equation' ".

PubMed

2016-06-01

The fractional uncertainty relation is a mathematical formulation of Heisenberg's uncertainty principle in the framework of fractional quantum mechanics. Two mistaken statements presented in the Comment have been revealed. The origin of each mistaken statement has been clarified and corrected statements have been made. A map between standard quantum mechanics and fractional quantum mechanics has been presented to emphasize the features of fractional quantum mechanics and to avoid misinterpretations of the fractional uncertainty relation. It has been shown that the fractional probability current equation is correct in the area of its applicability. Further studies have to be done to find meaningful quantum physics problems with involvement of the fractional probability current density vector and the extra term emerging in the framework of fractional quantum mechanics.

3. Revealing Numerical Solutions of a Differential Equation

ERIC Educational Resources Information Center

Glaister, P.

2006-01-01

In this article, the author considers a student exercise that involves determining the exact and numerical solutions of a particular differential equation. He shows how a typical student solution is at variance with a numerical solution, suggesting that the numerical solution is incorrect. However, further investigation shows that this numerical…

4. Rough differential equations with unbounded drift term

Riedel, S.; Scheutzow, M.

2017-01-01

We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply strong completeness and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the (semi)flow and a Freidlin-Wentzell-type large deviation result.

5. Numerical Study of Fractional Ensemble Average Transport Equations

Kim, S.; Park, Y.; Gyeong, C. B.; Lee, O.

2014-12-01

In this presentation, a newly developed theory is applied to the case of stationary and non-stationary stochastic advective flow field, and a numerical solution method is presented for the resulting fractional Fokker-Planck equation (fFPE), which describes the evolution of the probability density function (PDF) of contaminant concentration. The derived fFPE is evaluated for three different form: 1) purely advective form, 2) second-order moment form and 3) second-order cumulant form. The Monte Carlo analysis of the fractional governing equation is then performed in a stochastic flow field, generated by a fractional Brownian motion for the stationary and non-stationary stochastic advection, in order to provide a benchmark for the results obtained from the fFPEs. When compared to the Monte Carlo simulation based PDFs and their ensemble average, the second-order cumulant form gives a good fit in terms of the shape and mode of the PDF of the contaminant concentration. Therefore, it is quite promising that the non-Fickian transport behavior can be modeled by the derived fractional ensemble average transport equations either by means of the long memory in the underlying stochastic flow, or by means of the time-space non-stationarity of the underlying stochastic flow, or by means of the time and space fractional derivatives of the transport equations. This subject is supported by Korea Ministry of Environment as "The Eco Innovation Project : Non-point source pollution control research group"

6. Numerical study of fractional nonlinear Schrödinger equations.

PubMed

Klein, Christian; Sparber, Christof; Markowich, Peter

2014-12-08

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.

7. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

PubMed

Singh, Brajesh K; Srivastava, Vineet K

2015-04-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

8. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

PubMed Central

Singh, Brajesh K.; Srivastava, Vineet K.

2015-01-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639

9. Solving Parker's transport equation with stochastic differential equations on GPUs

Dunzlaff, P.; Strauss, R. D.; Potgieter, M. S.

2015-07-01

The numerical solution of transport equations for energetic charged particles in space is generally very costly in terms of time. Besides the use of multi-core CPUs and computer clusters in order to decrease the computation times, high performance calculations on graphics processing units (GPUs) have become available during the last years. In this work we introduce and describe a GPU-accelerated implementation of Parker's equation using Stochastic Differential Equations (SDEs) for the simulation of the transport of energetic charged particles with the CUDA toolkit, which is the focus of this work. We briefly discuss the set of SDEs arising from Parker's transport equation and their application to boundary value problems such as that of the Jovian magnetosphere. We compare the runtimes of the GPU code with a CPU version of the same algorithm. Compared to the CPU implementation (using OpenMP and eight threads) we find a performance increase of about a factor of 10-60, depending on the assumed set of parameters. Furthermore, we benchmark our simulation using the results of an existing SDE implementation of Parker's transport equation.

10. Fractional Feynman-Kac equation for non-brownian functionals.

PubMed

Turgeman, Lior; Carmi, Shai; Barkai, Eli

2009-11-06

We derive backward and forward fractional Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [Phys. Rev. Lett. 96, 230601 (2006)10.1103/PhysRevLett.96.230601] provide the correct fractional framework for the problem. For applications, we calculate the distribution of occupation times in half space and show how the statistics of anomalous functionals is related to weak ergodicity breaking.

11. The fractional Boussinesq equation of groundwater flow and its applications

Su, Ninghu

2017-04-01

This paper presents a set of fractional Boussinesq equations (fBEs) for groundwater flow in confined and unconfined aquifers and demonstrates the application of one of the fBEs for groundwater discharges known as recession curves. The fBEs are formulated with two-term distributed fractional orders in time and symmetrical fractional derivatives (SFD) in space applicable to both confined and unconfined aquifers. The SFD in theory consists of the forward fractional derivative (FFD) and the backward fractional derivative (BFD). The FFD represents the forward movement of water along the direction of mainstream flow while the BFD accounts for the backward motion of water in the direction opposite to the mainstream flow. The backward flow at the pore level can be referred to as the micro-scale backwater effect. The analogue of the backwater effect on a micro-scale using the BFD coincides with the wandering processes based on the continuous-time random walk (CTRW) theory which results in the fractional governing equation. With the analytical solutions of the fBE for given initial and boundary conditions of the first type for a finite depth, a set of formulae for groundwater recession has been derived using approximate solutions of the fBE. The examples of the applications of the recession curves are graphically illustrated and the effects of the orders of fractional derivatives on the geometry of the flow curves examined.

12. A new look at the fractionalization of the logistic equation

Ortigueira, Manuel; Bengochea, Gabriel

2017-02-01

The fractional version of the logistic equation will be studied in this paper. Motivated by unsuccessful previous papers, we showed how to obtain the correct solution. The algorithm is very simple. Its numerical implementation will be studied and exemplified using a Padé approximation.

13. Fractional Feynman-Kac equation for weak ergodicity breaking.

PubMed

Carmi, Shai; Barkai, Eli

2011-12-01

The continuous-time random walk (CTRW) is a model of anomalous subdiffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, ψ(τ) ~ τ(-(1+α)), leads to subdiffusion (x(2) ~ t(α)) for 0 < α < 1. In closed systems, the long stagnation periods cause time averages to divert from the corresponding ensemble averages, which is a manifestation of weak ergodicity breaking. The time average of a general observable U(t) = 1/t ∫(0)(t) U[x(τ)]dτ is a functional of the path and is described by the well-known Feynman-Kac equation if the motion is Brownian. Here, we derive forward and backward fractional Feynman-Kac equations for functionals of CTRW in a binding potential. We use our equations to study two specific time averages: the fraction of time spent by a particle in half-box, and the time average of the particle's position in a harmonic field. In both cases, we obtain the probability density function of the time averages for t → ∞ and the first two moments. Our results show that both the occupation fraction and the time-averaged position are random variables even for long times, except for α = 1, when they are identical to their ensemble averages. Using our fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.

14. Relationships between Students' Fractional Knowledge and Equation Writing

ERIC Educational Resources Information Center

Hackenberg, Amy J.; Lee, Mi Yeon

2015-01-01

To understand relationships between students' fractional knowledge and algebraic reasoning in the domain of equation writing, an interview study was conducted with 12 secondary school students, 6 students operating with each of 2 different multiplicative concepts. These concepts are based on how students coordinate composite units. Students…

15. A partial differential equation for pseudocontact shift.

PubMed

Charnock, G T P; Kuprov, Ilya

2014-10-07

It is demonstrated that pseudocontact shift (PCS), viewed as a scalar or a tensor field in three dimensions, obeys an elliptic partial differential equation with a source term that depends on the Hessian of the unpaired electron probability density. The equation enables straightforward PCS prediction and analysis in systems with delocalized unpaired electrons, particularly for the nuclei located in their immediate vicinity. It is also shown that the probability density of the unpaired electron may be extracted, using a regularization procedure, from PCS data.

16. Asymptotic stability of singularly perturbed differential equations

Artstein, Zvi

2017-02-01

Asymptotic stability is examined for singularly perturbed ordinary differential equations that may not possess a natural split into fast and slow motions. Rather, the right hand side of the equation is comprised of a singularly perturbed component and a regular one. The limit dynamics consists then of Young measures, with values being invariant measures of the fast contribution, drifted by the slow one. Relations between the asymptotic stability of the perturbed system and the limit dynamics are examined, and a Lyapunov functions criterion, based on averaging, is established.

17. On the Inclusion of Difference Equation Problems and Z Transform Methods in Sophomore Differential Equation Classes

ERIC Educational Resources Information Center

Savoye, Philippe

2009-01-01

In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.

18. Teaching Modeling with Partial Differential Equations: Several Successful Approaches

ERIC Educational Resources Information Center

Myers, Joseph; Trubatch, David; Winkel, Brian

2008-01-01

We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…

19. Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell.

PubMed

Santoro, P A; de Paula, J L; Lenzi, E K; Evangelista, L R

2011-09-21

The electrical response of an electrolytic cell in which the diffusion of mobile ions in the bulk is governed by a fractional diffusion equation of distributed order is analyzed. The boundary conditions at the electrodes limiting the sample are described by an integro-differential equation governing the kinetic at the interface. The analysis is carried out by supposing that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies the Poisson's equation. The results cover a rich variety of scenarios, including the ones connected to anomalous diffusion.

20. Observability of discretized partial differential equations

NASA Technical Reports Server (NTRS)

Cohn, Stephen E.; Dee, Dick P.

1988-01-01

It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.

1. LORENE: Spectral methods differential equations solver

Gourgoulhon, Eric; Grandclément, Philippe; Marck, Jean-Alain; Novak, Jérôme; Taniguchi, Keisuke

2016-08-01

LORENE (Langage Objet pour la RElativité NumériquE) solves various problems arising in numerical relativity, and more generally in computational astrophysics. It is a set of C++ classes and provides tools to solve partial differential equations by means of multi-domain spectral methods. LORENE classes implement basic structures such as arrays and matrices, but also abstract mathematical objects, such as tensors, and astrophysical objects, such as stars and black holes.

2. Partial differential equation models in macroeconomics.

PubMed

Achdou, Yves; Buera, Francisco J; Lasry, Jean-Michel; Lions, Pierre-Louis; Moll, Benjamin

2014-11-13

The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.

3. Stationary conditions for stochastic differential equations

NASA Technical Reports Server (NTRS)

1972-01-01

This is a preliminary study of possible necessary and sufficient conditions to insure stationarity in the solution process for a stochastic differential equation. It indirectly sheds some light on ergodicity properties and shows that the spectral density is generally inadequate as a statistical measure of the solution. Further work is proceeding on a more general theory which gives necessary and sufficient conditions in a form useful for applications.

4. A few remarks on ordinary differential equations

SciTech Connect

Desjardins, B.

1996-12-31

We present in this note existence and uniqueness results for solutions of ordinary differential equations and linear transport equations with discontinuous coefficients in a bounded open subset {Omega} of R{sup N} or in the whole space R{sup N} (N {ge} 1). R.J. Di Perna and P.L. Lions studied the case of vector fields b with coefficients in Sobolev spaces and bounded divergence. We want to show that similar results hold for more general b: we assume in the bounded autonomous case that b belongs to W{sup 1,1}({Omega}), b.n = 0 on {partial_derivative}{Omega}, and that there exists T{sub o} > O such that exp(T{sub o}{vert_bar}div b{vert_bar}) {element_of} L{sup 1}({Omega}). Furthermore, we establish results on transport equations with initial values in L{sup p} spaces (p > 1). 9 refs.

5. Group iterative methods for the solution of two-dimensional time-fractional diffusion equation

Balasim, Alla Tareq; Ali, Norhashidah Hj. Mohd.

2016-06-01

Variety of problems in science and engineering may be described by fractional partial differential equations (FPDE) in relation to space and/or time fractional derivatives. The difference between time fractional diffusion equations and standard diffusion equations lies primarily in the time derivative. Over the last few years, iterative schemes derived from the rotated finite difference approximation have been proven to work well in solving standard diffusion equations. However, its application on time fractional diffusion counterpart is still yet to be investigated. In this paper, we will present a preliminary study on the formulation and analysis of new explicit group iterative methods in solving a two-dimensional time fractional diffusion equation. These methods were derived from the standard and rotated Crank-Nicolson difference approximation formula. Several numerical experiments were conducted to show the efficiency of the developed schemes in terms of CPU time and iteration number. At the request of all authors of the paper an updated version of this article was published on 7 July 2016. The original version supplied to AIP Publishing contained an error in Table 1 and References 15 and 16 were incomplete. These errors have been corrected in the updated and republished article.

6. Numerical approximation of Lévy-Feller fractional diffusion equation via Chebyshev-Legendre collocation method

Sweilam, N. H.; Abou Hasan, M. M.

2016-08-01

This paper reports a new spectral algorithm for obtaining an approximate solution for the Lévy-Feller diffusion equation depending on Legendre polynomials and Chebyshev collocation points. The Lévy-Feller diffusion equation is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative. A new formula expressing explicitly any fractional-order derivatives, in the sense of Riesz-Feller operator, of Legendre polynomials of any degree in terms of Jacobi polynomials is proved. Moreover, the Chebyshev-Legendre collocation method together with the implicit Euler method are used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. Numerical results with comparisons are given to confirm the reliability of the proposed method for the Lévy-Feller diffusion equation.

7. Synchronization with propagation - The functional differential equations

2016-06-01

The structure represented by one or several oscillators couple to a one-dimensional transmission environment (e.g. a vibrating string in the mechanical case or a lossless transmission line in the electrical case) turned to be attractive for the research in the field of complex structures and/or complex behavior. This is due to the fact that such a structure represents some generalization of various interconnection modes with lumped parameters for the oscillators. On the other hand the lossless and distortionless propagation along transmission lines has generated several research in electrical, thermal, hydro and control engineering leading to the association of some functional differential equations to the basic initial boundary value problems. The present research is performed at the crossroad of the aforementioned directions. We shall associate to the starting models some functional differential equations - in most cases of neutral type - and make use of the general theorems for existence and stability of forced oscillations for functional differential equations. The challenges introduced by the analyzed problems for the general theory are emphasized, together with the implication of the results for various applications.

8. Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries.

PubMed

Jeon, Jae-Hyung; Metzler, Ralf

2010-02-01

Motivated by subdiffusive motion of biomolecules observed in living cells, we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time-averaged mean-squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular, we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single-particle trajectory data.

9. Approximate Solution of Time-Fractional Advection-Dispersion Equation via Fractional Variational Iteration Method

PubMed Central

İbiş, Birol

2014-01-01

This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs. PMID:24578662

10. Computational solutions of unified fractional reaction-diffusion equations with composite fractional time derivative

Saxena, R. K.; Mathai, A. M.; Haubold, H. J.

2015-10-01

This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function ϕ (x, t) which is a nonlinear function governing reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.

11. Constructing general partial differential equations using polynomial and neural networks.

PubMed

2016-01-01

Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.

12. Tempered fractional Feynman-Kac equation: Theory and examples.

PubMed

Wu, Xiaochao; Deng, Weihua; Barkai, Eli

2016-03-01

Functionals of Brownian and non-Brownian motions have diverse applications and attracted a lot of interest among scientists. This paper focuses on deriving the forward and backward fractional Feynman-Kac equations describing the distribution of the functionals of the space and time-tempered anomalous diffusion, belonging to the continuous time random walk class. Several examples of the functionals are explicitly treated, including the occupation time in half-space, the first passage time, the maximal displacement, the fluctuations of the occupation fraction, and the fluctuations of the time-averaged position.

13. The Influence of Fractional Diffusion in Fisher-KPP Equations

Cabré, Xavier; Roquejoffre, Jean-Michel

2013-06-01

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.

14. Numerical study of fractional nonlinear Schrödinger equations

PubMed Central

Klein, Christian; Sparber, Christof; Markowich, Peter

2014-01-01

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604

15. A modification of WKB method for fractional differential operators of Schrödinger's type

Sayevand, K.; Pichaghchi, K.

2016-08-01

In this paper, we were concerned with the description of the singularly perturbed differential equations within the scope of fractional calculus. However, we shall note that one of the main methods used to solve these problems is the so-called WKB method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the WKB to the scope of fractional derivative, we proposed a relatively new derivative called the local fractional derivative. By use of properties of local fractional derivative, we extend the WKB method in the scope of the fractional differential equation. By means of this extension, the WKB analysis based on the Borel resummation, for fractional differential operators of WKB type are investigated. The convergence and the Mittag-Leffler stability of the proposed approach is proven. The obtained results are in excellent agreement with the existing ones in open literature and it is shown that the present approach is very effective and accurate. Furthermore, we are mainly interested to construct the solution of fractional Schrödinger equation in the Mittag-Leffler form and how it leads naturally to this semi-classical approximation namely modified WKB.

16. Stability at systems of usual differential equations in virus dynamics

Schröer, H.

In this paper we discuss different models of differential equations systems, that describe virus dynamics in different situations (HIV-virus and Hepatitis B-virus). We inquire the stability of differential equations. We use theorems of the stability theory.

17. Lattice Boltzmann method for the fractional advection-diffusion equation.

PubMed

Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L

2016-04-01

Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

18. Lattice Boltzmann method for the fractional advection-diffusion equation

Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.

2016-04-01

Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

19. Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions

Jia, Jinhong; Wang, Hong

2015-07-01

Numerical methods for space-fractional diffusion equations often generate dense or even full stiffness matrices. Traditionally, these methods were solved via Gaussian type direct solvers, which requires O (N3) of computational work per time step and O (N2) of memory to store where N is the number of spatial grid points in the discretization. In this paper we develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite difference methods (both steady-state and time-dependent) space-fractional diffusion equations with fractional derivative boundary conditions in one space dimension. The method requires O (N) of memory and O (Nlog ⁡ N) of operations per iteration. Due to the application of effective preconditioners, significantly reduced numbers of iterations were achieved that further reduces the computational cost of the fast method. Numerical results are presented to show the utility of the method.

20. Solving Partial Differential Equations on Overlapping Grids

SciTech Connect

Henshaw, W D

2008-09-22

We discuss the solution of partial differential equations (PDEs) on overlapping grids. This is a powerful technique for efficiently solving problems in complex, possibly moving, geometry. An overlapping grid consists of a set of structured grids that overlap and cover the computational domain. By allowing the grids to overlap, grids for complex geometries can be more easily constructed. The overlapping grid approach can also be used to remove coordinate singularities by, for example, covering a sphere with two or more patches. We describe the application of the overlapping grid approach to a variety of different problems. These include the solution of incompressible fluid flows with moving and deforming geometry, the solution of high-speed compressible reactive flow with rigid bodies using adaptive mesh refinement (AMR), and the solution of the time-domain Maxwell's equations of electromagnetism.

1. Lectures on differential equations for Feynman integrals

Henn, Johannes M.

2015-04-01

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE.

2. Computer Corner. A Rich Differential Equation for Computer Demonstrations.

ERIC Educational Resources Information Center

Banks, Bernard W.

1990-01-01

Presents an example using a computer to illustrate concepts graphically in an introductory course on differential equations. Discusses the algorithms of the computer program displaying the solutions to an equation and the inclination field of the equation. (YP)

3. Solving Differential Equations in R: Package deSolve

EPA Science Inventory

In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...

4. The Klein-Gordon-Zakharov equations with the positive fractional power terms and their exact solutions

Zhang, Jinliang; Hu, Wuqiang; Ma, Yu

2016-12-01

In this paper, the famous Klein-Gordon-Zakharov equations are firstly generalized, the new special types of Klein-Gordon-Zakharov equations with the positive fractional power terms (gKGZE) are presented. In order to derive the exact solutions of new special gKGZE, the subsidiary higher order ordinary differential equations (sub-ODEs) with the positive fractional power terms are introduced, and with the aids of the Sub-ODE, the exact solutions of three special types of the gKGZE are derived, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of gKGZE satisfy certain constraint conditions.

5. Effective fractional acoustic wave equations in one-dimensional random multiscale media.

PubMed

Garnier, Josselin; Solna, Knut

2010-01-01

This paper considers multiple scattering of waves propagating in a non-lossy one-dimensional random medium with short- or long-range correlations. Using stochastic homogenization theory it is possible to show that pulse propagation is described by an effective deterministic fractional wave equation, which corresponds to an effective medium with a frequency-dependent attenuation that obeys a power law with an exponent between 0 and 2. The exponent is related to the Hurst parameter of the medium, which is a characteristic parameter of the correlation properties of the fluctuations of the random medium. Moreover the frequency-dependent attenuation is associated with a special frequency-dependent phase, which ensures that causality and Kramers-Kronig relations are satisfied. In the time domain the effective wave equation has the form of a linear integro-differential equation with a fractional derivative.

6. Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

7. Fast numerical solution for fractional diffusion equations by exponential quadrature rule

Zhang, Lu; Sun, Hai-Wei; Pang, Hong-Kui

2015-10-01

After spatial discretization to the fractional diffusion equation by the shifted Grünwald formula, it leads to a system of ordinary differential equations, where the resulting coefficient matrix possesses the Toeplitz-like structure. An exponential quadrature rule is employed to solve such a system of ordinary differential equations. The convergence by the proposed method is theoretically studied. In practical computation, the product of a Toeplitz-like matrix exponential and a vector is calculated by the shift-invert Arnoldi method. Meanwhile, the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift-invert Arnoldi method. Numerical results are given to demonstrate the efficiency of the proposed method.

8. Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations

Ma, Wen-Xiu; Zhou, Yuan; Dougherty, Rachael

2016-08-01

Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on f are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential equations on u, generated from taking a transformation of dependent variables u = 2(ln f)x.

9. Optimal q-homotopy analysis method for time-space fractional gas dynamics equation

Saad, K. M.; AL-Shareef, E. H.; Mohamed, Mohamed S.; Yang, Xiao-Jun

2017-01-01

It is well known that the homotopy analysis method is one of the most efficient methods for obtaining analytical or approximate semi-analytical solutions of both linear and non-linear partial differential equations. A more general form of HAM is introduced in this paper, which is called Optimal q-Homotopy Analysis Method (Oq-HAM). It has better convergence properties as compared with the usual HAM, due to the presence of fraction factor associated with the solution. The convergence of q-HAM is studied in details elsewhere (M.A. El-Tawil, Int. J. Contemp. Math. Sci. 8, 481 (2013)). Oq-HAM is applied to the non-linear homogeneous and non-homogeneous time and space fractional gas dynamics equations with initial condition. An optimal convergence region is determined through the residual error. By minimizing the square residual error, the optimal convergence control parameters can be obtained. The accuracy and efficiency of the proposed method are verified by comparison with the exact solution of the fractional gas dynamics equation. Also, it is shown that the Oq-HAM for the fractional gas dynamics equation is equivalent to the exact solution. We obtain graphical representations of the solutions using MATHEMATICA.

10. Comparison of fractional wave equations for power law attenuation in ultrasound and elastography.

PubMed

Holm, Sverre; Näsholm, Sven Peter

2014-04-01

A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional constitutive equations, whereas the former wave equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear wave elastography. In such applications, the wave equations based on constitutive equations are the viable ones.

11. On fractional differential inclusions with the Jumarie derivative

SciTech Connect

Kamocki, Rafał; Obczyński, Cezary

2014-02-15

In the paper, fractional differential inclusions with the Jumarie derivative are studied. We discuss the existence and uniqueness of a solution to such problems. Our study relies on standard variational methods.

12. Nonlocal diffusion second order partial differential equations

Benedetti, I.; Loi, N. V.; Malaguti, L.; Taddei, V.

2017-02-01

The paper deals with a second order integro-partial differential equation in Rn with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartman-type inequality and it involves a Scorza-Dragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces.

13. Stability and control of functional differential equations

Peet, Matthew Monnig

This thesis addresses the question of stability of systems defined by differential equations which contain nonlinearity and delay. In particular, we analyze the stability of a well-known delayed nonlinear implementation of a certain Internet congestion control protocol. We also describe a generalized methodology for proving stability of time-delay systems through the use of semidefinite programming. In Chapters 4 and 5, we consider an Internet congestion control protocol based on the decentralized gradient projection algorithm. For a certain class of utility function, this algorithm was shown to be globally convergent for some sufficiently small value of a gain parameter. Later work gave an explicit bound on this gain for a linearized version of the system. This thesis proves that this bound also implies stability of the original system. The proof is constructed within a generalized passivity framework. The dynamics of the system are separated into a linear, delayed component and a system defined by a nonlinear differential equation with discontinuity in the dynamics. Frequency-domain analysis is performed on the linear component and time-domain analysis is performed on the nonlinear discontinuous system. In Chapter 7, we describe a general methodology for proving stability of linear time-delay systems by computing solutions to an operator-theoretic version of the Lyapunov inequality via semidefinite programming. The result is stated in terms of a nested sequence of sufficient conditions which are of increasing accuracy. This approach is generalized to the case of parametric uncertainty by considering parameter-dependent Lyapunov functionals. Numerical examples are given to demonstrate convergence of the algorithm. In Chapter 8, this approach is generalized to nonlinear time-delay systems through the use of non-quadratic Lyapunov functionals.

14. Fault Detection in Differential Algebraic Equations

Scott, Jason Roderick

Fault detection and identification (FDI) is important in almost all real systems. Fault detection is the supervision of technical processes aimed at detecting undesired or unpermitted states (faults) and taking appropriate actions to avoid dangerous situations, or to ensure efficiency in a system. This dissertation develops and extends fault detection techniques for systems modeled by differential algebraic equations (DAEs). First, a passive, observer-based approach is developed and linear filters are constructed to identify faults by filtering residual information. The method presented here uses the least squares completion to compute an ordinary differential equation (ODE) that contains the solution of the DAE and applies the observer directly to this ODE. While observers have been applied to ODE models for the purpose of fault detection in the past, the use of observers on completions of DAEs is a new idea. Moreover, the resulting residuals are modified requiring additional analysis. Robustness with respect to disturbances is also addressed by a novel frequency filtering technique. Active detection, as opposed to passive detection where outputs are passively monitored, allows the injection of an auxiliary control signal to test the system. These algorithms compute an auxiliary input signal guaranteeing fault detection, assuming bounded noise. In the second part of this dissertation, a novel active detection approach for DAE models is developed by taking linear transformations of the DAEs and solving a bi-layer optimization problem. An efficient real-time detection algorithm is also provided, as is the extension to model uncertainty. The existence of a class of problems where the algorithm breaks down is revealed and an alternative algorithm that finds a nearly minimal auxiliary signal is presented. Finally, asynchronous signal design, that is, applying the test signal on a different interval than the observation window, is explored and discussed.

15. On the bound of the Lyapunov exponents for the fractional differential systems.

PubMed

Li, Changpin; Gong, Ziqing; Qian, Deliang; Chen, YangQuan

2010-03-01

In recent years, fractional(-order) differential equations have attracted increasing interests due to their applications in modeling anomalous diffusion, time dependent materials and processes with long range dependence, allometric scaling laws, and complex networks. Although an autonomous system cannot define a dynamical system in the sense of semigroup because of the memory property determined by the fractional derivative, we can still use the Lyapunov exponents to discuss its dynamical evolution. In this paper, we first define the Lyapunov exponents for fractional differential systems then estimate the bound of the corresponding Lyapunov exponents. For linear fractional differential system, the bounds of its Lyapunov exponents are conveniently derived which can be regarded as an example for the theoretical results established in this paper. Numerical example is also included which supports the theoretical analysis.

16. Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation

Parand, Kourosh; Mazaheri, Pooria; Yousefi, Hossein; Delkhosh, Mehdi

2017-02-01

In this paper, a new method based on Fractional order of Rational Jacobi (FRJ) functions is proposed that utilizes quasilinearization method to solve non-linear singular Thomas-Fermi equation on unbounded interval [0,∞). The equation is solved without domain truncation and variable changing. First, the quasilinearization method is used to convert the equation to the sequence of linear ordinary differential equations. Then, by using the FRJs collocation method the equations are solved. For the evaluation, comparison with some numerical solutions shows that the proposed solution is highly accurate.

17. Legendre-Tau approximations for functional differential equations

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1983-01-01

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

18. Periodic solutions for ordinary differential equations with sublinear impulsive effects

Qian, Dingbian; Li, Xinyu

2005-03-01

The continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects. The applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations, for instance, the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with Landesman-Lazer conditions.

19. Differential Equations Compatible with Boundary Rational qKZ Equation

Takeyama, Yoshihiro

2011-10-01

We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.

20. Compatible Spatial Discretizations for Partial Differential Equations

SciTech Connect

Arnold, Douglas, N, ed.

2004-11-25

From May 11--15, 2004, the Institute for Mathematics and its Applications held a hot topics workshop on Compatible Spatial Discretizations for Partial Differential Equations. The numerical solution of partial differential equations (PDE) is a fundamental task in science and engineering. The goal of the workshop was to bring together a spectrum of scientists at the forefront of the research in the numerical solution of PDEs to discuss compatible spatial discretizations. We define compatible spatial discretizations as those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum principles. A wide variety of discretization methods applied across a wide range of scientific and engineering applications have been designed to or found to inherit or mimic intrinsic spatial structure and reproduce fundamental properties of the solution of the continuous PDE model at the finite dimensional level. A profusion of such methods and concepts relevant to understanding them have been developed and explored: mixed finite element methods, mimetic finite differences, support operator methods, control volume methods, discrete differential forms, Whitney forms, conservative differencing, discrete Hodge operators, discrete Helmholtz decomposition, finite integration techniques, staggered grid and dual grid methods, etc. This workshop seeks to foster communication among the diverse groups of researchers designing, applying, and studying such methods as well as researchers involved in practical solution of large scale problems that may benefit from advancements in such discretizations; to help elucidate the relations between the different methods and concepts; and to generally advance our understanding in the area of compatible spatial discretization methods for PDE. Particular points of emphasis included: + Identification of intrinsic properties of PDE models that are critical for the fidelity of numerical

1. In-fiber all-optical fractional differentiator.

PubMed

2009-03-15

We demonstrate that an asymmetrical pi phase-shifted fiber Bragg grating operated in reflection can provide the required spectral response for implementing an all-optical fractional differentiator. There are different (but equivalent) ways to design it, e.g., by using different gratings lengths and keeping the same index modulation depth at both sides of the pi phase shift, or vice versa. Analytical expressions were found relating the fractional differentiator order with the grating parameters. The device shows a good accuracy calculating the fractional time derivatives of the complex field of an arbitrary input optical waveform. The introduced concept is supported by numerical simulations.

2. Space fractional Wigner equation and its semiclassical limit.

PubMed

Stickler, B A; Schachinger, E

2011-12-01

Manifestations of space fractional quantum mechanics (SFQM), as it was formulated by Laskin [Phys. Rev. E 62, 3135 (2000)], are deemed to offer a better physical interpretation of Lévy flight statistics on a quantum mechanical level. We start with the SFQM Schrödinger equation characterized by a Lévy flight index α∈ (1,2), perform a Wigner transform, and draw the limit h/Eτ → 0 (i.e., let the observed energy scale E go to infinity in comparison to the quantization given by h/τ). In order to obtain classical transport equations two possible substitutions for the terms |p|(α) and |p'|α which appear in von Neumann's equation are presented. It is demonstrated that they conform to the criteria for a successful Wigner transform. Their benefits and caveats are discussed in detail. We find, that, indeed, SFQM manifests itself in an anomalous kinetic term of the free particle's motion and, assuming an external potential diagonal in momentum space for the sake of simplicity, in corresponding anomalous terms in the resulting drift current. All our results reduce to the classical forms in the limit α = 2.

3. Space-fractional advection-dispersion equations by the Kansa method

Pang, Guofei; Chen, Wen; Fu, Zhuojia

2015-07-01

The paper makes the first attempt at applying the Kansa method, a radial basis function meshless collocation method, to the space-fractional advection-dispersion equations, which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer-order derivative. However, because of non-local property of integro-differential operator of space-fractional derivative, numerical solution of these novel models is very challenging and little has been reported in literature. It is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non-local property. Thus, the global methods appear to have certain advantages in numerical simulation of these non-local models because of their high accuracy and smaller size resultant matrix equation. Compared with the finite difference method, popular in the solution of fractional equations, the Kansa method is a recent meshless global technique and is promising for high-dimensional irregular domain problems. In this study, the resultant matrix of the Kansa method is accurately calculated by the Gauss-Jacobi quadrature rule. Numerical results show that the Kansa method is highly accurate and computationally efficient for space-fractional advection-dispersion problems.

4. From differential to difference equations for first order ODEs

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Walker, Kevin P.

1991-01-01

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

5. HEREDITARY DEPENDENCE IN THE THEORY OF DIFFERENTIAL EQUATIONS. PART I,

DTIC Science & Technology

A general class of differential equations with hereditary dependence is introduced which includes most equations of hereditary type encountered in...of solutions and dependence on initial data and parameters will be considered herein.

6. HEREDITARY DEPENDENCE IN THE THEORY OF DIFFERENTIAL EQUATIONS, PART II,

DTIC Science & Technology

A general class of differential equations with hereditary dependence is introduced which includes most equations of hereditary type encountered in...uniqueness of solutions and dependence on initial data and parameters are considered.

7. Dielectric metasurfaces solve differential and integro-differential equations.

PubMed

2017-04-01

Leveraging subwavelength resonant nanostructures, plasmonic metasurfaces have recently attracted much attention as a breakthrough concept for engineering optical waves both spatially and spectrally. However, inherent ohmic losses concomitant with low coupling efficiencies pose fundamental impediments over their practical applications. Not only can all-dielectric metasurfaces tackle such substantial drawbacks, but also their CMOS-compatible configurations support both Mie resonances that are invariant to the incident angle. Here, we report on a transmittive metasurface comprising arrayed silicon nanodisks embedded in a homogeneous dielectric medium to manipulate phase and amplitude of incident light locally and almost independently. By taking advantage of the interplay between the electric/magnetic resonances and employing general concepts of spatial Fourier transformation, a highly efficient metadevice is proposed to perform mathematical operations including solution of ordinary differential and integro-differential equations with constant coefficients. Our findings further substantiate dielectric metasurfaces as promising candidates for miniaturized, two-dimensional, and planar optical analog computing systems that are much thinner than their conventional lens-based counterparts.

8. Exact solutions to nonlinear delay differential equations of hyperbolic type

Vyazmin, Andrei V.; Sorokin, Vsevolod G.

2017-01-01

We consider nonlinear delay differential equations of hyperbolic type, including equations with varying transfer coefficients and varying delays. The equations contain one or two arbitrary functions of a single argument. We present new classes of exact generalized and functional separable solutions. All the solutions involve free parameters and can be suitable for solving certain model problems as well as testing numerical and approximate analytical methods for similar and more complex nonlinear differential-difference equations.

9. Stochastic symmetries of Wick type stochastic ordinary differential equations

Ünal, Gazanfer

2015-04-01

We consider Wick type stochastic ordinary differential equations with Gaussian white noise. We define the stochastic symmetry transformations and Lie equations in Kondratiev space (S)-1N. We derive the determining system of Wick type stochastic partial differential equations with Gaussian white noise. Stochastic symmetries for stochastic Bernoulli, Riccati and general stochastic linear equation in (S)-1N are obtained. A stochastic version of canonical variables is also introduced.

10. Bifurcation and stability for a nonlinear parabolic partial differential equation

NASA Technical Reports Server (NTRS)

Chafee, N.

1973-01-01

Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.

11. A complex Noether approach for variational partial differential equations

Naz, R.; Mahomed, F. M.

2015-10-01

Scalar complex partial differential equations which admit variational formulations are studied. Such a complex partial differential equation, via a complex dependent variable, splits into a system of two real partial differential equations. The decomposition of the Lagrangian of the complex partial differential equation in the real domain is shown to yield two real Lagrangians for the split system. The complex Maxwellian distribution, transonic gas flow, Maxwellian tails, dissipative wave and Klein-Gordon equations are considered. The Noether symmetries and gauge terms of the split system that correspond to both the Lagrangians are constructed by the Noether approach. In the case of coupled split systems, the same Noether symmetries are obtained. The Noether symmetries for the uncoupled split systems are different. The conserved vectors of the split system which correspond to both the Lagrangians are compared to the split conserved vectors of the complex partial differential equation for the examples. The split conserved vectors of the complex partial differential equation are the same as the conserved vectors of the split system of real partial differential equations in the case of coupled systems. Moreover a Noether-like theorem for the split system is proved which provides the Noether-like conserved quantities of the split system from knowledge of the Noether-like operators. An interesting result on the split characteristics and the conservation laws is shown as well. The Noether symmetries and gauge terms of the Lagrangian of the split system with the split Noether-like operators and gauge terms of the Lagrangian of the given complex partial differential equation are compared. Folklore suggests that the split Noether-like operators of a Lagrangian of a complex Euler-Lagrange partial differential equation are symmetries of the Lagrangian of the split system of real partial differential equations. This is not the case. They are proved to be the same if the

12. Robust estimation for ordinary differential equation models.

PubMed

Cao, J; Wang, L; Xu, J

2011-12-01

Applied scientists often like to use ordinary differential equations (ODEs) to model complex dynamic processes that arise in biology, engineering, medicine, and many other areas. It is interesting but challenging to estimate ODE parameters from noisy data, especially when the data have some outliers. We propose a robust method to address this problem. The dynamic process is represented with a nonparametric function, which is a linear combination of basis functions. The nonparametric function is estimated by a robust penalized smoothing method. The penalty term is defined with the parametric ODE model, which controls the roughness of the nonparametric function and maintains the fidelity of the nonparametric function to the ODE model. The basis coefficients and ODE parameters are estimated in two nested levels of optimization. The coefficient estimates are treated as an implicit function of ODE parameters, which enables one to derive the analytic gradients for optimization using the implicit function theorem. Simulation studies show that the robust method gives satisfactory estimates for the ODE parameters from noisy data with outliers. The robust method is demonstrated by estimating a predator-prey ODE model from real ecological data.

13. Parameter Estimation of Partial Differential Equation Models.

PubMed

Xun, Xiaolei; Cao, Jiguo; Mallick, Bani; Carroll, Raymond J; Maity, Arnab

2013-01-01

Partial differential equation (PDE) models are commonly used to model complex dynamic systems in applied sciences such as biology and finance. The forms of these PDE models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system. Parameters in PDE models often have interesting scientific interpretations, but their values are often unknown, and need to be estimated from the measurements of the dynamic system in the present of measurement errors. Most PDEs used in practice have no analytic solutions, and can only be solved with numerical methods. Currently, methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values, and thus the computational load is high. In this article, we propose two methods to estimate parameters in PDE models: a parameter cascading method and a Bayesian approach. In both methods, the underlying dynamic process modeled with the PDE model is represented via basis function expansion. For the parameter cascading method, we develop two nested levels of optimization to estimate the PDE parameters. For the Bayesian method, we develop a joint model for data and the PDE, and develop a novel hierarchical model allowing us to employ Markov chain Monte Carlo (MCMC) techniques to make posterior inference. Simulation studies show that the Bayesian method and parameter cascading method are comparable, and both outperform other available methods in terms of estimation accuracy. The two methods are demonstrated by estimating parameters in a PDE model from LIDAR data.

14. Electrocardiogram classification using delay differential equations

Lainscsek, Claudia; Sejnowski, Terrence J.

2013-06-01

Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.

15. A Geometric Treatment of Implicit Differential-Algebraic Equations

Rabier, P. J.; Rheinboldt, W. C.

A differential-geometric approach for proving the existence and uniqueness of implicit differential-algebraic equations is presented. It provides for a significant improvement of an earlier theory developed by the authors as well as for a completely intrinsic definition of the index of such problems. The differential-algebraic equation is transformed into an explicit ordinary differential equation by a reduction process that can be abstractly defined for specific submanifolds of tangent bundles here called reducible π-submanifolds. Local existence and uniqueness results for differential-algebraic equations then follow directly from the final stage of this reduction by means of an application of the standard theory of ordinary differential equations.

16. A parallel algorithm for the two-dimensional time fractional diffusion equation with implicit difference method.

PubMed

Gong, Chunye; Bao, Weimin; Tang, Guojian; Jiang, Yuewen; Liu, Jie

2014-01-01

It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE) with iterative implicit finite difference method is O(M(x)M(y)N(2)). In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16-4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.

17. A Spectral Legendre-Gauss-Lobatto Collocation Method for a Space-Fractional Advection Diffusion Equations with Variable Coefficients

Bhrawy, A. H.; Baleanu, D.

2013-10-01

An efficient Legendre-Gauss-Lobatto collocation (L-GL-C) method is applied to solve the space-fractional advection diffusion equation with nonhomogeneous initial-boundary conditions. The Legendre-Gauss-Lobatto points are used as collocation nodes for spatial fractional derivatives as well as the Caputo fractional derivative. This approach is reducing the problem to the solution of a system of ordinary differential equations in time which can be solved by using any standard numerical techniques. The proposed numerical solutions when compared with the exact solutions reveal that the obtained solution produces highly accurate results. The results show that the proposed method has high accuracy and is efficient for solving the space-fractional advection diffusion equation.

18. Face recognition with histograms of fractional differential gradients

Yu, Lei; Ma, Yan; Cao, Qi

2014-05-01

It has proved that fractional differentiation can enhance the edge information and nonlinearly preserve textural detailed information in an image. This paper investigates its ability for face recognition and presents a local descriptor called histograms of fractional differential gradients (HFDG) to extract facial visual features. HFDG encodes a face image into gradient patterns using multiorientation fractional differential masks, from which histograms of gradient directions are computed as the face representation. Experimental results on Yale, face recognition technology (FERET), Carnegie Mellon University pose, illumination, and expression (CMU PIE), and A. Martinez and R. Benavente (AR) databases validate the feasibility of the proposed method and show that HFDG outperforms local binary patterns (LBP), histograms of oriented gradients (HOG), enhanced local directional patterns (ELDP), and Gabor feature-based methods.

19. Comment on "Fractional quantum mechanics" and "Fractional Schrödinger equation".

PubMed

Wei, Yuchuan

2016-06-01

In this Comment we point out some shortcomings in two papers [N. Laskin, Phys. Rev. E 62, 3135 (2000)10.1103/PhysRevE.62.3135; N. Laskin, Phys. Rev. E 66, 056108 (2002)10.1103/PhysRevE.66.056108]. We prove that the fractional uncertainty relation does not hold generally. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.e., a particle can teleport from a place to another. Since the relativistic kinetic energy can be viewed as an approximate realization of the fractional kinetic energy, the particle teleportation should be an observable relativistic effect in quantum mechanics. With the help of this concept, superconductivity could be viewed as the teleportation of electrons from one side of a superconductor to another and superfluidity could be viewed as the teleportation of helium atoms from one end of a capillary tube to the other. We also point out how to teleport a particle to an arbitrary destination.

20. Comment on "Fractional quantum mechanics" and "Fractional Schrödinger equation"

Wei, Yuchuan

2016-06-01

In this Comment we point out some shortcomings in two papers [N. Laskin, Phys. Rev. E 62, 3135 (2000), 10.1103/PhysRevE.62.3135; N. Laskin, Phys. Rev. E 66, 056108 (2002), 10.1103/PhysRevE.66.056108]. We prove that the fractional uncertainty relation does not hold generally. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.e., a particle can teleport from a place to another. Since the relativistic kinetic energy can be viewed as an approximate realization of the fractional kinetic energy, the particle teleportation should be an observable relativistic effect in quantum mechanics. With the help of this concept, superconductivity could be viewed as the teleportation of electrons from one side of a superconductor to another and superfluidity could be viewed as the teleportation of helium atoms from one end of a capillary tube to the other. We also point out how to teleport a particle to an arbitrary destination.

1. Parameter Estimates in Differential Equation Models for Chemical Kinetics

ERIC Educational Resources Information Center

Winkel, Brian

2011-01-01

We discuss the need for devoting time in differential equations courses to modelling and the completion of the modelling process with efforts to estimate the parameters in the models using data. We estimate the parameters present in several differential equation models of chemical reactions of order n, where n = 0, 1, 2, and apply more general…

2. Nonlinear partial differential equations: Integrability, geometry and related topics

Krasil'shchik, Joseph; Rubtsov, Volodya

2017-03-01

Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.

3. Sourcing for Parameter Estimation and Study of Logistic Differential Equation

ERIC Educational Resources Information Center

Winkel, Brian J.

2012-01-01

This article offers modelling opportunities in which the phenomena of the spread of disease, perception of changing mass, growth of technology, and dissemination of information can be described by one differential equation--the logistic differential equation. It presents two simulation activities for students to generate real data, as well as…

4. Nonstandard Topics for Student Presentations in Differential Equations

ERIC Educational Resources Information Center

LeMasurier, Michelle

2006-01-01

An interesting and effective way to showcase the wide variety of fields to which differential equations can be applied is to have students give short oral presentations on a specific application. These talks, which have been presented by 30-40 students per year in our differential equations classes, provide exposure to a diverse array of topics…

5. Monograph - The Numerical Integration of Ordinary Differential Equations.

ERIC Educational Resources Information Center

Hull, T. E.

The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…

6. BIFURCATIONS OF RANDOM DIFFERENTIAL EQUATIONS WITH BOUNDED NOISE ON SURFACES

PubMed Central

Homburg, Ale Jan; Young, Todd R.

2011-01-01

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations. PMID:22211081

7. Optimal moving grids for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Wathen, A. J.

1989-01-01

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of partial differential equation solutions in the least squares norm.

8. Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics

Mirzazadeh, Mohammad; Ekici, Mehmet; Sonmezoglu, Abdullah; Ortakaya, Sami; Eslami, Mostafa; Biswas, Anjan

2016-05-01

This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi's elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.

9. The Radially Symmetric Euler Equations as an Exterior Differential System

Baty, Roy; Ramsey, Scott; Schmidt, Joseph

2016-11-01

This work develops the Euler equations as an exterior differential system in radially symmetric coordinates. The Euler equations are studied for unsteady, compressible, inviscid fluids in one-dimensional, converging flow fields with a general equation of state. The basic geometrical constructions (for example, the differential forms, tangent planes, jet space, and differential ideal) used to define and analyze differential equations as systems of exterior forms are reviewed and discussed for converging flows. Application of the Frobenius theorem to the question of the existence of solutions to radially symmetric converging flows is also reviewed and discussed. The exterior differential system is further applied to derive and analyze the general family of characteristic vector fields associated with the one-dimensional inviscid flow equations.

10. A New Factorisation of a General Second Order Differential Equation

ERIC Educational Resources Information Center

Clegg, Janet

2006-01-01

A factorisation of a general second order ordinary differential equation is introduced from which the full solution to the equation can be obtained by performing two integrations. The method is compared with traditional methods for solving these type of equations. It is shown how the Green's function can be derived directly from the factorisation…

11. The method of averages applied to the KS differential equations

NASA Technical Reports Server (NTRS)

Graf, O. F., Jr.; Mueller, A. C.; Starke, S. E.

1977-01-01

A new approach for the solution of artificial satellite trajectory problems is proposed. The basic idea is to apply an analytical solution method (the method of averages) to an appropriate formulation of the orbital mechanics equations of motion (the KS-element differential equations). The result is a set of transformed equations of motion that are more amenable to numerical solution.

12. Intuitive Understanding of Solutions of Partially Differential Equations

ERIC Educational Resources Information Center

Kobayashi, Y.

2008-01-01

This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based…

13. Solution of fractional kinetic equation by a class of integral transform of pathway type

Kumar, Dilip

2013-04-01

Solutions of fractional kinetic equations are obtained through an integral transform named Pα-transform introduced in this paper. The Pα-transform is a binomial type transform containing many class of transforms including the well known Laplace transform. The paper is motivated by the idea of pathway model introduced by Mathai [Linear Algebra Appl. 396, 317-328 (2005), 10.1016/j.laa.2004.09.022]. The composition of the transform with differential and integral operators are proved along with convolution theorem. As an illustration of applications to the general theory of differential equations, a simple differential equation is solved by the new transform. Being a new transform, the Pα-transform of some elementary functions as well as some generalized special functions such as H-function, G-function, Wright generalized hypergeometric function, generalized hypergeometric function, and Mittag-Leffler function are also obtained. The results for the classical Laplace transform is retrieved by letting α → 1.

14. Real-time optical laboratory solution of parabolic differential equations

NASA Technical Reports Server (NTRS)

Casasent, David; Jackson, James

1988-01-01

An optical laboratory matrix-vector processor is used to solve parabolic differential equations (the transient diffusion equation with two space variables and time) by an explicit algorithm. This includes optical matrix-vector nonbase-2 encoded laboratory data, the combination of nonbase-2 and frequency-multiplexed data on such processors, a high-accuracy optical laboratory solution of a partial differential equation, new data partitioning techniques, and a discussion of a multiprocessor optical matrix-vector architecture.

15. Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise

SciTech Connect

Shu, Ji E-mail: 530282863@qq.com; Li, Ping E-mail: 530282863@qq.com; Zhang, Jia; Liao, Ou

2015-10-15

This paper is concerned with the stochastic coupled fractional Ginzburg-Landau equation with additive noise. We first transform the stochastic coupled fractional Ginzburg-Landau equation into random equations whose solutions generate a random dynamical system. Then we prove the existence of random attractor for random dynamical system.

16. Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations

SciTech Connect

Fuhrman, Marco Tessitore, Gianmario

2005-05-15

We study a forward-backward system of stochastic differential equations in an infinite-dimensional framework and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions.The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction-diffusion equations),where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black-Scholes or Hamilton-Jacobi-Bellman type.

17. Algebraic and geometric structures of analytic partial differential equations

Kaptsov, O. V.

2016-11-01

We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.

18. Solving Differential Equations Analytically. Elementary Differential Equations. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Unit 335.

ERIC Educational Resources Information Center

Goldston, J. W.

This unit introduces analytic solutions of ordinary differential equations. The objective is to enable the student to decide whether a given function solves a given differential equation. Examples of problems from biology and chemistry are covered. Problem sets, quizzes, and a model exam are included, and answers to all items are provided. The…

19. In-out intermittency in partial differential equation and ordinary differential equation models.

PubMed

Covas, Eurico; Tavakol, Reza; Ashwin, Peter; Tworkowski, Andrew; Brooke, John M.

2001-06-01

We find concrete evidence for a recently discovered form of intermittency, referred to as in-out intermittency, in both partial differential equation (PDE) and ordinary differential equation (ODE) models of mean field dynamos. This type of intermittency [introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 (1999)] occurs in systems with invariant submanifolds and, as opposed to on-off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in-out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities. (c) 2001 American Institute of Physics.

20. Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel

Gómez-Aguilar, J. F.

2017-01-01

In this paper, using the fractional operators with Mittag-Leffler kernel in Caputo and Riemann-Liouville sense the space-time fractional diffusion equation is modified, the fractional equation will be examined separately; with fractional spatial derivative and fractional temporal derivative. For the study cases, the order considered is 0 < β , γ ≤ 1 respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation, these parameters related to equation results in a fractal space-time geometry provide a new family of solutions for the diffusive processes. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, viscoelastic materials, material heterogeneities and media with different scales.

1. Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

SciTech Connect

Zhang, Xiao; Wei, Chaozhen; Liu, Yingming; Luo, Maokang

2014-11-15

In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same.

2. Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation

Ye, Zhuan; Xu, Xiaojing

2016-06-01

In this paper, we study the 2D Boussinesq equations with fractional Laplacian dissipation. In particular, we prove the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The main ingredient of the proof is the utilization of the Hölder estimates for advection fractional-diffusion equations as well as Littlewood-Paley technique.

3. A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

Bhrawy, A. H.; Abdelkawy, M. A.

2015-08-01

A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations (TFSEs) subject to initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method for spatial discretization; an expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the TFSE is investigated. In addition, the Legendre-Gauss-Lobatto quadrature rule is established to treat the nonlocal conservation conditions. Thereby, the expansion coefficients are then determined by reducing the TFSE with its nonlocal conditions to a system of fractional differential equations (SFDEs) for these coefficients. The second step is to propose a shifted Legendre Gauss-Radau collocation (SL-GR-C) scheme, for temporal discretization, to reduce such system into a system of algebraic equations which is far easier to be solved. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to solve the two-dimensional TFSE. Numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms. By selecting relatively limited Legendre Gauss-Lobatto and Gauss-Radau collocation nodes, we are able to get very accurate approximations, demonstrating the utility and high accuracy of the new approach over other numerical methods.

4. A neuro approach to solve fuzzy Riccati differential equations

SciTech Connect

Shahrir, Mohammad Shazri; Kumaresan, N. Kamali, M. Z. M.; Ratnavelu, Kurunathan

2015-10-22

There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.

5. A neuro approach to solve fuzzy Riccati differential equations

Shahrir, Mohammad Shazri; Kumaresan, N.; Kamali, M. Z. M.; Ratnavelu, Kurunathan

2015-10-01

There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.

6. Traveling wave solution of fractional KdV-Burger-Kuramoto equation describing nonlinear physical phenomena

Gupta, A. K.; Ray, S. Saha

2014-09-01

In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK) equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.

7. Numerical integration of ordinary differential equations of various orders

NASA Technical Reports Server (NTRS)

Gear, C. W.

1969-01-01

Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.

8. Systems of Differential Equations with Skew-Symmetric, Orthogonal Matrices

ERIC Educational Resources Information Center

Glaister, P.

2008-01-01

The solution of a system of linear, inhomogeneous differential equations is discussed. The particular class considered is where the coefficient matrix is skew-symmetric and orthogonal, and where the forcing terms are sinusoidal. More general matrices are also considered.

9. Comparison theorems for neutral stochastic functional differential equations

Bai, Xiaoming; Jiang, Jifa

2016-05-01

The comparison theorems under Wu and Freedman's order are proved for neutral stochastic functional differential equations with finite or infinite delay whose drift terms satisfy the quasimonotone condition and diffusion term is the same.

10. Uniqueness and existence results for ordinary differential equations

Cid, J. Angel; Heikkila, Seppo; Pouso, Rodrigo Lopez

2006-04-01

We establish some uniqueness and existence results for first-order ordinary differential equations with constant-signed discontinuous nonlinear parts. Several examples are given to illustrate the applicability of our work.

11. Variational integrators for nonvariational partial differential equations

Kraus, Michael; Maj, Omar

2015-08-01

Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.

12. Transformation matrices between non-linear and linear differential equations

NASA Technical Reports Server (NTRS)

Sartain, R. L.

1983-01-01

In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.

13. Fractional Schroedinger equation for a particle moving in a potential well

SciTech Connect

Luchko, Yuri

2013-01-15

In this paper, the fractional Schroedinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, 'On the nonlocality of the fractional Schroedinger equation,' J. Math. Phys. 51, 062102 (2010)] and [S. S. Bayin, 'On the consistency of the solutions of the space fractional Schroedinger equation,' J. Math. Phys. 53, 042105 (2012)] published in this journal, controversial opinions regarding solutions to the fractional Schroedinger equation for a particle moving in the infinite potential well that were derived by Laskin ['Fractals and quantum mechanics,' Chaos 10, 780-790 (2000)] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schroedinger equation for a particle moving in the infinite potential well given by Laskin ['Fractals and quantum mechanics,' Chaos 10, 780-790 (2000)] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. ['On the nonlocality of the fractional Schroedinger equation,' J. Math. Phys. 51, 062102 (2010)].

14. Classification of five-point differential-difference equations

Garifullin, R. N.; Yamilov, R. I.; Levi, D.

2017-03-01

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh–Narita–Bogoyavlensky and the discrete Sawada–Kotera equations. The resulting list contains 17 equations, some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries.

15. Varieties of operator manipulation. [for solving differential equations and calculating finite differences

NASA Technical Reports Server (NTRS)

Doohovskoy, A.

1977-01-01

A change in MACSYMA syntax is proposed to accommodate the operator manipulators necessary to implement direct and indirect methods for the solution of differential equations, calculus of finite differences, and the fractional calculus, as well as their modern counterparts. To illustrate the benefits and convenience of this syntax extension, an example is given to show how MACSYMA's pattern-matching capability can be used to implement a particular set of operator identities which can then be used to obtain exact solutions to nonlinear differential equations.

16. Cosmic ray anisotropy in fractional differential models of anomalous diffusion

SciTech Connect

Uchaikin, V. V.

2013-06-15

The problem of galactic cosmic ray anisotropy is considered in two versions of the fractional differential model for anomalous diffusion. The simplest problem of cosmic ray propagation from a point instantaneous source in an unbounded medium is used as an example to show that the transition from the standard diffusion model to the Lagutin-Uchaikin fractional differential model (with characteristic exponent {alpha} = 3/5 and a finite velocity of free particle motion), which gives rise to a knee in the energy spectrum at 10{sup 6} GeV, increases the anisotropy coefficient only by 20%, while the anisotropy coefficient in the Lagutin-Tyumentsev model (with exponents {alpha} = 0.3 and {beta} = 0.8, a long stay of particles in traps, and an infinite velocity of their jumps) is close to one. This is because the parameters of the Lagutin-Tyumentsev model have been chosen improperly.

17. International Conference on Multiscale Methods and Partial Differential Equations.

SciTech Connect

Thomas Hou

2006-12-12

The International Conference on Multiscale Methods and Partial Differential Equations (ICMMPDE for short) was held at IPAM, UCLA on August 26-27, 2005. The conference brought together researchers, students and practitioners with interest in the theoretical, computational and practical aspects of multiscale problems and related partial differential equations. The conference provided a forum to exchange and stimulate new ideas from different disciplines, and to formulate new challenging multiscale problems that will have impact in applications.

18. Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations

NASA Technical Reports Server (NTRS)

Geddes, K. O.

1977-01-01

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

19. Stability analysis of linear fractional differential system with distributed delays

Veselinova, Magdalena; Kiskinov, Hristo; Zahariev, Andrey

2015-11-01

In the present work we study the Cauchy problem for linear incommensurate fractional differential system with distributed delays. For the autonomous case with distributed delays with derivatives in Riemann-Liouville or Caputo sense, we establish sufficient conditions under which the zero solution is globally asymptotic stable. The established conditions coincide with the conditions which guaranty the same result in the particular case of system with constant delays and for the case of system without delays in the commensurate case too.

20. Effect of Differential Item Functioning on Test Equating

ERIC Educational Resources Information Center

Kabasakal, Kübra Atalay; Kelecioglu, Hülya

2015-01-01

This study examines the effect of differential item functioning (DIF) items on test equating through multilevel item response models (MIRMs) and traditional IRMs. The performances of three different equating models were investigated under 24 different simulation conditions, and the variables whose effects were examined included sample size, test…

1. Student Difficulties with Units in Differential Equations in Modelling Contexts

ERIC Educational Resources Information Center

Rowland, David R.

2006-01-01

First-year undergraduate engineering students' understanding of the units of factors and terms in first-order ordinary differential equations used in modelling contexts was investigated using diagnostic quiz questions. Few students appeared to realize that the units of each term in such equations must be the same, or if they did, nevertheless…

2. The Use of Kruskal-Newton Diagrams for Differential Equations

SciTech Connect

T. Fishaleck and R.B. White

2008-02-19

The method of Kruskal-Newton diagrams for the solution of differential equations with boundary layers is shown to provide rapid intuitive understanding of layer scaling and can result in the conceptual simplification of some problems. The method is illustrated using equations arising in the theory of pattern formation and in plasma physics.

3. Solution of partial differential equations by agent-based simulation

Szilagyi, Miklos N.

2014-01-01

The purpose of this short note is to demonstrate that partial differential equations can be quickly solved by agent-based simulation with high accuracy. There is no need for the solution of large systems of algebraic equations. This method is especially useful for quick determination of potential distributions and demonstration purposes in teaching electromagnetism.

4. Canonical coordinates for partial differential equations

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

Necessary and sufficient conditions are found under which operators of the form Sigma(m, j=1) X(2)sub j + X sub 0 can be made constant coefficient. In addition, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type.

5. The fractional coupled KdV equations: Exact solutions and white noise functional approach

Hossam, A. Ghany; S. Okb El Bab, A.; M. Zabel, A.; Abd-Allah, Hyder

2013-08-01

Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the modified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.

6. Similarity solution to fractional nonlinear space-time diffusion-wave equation

Costa, F. Silva; Marão, J. A. P. F.; Soares, J. C. Alves; de Oliveira, E. Capelas

2015-03-01

In this article, the so-called fractional nonlinear space-time wave-diffusion equation is presented and discussed. This equation is solved by the similarity method using fractional derivatives in the Caputo, Riesz-Feller, and Riesz senses. Some particular cases are presented and the corresponding solutions are shown by means of 2-D and 3-D plots.

7. Time-domain comparisons of power law attenuation in causal and noncausal time-fractional wave equations.

PubMed

Zhao, Xiaofeng; McGough, Robert J

2016-05-01

The attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different causal and noncausal fractional partial differential equations. To demonstrate some of the similarities and differences that are observed in three related time-fractional partial differential equations, time-domain Green's functions are calculated numerically for the power law wave equation, the Szabo wave equation, and for the Caputo wave equation. These Green's functions are evaluated for water with a power law exponent of y = 2, breast with a power law exponent of y = 1.5, and liver with a power law exponent of y = 1.139. Simulation results show that the noncausal features of the numerically calculated time-domain response are only evident very close to the source and that these causal and noncausal time-domain Green's functions converge to the same result away from the source. When noncausal time-domain Green's functions are convolved with a short pulse, no evidence of noncausal behavior remains in the time-domain, which suggests that these causal and noncausal time-fractional models are equally effective for these numerical calculations.

8. Residual power series method for fractional Burger types equations

Kumar, Amit; Kumar, Sunil

2016-12-01

We present an analytic algorithm to solve the generalized Berger-Fisher (B-F) equation, B-F equation, generalized Fisher equation and Fisher equation by using residual power series method (RPSM), which is based on the generalized Taylor's series formula together with the residual error function. In all the cases obtained results are verified through the different graphical representation. Comparison of the results obtained by the present method with exact solution reveals that the accuracy and fast convergence of the proposed method.

9. Optimal moving grids for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Wathen, A. J.

1992-01-01

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm are reported.

10. Similarity analysis of differential equations by Lie group.

NASA Technical Reports Server (NTRS)

Na, T. Y.; Hansen, A. G.

1971-01-01

Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations. A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations.

11. On solutions of polynomial growth of ordinary differential equations

van den Berg, I. P.

We present a theorem on the existence of solutions of polynomial growth of ordinary differential equations of type E: {dY}/{dX} = F(X, Y) , where F is of class C1. We show that the asymptotic behaviour of these solutions and the variation of neighbouring solutions are obtained by solving an asymptotic functional equation related to E, and that this method has practical value. The theorem is standard; its nonstandard proof uses macroscope and microscope techniques. The result is an extension of results by F. and M. Diener and G. Reeb on solutions of polynomial growth of rational differential equations.

12. In silico ordinary differential equation/partial differential equation hemodialysis model estimates methadone removal during dialysis

PubMed Central

Linares, Oscar A; Schiesser, William E; Fudin, Jeffrey; Pham, Thien C; Bettinger, Jeffrey J; Mathew, Roy O; Daly, Annemarie L

2015-01-01

Background There is a need to have a model to study methadone’s losses during hemodialysis to provide informed methadone dose recommendations for the practitioner. Aim To build a one-dimensional (1-D), hollow-fiber geometry, ordinary differential equation (ODE) and partial differential equation (PDE) countercurrent hemodialyzer model (ODE/PDE model). Methodology We conducted a cross-sectional study in silico that evaluated eleven hemodialysis patients. Patients received a ceiling dose of methadone hydrochloride 30 mg/day. Outcome measures included: the total amount of methadone removed during dialysis; methadone’s overall intradialytic mass transfer rate coefficient, km; and, methadone’s removal rate, jME. Each metric was measured at dialysate flow rates of 250 mL/min and 800 mL/min. Results The ODE/PDE model revealed a significant increase in the change of methadone’s mass transfer with increased dialysate flow rate, %Δkm=18.56, P=0.02, N=11. The total amount of methadone mass transferred across the dialyzer membrane with high dialysate flow rate significantly increased (0.042±0.016 versus 0.052±0.019 mg/kg, P=0.02, N=11). This was accompanied by a small significant increase in methadone’s mass transfer rate (0.113±0.002 versus 0.014±0.002 mg/kg/h, P=0.02, N=11). The ODE/PDE model accurately predicted methadone’s removal during dialysis. The absolute value of the prediction errors for methadone’s extraction and throughput were less than 2%. Conclusion ODE/PDE modeling of methadone’s hemodialysis is a new approach to study methadone’s removal, in particular, and opioid removal, in general, in patients with end-stage renal disease on hemodialysis. ODE/PDE modeling accurately quantified the fundamental phenomena of methadone’s mass transfer during hemodialysis. This methodology may lead to development of optimally designed intradialytic opioid treatment protocols, and allow dynamic monitoring of outflow plasma opioid concentrations for model

13. Lie symmetry analysis and soliton solutions of time-fractional K ( m, n) equation

Wang, G. W.; Hashemi, M. S.

2017-01-01

In this note, method of Lie symmetries is applied to investigate symmetry properties of time-fractional K( m, n) equation with the Riemann-Liouville derivatives. Reduction of time-fractional K( m, n) equation is done by virtue of the Erdélyi-Kober fractional derivative which depends on a parameter α. Then soliton solutions are extracted by means of a transformation.

14. NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION

PubMed Central

Liu, F.; Meerschaert, M.M.; McGough, R.J.; Zhuang, P.; Liu, Q.

2013-01-01

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian. PMID:23772179

15. Solving constant-coefficient differential equations with dielectric metamaterials

Zhang, Weixuan; Qu, Che; Zhang, Xiangdong

2016-07-01

Recently, the concept of metamaterial analog computing has been proposed (Silva et al 2014 Science 343 160-3). Some mathematical operations such as spatial differentiation, integration, and convolution, have been performed by using designed metamaterial blocks. Motivated by this work, we propose a practical approach based on dielectric metamaterial to solve differential equations. The ordinary differential equation can be solved accurately by the correctly designed metamaterial system. The numerical simulations using well-established numerical routines have been performed to successfully verify all theoretical analyses.

16. Generating functionals and Lagrangian partial differential equations

SciTech Connect

Vankerschaver, Joris; Liao, Cuicui; Leok, Melvin

2013-08-15

The main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values. We review the concept of Type-I/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi's solution to the Hamilton–Jacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the so-called multisymplectic form formula. As an example of the latter, we show that Lorentz's reciprocity principle in electromagnetism is a particular instance of the multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges.

17. Spatial dynamics for lattice differential equations with a shifting habitat

Hu, Changbing; Li, Bingtuan

2015-09-01

We study a lattice differential equation model that describes the growth and spread of a species in a shifting habitat. We show that the long term behavior of solutions depends on the speed of the shifting habitat edge c and a number c* (∞) that is determined by the maximum linearized growth rate and the diffusion coefficient. We demonstrate that if c >c* (∞) then the species will become extinct in the habitat, and that if c equations involving modified Bessel functions, for which new asymptotic estimates are provided. To the best of our knowledge, this is the first time that the classical Bessel functions are used to describe the solutions of lattice differential equations, and this approach possesses its own interest in further studying lattice differential equations.

18. A Difference Differential Equation of Euler-Cauchy Type

Bradley, David M.; Diamond, Harold G.

1997-08-01

We study a class of advanced argument linear difference differential equations analogous to Euler-Cauchy ordinary differential equations. Solutions of two equations of this type have arisen as adjoint functions in sieve theory, and they are also of use in control theory. Here we study the problem in a general setting. Subject to mild assumptions, each of our equations is shown to have a unique solution which is analytic in the right half-plane. In some cases the solution is a polynomial, and in others it has an asymptotic expansion. Finally, the solution is shown to have a representation as an exponential of a Hellinger type integro-differential operator acting on a monomial.

19. 1/f noise from nonlinear stochastic differential equations

Ruseckas, J.; Kaulakys, B.

2010-03-01

We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of 1/fβ noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating 1/fβ noise, and provides further insights into the origin of 1/fβ noise.

20. The two modes extension to the Berk-Breizman equation: Delayed differential equations and asymptotic solutions

SciTech Connect

Marczynski, Slawomir

2011-09-15

The integro-differential Berk-Breizman (BB) equation, describing the evolution of particle-driven wave mode is transformed into a simple delayed differential equation form {nu}{partial_derivative}a({tau})/{partial_derivative}{tau}=a({tau}) -a{sup 2}({tau}- 1) a({tau}- 2). This transformation is also applied to the two modes extension of the BB theory. The obtained solutions are presented together with the derived asymptotic analytical solutions and the numerical results.

1. Dedalus: Flexible framework for spectrally solving differential equations

Burns, Keaton J.; Vasil, Geoffrey M.; Oishi, Jeffrey S.; Lecoanet, Daniel; Brown, Benjamin

2016-03-01

Dedalus solves differential equations using spectral methods. It implements flexible algorithms to solve initial-value, boundary-value, and eigenvalue problems with broad ranges of custom equations and spectral domains. Its primary features include symbolic equation entry, multidimensional parallelization, implicit-explicit timestepping, and flexible analysis with HDF5. The code is written primarily in Python and features an easy-to-use interface. The numerical algorithm produces highly sparse systems for many equations which are efficiently solved using compiled libraries and MPI.

2. Modified fractional variational iteration method for solving the generalized time-space fractional Schrödinger equation.

PubMed

Hong, Baojian; Lu, Dianchen

2014-01-01

Based on He's variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schrödinger equations.

3. Construction and accuracy of partial differential equation approximations to the chemical master equation.

PubMed

Grima, Ramon

2011-11-01

The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.

4. Governing equations of transient soil water flow and soil water flux in multi-dimensional fractional anisotropic media and fractional time

Kavvas, M. Levent; Ercan, Ali; Polsinelli, James

2017-03-01

In this study dimensionally consistent governing equations of continuity and motion for transient soil water flow and soil water flux in fractional time and in fractional multiple space dimensions in anisotropic media are developed. Due to the anisotropy in the hydraulic conductivities of natural soils, the soil medium within which the soil water flow occurs is essentially anisotropic. Accordingly, in this study the fractional dimensions in two horizontal and one vertical directions are considered to be different, resulting in multi-fractional multi-dimensional soil space within which the flow takes place. Toward the development of the fractional governing equations, first a dimensionally consistent continuity equation for soil water flow in multi-dimensional fractional soil space and fractional time is developed. It is shown that the fractional soil water flow continuity equation approaches the conventional integer form of the continuity equation as the fractional derivative powers approach integer values. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-dimensional fractional soil space and fractional time, a dimensionally consistent equation is also developed. Again, it is shown that this fractional water flux equation approaches the conventional Darcy equation as the fractional derivative powers approach integer values. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-dimensional fractional soil space and fractional time is obtained. It is shown that this equation approaches the conventional Richards equation as the fractional derivative powers approach integer values. Then by the introduction of the Brooks-Corey constitutive relationships for soil water into the fractional transient soil water flow equation, an explicit form of the equation is obtained in multi-dimensional fractional soil space and fractional time. The

5. Kramers' escape problem for fractional Klein-Kramers equation with tempered α-stable waiting times.

PubMed

Gajda, Janusz; Magdziarz, Marcin

2011-08-01

In this paper we extend the subdiffusive Klein-Kramers model, in which the waiting times are modeled by the α-stable laws, to the case of waiting times belonging to the class of tempered α-stable distributions. We introduce a generalized version of the Klein-Kramers equation, in which the fractional Riemman-Liouville derivative is replaced with a more general integro-differential operator. This allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled. Taking advantage of the corresponding Langevin equation, we study some properties of the tempered dynamics, in particular, we approximate solutions of the tempered Klein-Kramers equation via Monte Carlo methods. Also, we study the distribution of the escape time from the potential well and compare it to the classical results in the Kramers escape theory. Finally, we derive the analytical formula for the first-passage-time distribution for the case of free particles. We show that the well-known Sparre Andersen scaling holds also for the tempered subdiffusion.

6. Fractional Trajectories: Decorrelation Versus Friction

DTIC Science & Technology

2013-07-27

from the integration of fractional differential equations in time. In Section 2 we provide a general demonstration of the new perspective on fractional ...section we demonstrate the equivalence between a fractional trajectory that is the solution of a Caputo fractional differential equation , and the... fractional differential equation dα dtα V(t) = OV(t), (1) where 0 < α < 1 and O is an operator, either linear or nonlinear, acting on the vector V(t

7. Scattering problems in the fractional quantum mechanics governed by the 2D space-fractional Schrödinger equation

SciTech Connect

Dong, Jianping

2014-03-15

The 2D space-fractional Schrödinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.

8. Mackey-Glass equation driven by fractional Brownian motion

Nguyen, Dung Tien

2012-11-01

In this paper we introduce a fractional stochastic version of the Mackey-Glass model which is a potential candidate to model objects in biology and finance. By a semi-martingale approximate approach we find an semi-analytical expression for the solution.

9. A perturbative solution to metadynamics ordinary differential equation.

PubMed

Tiwary, Pratyush; Dama, James F; Parrinello, Michele

2015-12-21

Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.

10. Entropy and convexity for nonlinear partial differential equations

PubMed Central

Ball, John M.; Chen, Gui-Qiang G.

2013-01-01

Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue. PMID:24249768

11. Numerical diagnostics of solution blowup in differential equations

Belov, A. A.

2017-01-01

New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.

12. Entropy and convexity for nonlinear partial differential equations.

PubMed

Ball, John M; Chen, Gui-Qiang G

2013-12-28

Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.

13. Reproducibility of Differential Proteomic Technologies in CPTAC Fractionated Xenografts

PubMed Central

2015-01-01

The NCI Clinical Proteomic Tumor Analysis Consortium (CPTAC) employed a pair of reference xenograft proteomes for initial platform validation and ongoing quality control of its data collection for The Cancer Genome Atlas (TCGA) tumors. These two xenografts, representing basal and luminal-B human breast cancer, were fractionated and analyzed on six mass spectrometers in a total of 46 replicates divided between iTRAQ and label-free technologies, spanning a total of 1095 LC–MS/MS experiments. These data represent a unique opportunity to evaluate the stability of proteomic differentiation by mass spectrometry over many months of time for individual instruments or across instruments running dissimilar workflows. We evaluated iTRAQ reporter ions, label-free spectral counts, and label-free extracted ion chromatograms as strategies for data interpretation (source code is available from http://homepages.uc.edu/~wang2x7/Research.htm). From these assessments, we found that differential genes from a single replicate were confirmed by other replicates on the same instrument from 61 to 93% of the time. When comparing across different instruments and quantitative technologies, using multiple replicates, differential genes were reproduced by other data sets from 67 to 99% of the time. Projecting gene differences to biological pathways and networks increased the degree of similarity. These overlaps send an encouraging message about the maturity of technologies for proteomic differentiation. PMID:26653538

14. Reproducibility of Differential Proteomic Technologies in CPTAC Fractionated Xenografts

SciTech Connect

Tabb, David L.; Wang, Xia; Carr, Steven A.; Clauser, Karl R.; Mertins, Philipp; Chambers, Matthew C.; Holman, Jerry D.; Wang, Jing; Zhang, Bing; Zimmerman, Lisa J.; Chen, Xian; Gunawardena, Harsha P.; Davies, Sherri R.; Ellis, Matthew J. C.; Li, Shunqiang; Townsend, R. Reid; Boja, Emily S.; Ketchum, Karen A.; Kinsinger, Christopher R.; Mesri, Mehdi; Rodriguez, Henry; Liu, Tao; Kim, Sangtae; McDermott, Jason E.; Payne, Samuel H.; Petyuk, Vladislav A.; Rodland, Karin D.; Smith, Richard D.; Yang, Feng; Chan, Daniel W.; Zhang, Bai; Zhang, Hui; Zhang, Zhen; Zhou, Jian-Ying; Liebler, Daniel C.

2016-03-04

The NCI Clinical Proteomic Tumor Analysis Consortium (CPTAC) employed a pair of reference xenograft proteomes for initial platform validation and ongoing quality control of its data collection for The Cancer Genome Atlas (TCGA) tumors. These two xenografts, representing basal and luminal-B human breast cancer, were fractionated and analyzed on six mass spectrometers in a total of 46 replicates divided between iTRAQ and label-free technologies, spanning a total of 1095 LC-MS/MS experiments. These data represent a unique opportunity to evaluate the stability of proteomic differentiation by mass spectrometry over many months of time for individual instruments or across instruments running dissimilar workflows. We evaluated iTRAQ reporter ions, label-free spectral counts, and label-free extracted ion chromatograms as strategies for data interpretation. From these assessments we found that differential genes from a single replicate were confirmed by other replicates on the same instrument from 61-93% of the time. When comparing across different instruments and quantitative technologies, differential genes were reproduced by other data sets from 67-99% of the time. Projecting gene differences to biological pathways and networks increased the similarities. These overlaps send an encouraging message about the maturity of technologies for proteomic differentiation.

15. Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation.

PubMed

Müller, Eike H; Scheichl, Rob; Shardlow, Tony

2015-04-08

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

16. Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

PubMed Central

Müller, Eike H.; Scheichl, Rob; Shardlow, Tony

2015-01-01

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy. PMID:27547075

17. Algebraic Riccati equations in zero-sum differential games

NASA Technical Reports Server (NTRS)

Johnson, T. L.; Chao, A.

1974-01-01

The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.

18. Numerical integration of asymptotic solutions of ordinary differential equations

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.

1989-01-01

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

19. Advanced methods for the solution of differential equations

NASA Technical Reports Server (NTRS)

Goldstein, M. E.; Braun, W. H.

1973-01-01

This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs.

20. Difference equations versus differential equations, a possible equivalence for the Rössler system?

Letellier, Christophe; Elaydi, Saber; Aguirre, Luis A.; Alaoui, Aziz

2004-08-01

When a set of nonlinear differential equations is investigated, most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete. In numerical investigations a fourth-order Runge-Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of difference equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps. In some cases, it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps. Nonstandard schemes as introduced by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1994] allow to obtain more robust difference equations. In this paper, using such nonstandard scheme, we propose some difference equations as discrete analogues of the Rössler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the Rössler system as long as the time step is less than the threshold value associated with the Nyquist criterion.

1. Master integrals for splitting functions from differential equations in QCD

Gituliar, Oleksandr

2016-02-01

A method for calculating phase-space master integrals for the decay process 1 → n masslesspartonsinQCDusingintegration-by-partsanddifferentialequationstechniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G) HPLs as a series in the dimensional regulator ɛ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.

2. GHM method for obtaining rationalsolutions of nonlinear differential equations.

PubMed

Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo

2015-01-01

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.

3. Samples of noncommutative products in certain differential equations

Légaré, M.

2010-11-01

A set of associative noncommutative products is considered in different differential equations of the ordinary and partial types. A method of separation of variables is considered for a large set of those systems. The products involved include for example some * products and some products based on Nijenhuis tensors, which are embedded in the differential equations of the Laplace/Poisson, Lax and Schrödinger styles. A comment on the *-products of Reshetikhin-Jambor-Sykora type is also given in relation to *-products of Vey type.

4. Analytic solution for Telegraph equation by differential transform method

Biazar, J.; Eslami, M.

2010-06-01

In this article differential transform method (DTM) is considered to solve Telegraph equation. This method is a powerful tool for solving large amount of problems (Zhou (1986) [1], Chen and Ho (1999) [2], Jang et al. (2001) [3], Kangalgil and Ayaz (2009) [4], Ravi Kanth and Aruna (2009) [5], Arikoglu and Ozkol (2007) [6]). Using differential transform method, it is possible to find the exact solution or a closed approximate solution of an equation. To illustrate the ability and reliability of the method some examples are provided. The results reveal that the method is very effective and simple.

5. Strongly differentiable solutions of the discrete coagulation-fragmentation equation

McBride, A. C.; Smith, A. L.; Lamb, W.

2010-08-01

We examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters. In contrast to previous investigations, our analysis does not involve finite-dimensional truncations of the system. Instead, we treat the problem as an infinite-dimensional differential equation, posed in an appropriate Banach space, and apply perturbation results from the theory of strongly continuous semigroups of operators. The existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates.

6. On Existence and Uniqueness Results for Nonsmooth Implicit Differential Equations

You, Xiong; Wu, Xinyuan; Chen, Zhaoxia; Yang, Hongli; Fang, Yonglei

2008-09-01

The classical implicit function theorem gives conditions that the function is Fréchet differentiable and the derivative is surjective. In this short article they are generalized to conditions of Lipschitz and monotone type. The newly obtained implicit function theorems are used to derive two sets of sufficient conditions for the existence and uniqueness of solutions to the initial value problems of nonsmooth implicit differential equations.

7. Power series solutions of ordinary differential equations in MACSYMA

NASA Technical Reports Server (NTRS)

Lafferty, E. L.

1977-01-01

A program is described which extends the differential equation solving capability of MACSYMA to power series solutions and is available via the SHARE library. The program is directed toward those classes of equations with variable coefficients (in particular, those with singularities) and uses the method of Frobenius. Probably the most important distinction between this package and others currently available or being developed is that, wherever possible, this program will attempt to provide a complete solution to the equation rather than an approximation, i.e., a finite number of terms. This solution will take the form of a sum of infinite series.

8. Solutions of fractional reaction-diffusion equations in terms of the H-function

Haubold, H. J.; Mathai, A. M.; Saxena, R. K.

2007-12-01

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction-diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation containing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space-fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form.

9. Excitability in a stochastic differential equation model for calcium puffs.

PubMed

Rüdiger, S

2014-06-01

Calcium dynamics are essential to a multitude of cellular processes. For many cell types, localized discharges of calcium through small clusters of intracellular channels are building blocks for all spatially extended calcium signals. Because of the large noise amplitude, the validity of noise-approximating model equations for this system has been questioned. Here we revisit the master equations for local calcium release, examine the multiple scales of calcium concentrations in the cluster domain, and derive adapted stochastic differential equations. We show by comparison of discrete and continuous trajectories that the Langevin equations can be made consistent with the master equations even for very small channel numbers. In its deterministic limit, the model reveals that excitability, a dynamical phenomenon observed in many natural systems, is at the core of calcium puffs. The model also predicts a bifurcation from transient to sustained release which may link local and global calcium signals in cells.

10. Communications: The fractional Stokes-Einstein equation: application to water.

PubMed

Harris, Kenneth R

2010-06-21

Previously [K. R. Harris, J. Chem. Phys. 131, 054503 (2009)] it was shown that both real and model liquids fit the fractional form of the Stokes-Einstein relation [fractional Stokes-Einstein (FSE)] over extended ranges of temperature and density. For example, the self-diffusion coefficient and viscosity of the Lennard-Jones fluid fit the relation (D/T) = (1/eta)(t) with t = (0.921+/-0.003) and a range of molecular and ionic liquids for which high pressure data are available behave similarly, with t values between 0.79 and 1. At atmospheric pressure, normal and heavy water were also found to fit FSE from 238 to 363 K and from 242 to 328 K, respectively, but with distinct transitions in the supercooled region at about 258 and 265 K, respectively, from t = 0.94 (high temperature) to 0.67 (low temperature). Here the recent self-diffusion data of Yoshida et al. [J. Chem. Phys. 129, 214501 (2008)] for the saturation line are used to extend the high temperature fit to FSE to 623 K for both isotopomers. The FSE transition temperature in bulk water can be contrasted with much lower values reported in the literature for confined water.

11. Spectral Deferred Corrections for Parabolic Partial Differential Equations

DTIC Science & Technology

2015-06-08

linear differential equation ϕ′(t) = λϕ(t), t ≥ 0 ϕ(0) = 1, (3.31) where λ ∈ C, has exact solution ϕ(t) = eλt. (3.32) Traditionally, for a fixed time step...the second-order differentiation matrix with 16 subintervals and 16 points per subinterval. From Figure 5.2, this matrix approximates the exact ...We describe a new class of algorithms for the solution of parabolic partial differential equa- tions (PDEs). This class of schemes is based on three

12. Solving Second-Order Differential Equations with Variable Coefficients

ERIC Educational Resources Information Center

Wilmer, A., III; Costa, G. B.

2008-01-01

A method is developed in which an analytical solution is obtained for certain classes of second-order differential equations with variable coefficients. By the use of transformations and by repeated iterated integration, a desired solution is obtained. This alternative method represents a different way to acquire a solution from classic power…

13. Building Context with Tumor Growth Modeling Projects in Differential Equations

ERIC Educational Resources Information Center

Beier, Julie C.; Gevertz, Jana L.; Howard, Keith E.

2015-01-01

The use of modeling projects serves to integrate, reinforce, and extend student knowledge. Here we present two projects related to tumor growth appropriate for a first course in differential equations. They illustrate the use of problem-based learning to reinforce and extend course content via a writing or research experience. Here we discuss…

14. Integration of CAS in the Didactics of Differential Equations.

ERIC Educational Resources Information Center

Balderas Puga, Angel

In this paper are described some features of the intensive use of math software, primarily DERIVE, in the context of modeling in an introductory university course in differential equations. Different aspects are detailed: changes in the curriculum that included not only course contents, but also the sequence of introduction to various topics and…

15. On Polynomial Solutions of Linear Differential Equations with Polynomial Coefficients

ERIC Educational Resources Information Center

Si, Do Tan

1977-01-01

Demonstrates a method for solving linear differential equations with polynomial coefficients based on the fact that the operators z and D + d/dz are known to be Hermitian conjugates with respect to the Bargman and Louck-Galbraith scalar products. (MLH)

16. Phaser-Based Courseware for Ordinary Differential Equations.

ERIC Educational Resources Information Center

Zia, Lee l.

1991-01-01

Presented are classroom-tested examples of instructional materials (courseware) for ordinary differential equations using the software package PHASER. All of the examples include in-class demonstration techniques and commentaries for instructor use, student homework and laboratory exercises, and suggestions for in-class examination questions. (JJK)

17. Numerical Aspects of Solving Differential Equations: Laboratory Approach for Students.

ERIC Educational Resources Information Center

Witt, Ana

1997-01-01

Describes three labs designed to help students in a first course on ordinary differential equations with three of the most common numerical difficulties they might encounter when solving initial value problems with a numerical software package. The goal of these labs is to help students advance to independent work on common numerical anomalies.…

18. Do Students Really Understand What an Ordinary Differential Equation Is?

ERIC Educational Resources Information Center

Arslan, Selahattin

2010-01-01

Differential equations (DEs) are important in mathematics as well as in science and the social sciences. Thus, the study of DEs has been included in various courses in different departments in higher education. The importance of DEs has attracted the attention of many researchers who have generally focussed on the content and instruction of DEs.…

19. Parameter Estimates in Differential Equation Models for Population Growth

ERIC Educational Resources Information Center

Winkel, Brian J.

2011-01-01

We estimate the parameters present in several differential equation models of population growth, specifically logistic growth models and two-species competition models. We discuss student-evolved strategies and offer "Mathematica" code for a gradient search approach. We use historical (1930s) data from microbial studies of the Russian biologist,…

20. A Simple Derivation of Kepler's Laws without Solving Differential Equations

ERIC Educational Resources Information Center

Provost, J.-P.; Bracco, C.

2009-01-01

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple…

1. Climate Modeling in the Calculus and Differential Equations Classroom

ERIC Educational Resources Information Center

Kose, Emek; Kunze, Jennifer

2013-01-01

Students in college-level mathematics classes can build the differential equations of an energy balance model of the Earth's climate themselves, from a basic understanding of the background science. Here we use variable albedo and qualitative analysis to find stable and unstable equilibria of such a model, providing a problem or perhaps a…

2. A Second-Year Undergraduate Course in Applied Differential Equations.

ERIC Educational Resources Information Center

Fahidy, Thomas Z.

1991-01-01

Presents the framework for a chemical engineering course using ordinary differential equations to solve problems with the underlying strategy of concisely discussing the theory behind each solution technique without extensions to formal proofs. Includes typical class illustrations, student responses to this strategy, and reaction of the…

3. Neumann problems for second order ordinary differential equations across resonance

Yong, Li; Huaizhong, Wang

1995-05-01

This paper deals with the existence-uniqueness problem to Neumann problems for second order ordinary differential equations probably across resonance. By the optimal control theory method, some global optimality results about the unique solvability for such boundary value problems are established.

4. Solving Differential Matrix Riccati Equations by a piecewise-linearized method based on the conmutant equation

Ibáñez, Javier; Hernández, Vicente

2009-11-01

Differential Matrix Riccati Equations play a fundamental role in control theory, for example, in optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a piecewise-linearized method based on the conmutant equation to solve Differential Matrix Riccati Equations (DMREs) is described. This method is applied to develop two algorithms which solve these equations: one for time-varying DMREs and another for time-invariant DMREs, also MATLAB implementations of the above algorithms are developed. Since MATLAB does not have functions which solve DMREs, two algorithms based on a BDF method are also developed. All implemented algorithms have been compared, under equal conditions, at both precision and computational costs. Experimental results show the advantages of solving non-stiff DMREs and in particular stiff DMREs by the proposed algorithms.

5. Improved stochastic approximation methods for discretized parabolic partial differential equations

Guiaş, Flavius

2016-12-01

We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).

6. Differential equation based method for accurate approximations in optimization

NASA Technical Reports Server (NTRS)

Pritchard, Jocelyn I.; Adelman, Howard M.

1990-01-01

A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.

7. [Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (1)].

PubMed

Murase, Kenya

2014-01-01

Utilization of differential equations and methods for solving them in medical physics are presented. First, the basic concept and the kinds of differential equations were overviewed. Second, separable differential equations and well-known first-order and second-order differential equations were introduced, and the methods for solving them were described together with several examples. In the next issue, the symbolic and series expansion methods for solving differential equations will be mainly introduced.

8. Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains

Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F.

2017-02-01

In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.

9. A differential equation for the Generalized Born radii.

PubMed

Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro

2013-06-28

The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear second-order partial differential equation for the Generalized Born radius, which may be solved using local iterative algorithms. The equation is derived under the assumption that the usual GB approximation to the reaction field obeys Laplace's equation. The equation admits as particular solutions the correct GB radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall agreement with other reference GB models and "perfect" Poisson-Boltzmann based values.

10. Learning partial differential equations via data discovery and sparse optimization

Schaeffer, Hayden

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

11. Learning partial differential equations via data discovery and sparse optimization.

PubMed

Schaeffer, Hayden

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

12. Packing fraction of crystalline structures of binary hard spheres: a general equation and application to amorphization.

PubMed

Brouwers, H J H

2008-07-01

In a previous paper analytical equations were derived for the packing fraction of crystalline structures consisting of bimodal randomly placed hard spheres [H. J. H. Brouwers, Phys. Rev. E 76, 041304 (2007)]. The bimodal packing fraction was derived for the three crystalline cubic systems: viz., face-centered cubic, body-centered cubic, and simple cubic. These three equations appeared also to be applicable to all 14 Bravais lattices. Here it is demonstrated, accounting for the number of distorted bonds in the building blocks and using graph theory, that one general packing equation can be derived, valid again for all lattices. This expression is validated and applied to the process of amorphization.

13. A semi-discrete finite element method for a class of time-fractional diffusion equations.

PubMed

Sun, HongGuang; Chen, Wen; Sze, K Y

2013-05-13

As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly used in the related mathematical descriptions. These models usually involve long-time-range computation, which is a critical obstacle for their application; improvement of computational efficiency is of great significance. In this paper, a semi-discrete method is presented for solving a class of time-fractional diffusion equations that overcome the critical long-time-range computation problem. In the procedure, the spatial domain is discretized by the finite element method, which reduces the fractional diffusion equations to approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time-range computation can effectively be minimized. To illustrate its efficiency and simplicity, four examples are presented. In addition, the method is used to solve the time-fractional advection-diffusion equation characterizing the bromide transport process in a fractured granite aquifer. The prediction closely agrees with the experimental data, and the heavy-tail decay of the anomalous transport process is well represented.

14. Solution of Some Types of Differential Equations: Operational Calculus and Inverse Differential Operators

PubMed Central

Zhukovsky, K.

2014-01-01

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated. PMID:24892051

15. On connections between stochastic differential inclusions and set-valued stochastic differential equations driven by semimartingales

Michta, Mariusz

2017-02-01

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations with respect to semimartingale integrators. We present new connections between their solutions. In particular, we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set-valued stochastic equations. We also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. The results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases.

16. Linking multiple relaxation, power-law attenuation, and fractional wave equations.

PubMed

Näsholm, Sven Peter; Holm, Sverre

2011-11-01

The acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media, e.g., biological tissue, polymers, rocks, and rubber. Recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power-law attenuation. On the other hand, a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by Nachman et al. [J. Acoust. Soc. Am. 88, 1584-1595 (1990)]. Through assumption of a continuum of relaxation mechanisms, each with an effective compressibility described by a distribution related to the Mittag-Leffler function, this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation. This work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling.

17. The reservoir model: a differential equation model of psychological regulation.

PubMed

Deboeck, Pascal R; Bergeman, C S

2013-06-01

Differential equation models can be used to describe the relationships between the current state of a system of constructs (e.g., stress) and how those constructs are changing (e.g., based on variable-like experiences). The following article describes a differential equation model based on the concept of a reservoir. With a physical reservoir, such as one for water, the level of the liquid in the reservoir at any time depends on the contributions to the reservoir (inputs) and the amount of liquid removed from the reservoir (outputs). This reservoir model might be useful for constructs such as stress, where events might "add up" over time (e.g., life stressors, inputs), but individuals simultaneously take action to "blow off steam" (e.g., engage coping resources, outputs). The reservoir model can provide descriptive statistics of the inputs that contribute to the "height" (level) of a construct and a parameter that describes a person's ability to dissipate the construct. After discussing the model, we describe a method of fitting the model as a structural equation model using latent differential equation modeling and latent distribution modeling. A simulation study is presented to examine recovery of the input distribution and output parameter. The model is then applied to the daily self-reports of negative affect and stress from a sample of older adults from the Notre Dame Longitudinal Study on Aging.

18. Integro-differential equation of non-local wave interaction

SciTech Connect

Engibaryan, N B; Khachatryan, Aghavard Kh

2007-06-30

The integro-differential equation d{sup 2}f/dx{sup 2} + Af = {integral}{sub 0}{sup {infinity}}K(x-t)f(t)dt + g(x) with kernel K(x)={lambda}{integral}{sub a}{sup {infinity}}e{sup -|x|p}G(p)dp, a{>=}0, is considered, in which A>0, {lambda} element of 9-{infinity},{infinity}), G(p){>=}0, 2{integral}{sub a}{sup {infinity}}1/p g(p)dp=1. These equations arise, in particular, in the theory of non-local wave interaction. A factorization method of their analysis and solution is developed. Bibliography: 9 titles.

19. Numerical Solution of a Nonlinear Integro-Differential Equation

Buša, Ján; Hnatič, Michal; Honkonen, Juha; Lučivjanský, Tomáš

2016-02-01

A discretization algorithm for the numerical solution of a nonlinear integrodifferential equation modeling the temporal variation of the mean number density a(t) in the single-species annihilation reaction A + A → 0 is discussed. The proposed solution for the two-dimensional case (where the integral entering the equation is divergent) uses regularization and then finite differences for the approximation of the differential operator together with a piecewise linear approximation of a(t) under the integral. The presented numerical results point to basic features of the behavior of the number density function a(t) and suggest further improvement of the proposed algorithm.

20. Connecting orbits for nonlinear differential equations at resonance

Kokocki, Piotr

We study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the Conley index of associated semiflow with respect to appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of the well-known in literature Landesman-Lazer and strong resonance conditions. Obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity.

1. Computer transformation of partial differential equations into any coordinate system

NASA Technical Reports Server (NTRS)

Sullivan, R. D.

1977-01-01

The use of tensors to provide a compact way of writing partial differential equations in a form valid in all coordinate systems is discussed. In order to find solutions to the equations with their boundary conditions they must be expressed in terms of the coordinate system under consideration. The process of arriving at these expressions from the tensor formulation was automated by a software system, TENSR. An allied system that analyzes the resulting expressions term by term and drops those that are negligible is also described.

2. Solution of partial differential equations on vector and parallel computers

NASA Technical Reports Server (NTRS)

Ortega, J. M.; Voigt, R. G.

1985-01-01

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.

3. Constructive Development of the Solutions of Linear Equations in Introductory Ordinary Differential Equations

ERIC Educational Resources Information Center

Mallet, D. G.; McCue, S. W.

2009-01-01

The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to…

4. Richards' Equation and its Constitutive Relations as a System of Differential-Algebraic Equations

Murray, S. K.; Mead, J. L.

2007-12-01

Richards' Equation is commonly used to understand how water flows in unsaturated soils. We present a new formulation of Richards' Equation which will allow us to incorporate model and observation errors. In addition, we can address spatial and temporal inconsistencies existing between the model and observations. There are two basic formulations for Richards' Equation: the pressure head form and the mixed form, the latter of which explicitly incorporates soil moisture content. The mixed form is typically solved using HYDRUS, a freely available program that uses finite elements with Picard iteration to handle the nonlinearities. However, recent results suggest considering Richards' Equation as a differential-algebraic equation (DAE), where the algebraic models for soil moisture content (van Genuchten's equation) is solved simultaneously with Richards' Equation (Kees, et. al., 2002). This formulation can give more accurate forward model solutions, however, we note that it also allows us to consider the uncertainties in the pressure head ψ and the soil moister content θ during the inversion process. We extend the DAE formulation to include the algebraic constraint for hydraulic conductivity K, so that its uncertainty can also be considered in an inversion. This poster focuses on the efficiency and accuracy of the forward numerical solution of this particular DAE formulation of Richards' Equation and how it compares to other forward solutions, such as HYDRUS.

5. On an exterior boundary value problem for the Laplace equation with boundary operator of fractional order

Turmetov, B. Kh.

2016-12-01

In the paper in a class of regular harmonic functions we study properties of some integro-differential operators that generalize the operators of fractional differentiation in Hadamard sense. These operators transfer regular harmonic functions to the same function, and are inverse to the regular harmonic functions. Boundary value problem with the boundary operator of fractional order is studied in the exterior of the unit sphere. The considered problem generalizes the well-known Neumann problem on boundary operators of fractional order. We prove a theorem on existence and uniqueness of solutions of the problem. Moreover, an integral representation of the problem solution is obtained.

6. Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition.

PubMed

2002-12-01

We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

7. Data-driven discovery of partial differential equations

Rudy, Samuel; Brunton, Steven; Proctor, Joshua; Kutz, J. Nathan

2016-11-01

Fluid dynamics is inherently governed by spatial-temporal interactions which can be characterized by partial differential equations (PDEs). Emerging sensor and measurement technologies allowing for rich, time-series data collection motivate new data-driven methods for discovering governing equations. We present a novel computational technique for discovering governing PDEs from time series measurements. A library of candidate terms for the PDE including nonlinearities and partial derivatives is computed and sparse regression is then used to identify a subset which accurately reflects the measured dynamics. Measurements may be taken either in a Eulerian framework to discover field equations or in a Lagrangian framework to study a single stochastic trajectory. The method is shown to be robust, efficient, and to work on a variety of canonical equations. Data collected from a simulation of a flow field around a cylinder is used to accurately identify the Navier-Stokes vorticity equation and the Reynolds number to within 1%. A single trace of Brownian motion is also used to identify the diffusion equation. Our method provides a novel approach towards data enabled science where spatial-temporal information bolsters classical machine learning techniques to identify physical laws.

8. Analytic solution of differential equation for gyroscope's motions

Tyurekhodjaev, Abibulla N.; Mamatova, Gulnar U.

2016-08-01

Problems of motion of a rigid body with a fixed point are one of the urgent problems in classical mechanics. A feature of this problem is that, despite the important results achieved by outstanding mathematicians in the last two centuries, there is still no complete solution. This paper obtains an analytical solution of the problem of motion of an axisymmetric rigid body with variable inertia moments in resistant environment described by the system of nonlinear differential equations of L. Euler, involving the partial discretization method for nonlinear differential equations, which was built by A. N. Tyurekhodjaev based on the theory of generalized functions. To such problems belong gyroscopic instruments, in particular, and especially gyroscopes.

9. An ordinary differential equation based solution path algorithm.

PubMed

Wu, Yichao

2011-01-01

Efron, Hastie, Johnstone and Tibshirani (2004) proposed Least Angle Regression (LAR), a solution path algorithm for the least squares regression. They pointed out that a slight modification of the LAR gives the LASSO (Tibshirani, 1996) solution path. However it is largely unknown how to extend this solution path algorithm to models beyond the least squares regression. In this work, we propose an extension of the LAR for generalized linear models and the quasi-likelihood model by showing that the corresponding solution path is piecewise given by solutions of ordinary differential equation systems. Our contribution is twofold. First, we provide a theoretical understanding on how the corresponding solution path propagates. Second, we propose an ordinary differential equation based algorithm to obtain the whole solution path.

10. Numerical solution of three-dimensional magnetic differential equations

SciTech Connect

Reiman, A.H.; Greenside, H.S.

1987-02-01

A computer code is described that solves differential equations of the form B . del f = h for a single-valued solution f, given a toroidal three-dimensional divergence-free field B and a single-valued function h. The code uses a new algorithm that Fourier decomposes a given function in a set of flux coordinates in which the field lines are straight. The algorithm automatically adjusts the required integration lengths to compensate for proximity to low order rational surfaces. Applying this algorithm to the Cartesian coordinates defines a transformation to magnetic coordinates, in which the magnetic differential equation can be accurately solved. Our method is illustrated by calculating the Pfirsch-Schlueter currents for a stellarator.

11. Model Predictive Control for Nonlinear Parabolic Partial Differential Equations

Hashimoto, Tomoaki; Yoshioka, Yusuke; Ohtsuka, Toshiyuki

In this study, the optimal control problem of nonlinear parabolic partial differential equations (PDEs) is investigated. Optimal control of nonlinear PDEs is an open problem with applications that include fluid, thermal, biological, and chemically-reacting systems. Model predictive control with a fast numerical solution method has been well established to solve the optimal control problem of nonlinear systems described by ordinary differential equations. In this study, we develop a design method of the model predictive control for nonlinear systems described by parabolic PDEs. Our approach is a direct infinite dimensional extension of the model predictive control method for finite-dimensional systems. The objective of this paper is to develop an efficient algorithm for numerically solving the model predictive control problem of nonlinear parabolic PDEs. The effectiveness of the proposed method is verified by numerical simulations.

12. Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations

Junaid, Ali Khan; Muhammad, Asif Zahoor Raja; Ijaz Mansoor, Qureshi

2011-02-01

We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.

13. Multigrid methods for differential equations with highly oscillatory coefficients

NASA Technical Reports Server (NTRS)

Engquist, Bjorn; Luo, Erding

1993-01-01

New coarse grid multigrid operators for problems with highly oscillatory coefficients are developed. These types of operators are necessary when the characters of the differential equations on coarser grids or longer wavelengths are different from that on the fine grid. Elliptic problems for composite materials and different classes of hyperbolic problems are practical examples. The new coarse grid operators can be constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. Convergence analysis based on the homogenization theory is given for elliptic problems with periodic coefficients and some hyperbolic problems. These are classes of equations for which there exists a fairly complete theory for the interaction between shorter and longer wavelengths in the problems. Numerical examples are presented.

14. A convex penalty for switching control of partial differential equations

DOE PAGES

Clason, Christian; Rund, Armin; Kunisch, Karl; ...

2016-01-19

A convex penalty for promoting switching controls for partial differential equations is introduced; such controls consist of an arbitrary number of components of which at most one should be simultaneously active. Using a Moreau–Yosida approximation, a family of approximating problems is obtained that is amenable to solution by a semismooth Newton method. In conclusion, the efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples.

15. Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations

DTIC Science & Technology

2014-07-01

non- linear hybrid systems by linear algebraic methods. In Radhia Cousot and Matthieu Martel, editors, SAS, volume 6337 of LNCS, pages 373–389. Springer...Tarski. A decision method for elementary algebra and geometry. Bulletin of the American Mathematical Society, 59, 1951. [36] Wolfgang Walter. Ordinary...Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 July 2014

16. Advanced-Retarded Differential Equations in Quantum Photonic Systems

Alvarez-Rodriguez, Unai; Perez-Leija, Armando; Egusquiza, Iñigo L.; Gräfe, Markus; Sanz, Mikel; Lamata, Lucas; Szameit, Alexander; Solano, Enrique

2017-02-01

We propose the realization of photonic circuits whose dynamics is governed by advanced-retarded differential equations. Beyond their mathematical interest, these photonic configurations enable the implementation of quantum feedback and feedforward without requiring any intermediate measurement. We show how this protocol can be applied to implement interesting delay effects in the quantum regime, as well as in the classical limit. Our results elucidate the potential of the protocol as a promising route towards integrated quantum control systems on a chip.

17. Higher order matrix differential equations with singular coefficient matrices

SciTech Connect

Fragkoulis, V. C.; Kougioumtzoglou, I. A.; Pantelous, A. A.; Pirrotta, A.

2015-03-10

In this article, the class of higher order linear matrix differential equations with constant coefficient matrices and stochastic process terms is studied. The coefficient of the highest order is considered to be singular; thus, rendering the response determination of such systems in a straightforward manner a difficult task. In this regard, the notion of the generalized inverse of a singular matrix is used for determining response statistics. Further, an application relevant to engineering dynamics problems is included.

18. Neural network differential equation and plasma equilibrium solver

van Milligen, B. Ph.; Tribaldos, V.; Jiménez, J. A.

1995-11-01

A new generally applicable method to solve differential equations, based on neural networks, is proposed. Straightforward to implement, finite differences and coordinate transformations are not used. The neural network provides a flexible and compact base for representing the solution, found through the global minimization of an error functional. As a proof of principle, a two-dimensional ideal magnetohydrodynamic plasma equilibrium is solved. Since no particular topology is assumed, the technique is especially promising for the three-dimensional plasma equilibrium problem.

19. On approximating hereditary dynamics by systems of ordinary differential equations

NASA Technical Reports Server (NTRS)

Cliff, E. M.; Burns, J. A.

1978-01-01

The paper deals with methods of obtaining approximate solutions to linear retarded functional differential equations (hereditary systems). The basic notion is to project the infinite dimensional space of initial functions for the hereditary system onto a finite dimensional subspace. Within this framework, two particular schemes are discussed. The first uses well-known piecewise constant approximations, while the second is a new method based on piecewise linear approximating functions. Numerical results are given.

20. Spectral methods for time dependent partial differential equations

NASA Technical Reports Server (NTRS)

Gottlieb, D.; Turkel, E.

1983-01-01

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

1. A differential delay equation arising from the sieve of Eratosthenes

NASA Technical Reports Server (NTRS)

Cheer, A. Y.; Goldston, D. A.

1990-01-01

Consideration is given to the differential delay equation introduced by Buchstab (1937) in connection with an asymptotic formula for the uncanceled terms in the sieve of Eratosthenes. Maier (1985) used this result to show there is unexpected irreqularity in the distribution of primes in short intervals. The function omega(u) is studied in this paper using numerical and analytical techniques. The results are applied to give some numerical constants in Maier's theorem.

2. Advanced-Retarded Differential Equations in Quantum Photonic Systems

PubMed Central

Alvarez-Rodriguez, Unai; Perez-Leija, Armando; Egusquiza, Iñigo L.; Gräfe, Markus; Sanz, Mikel; Lamata, Lucas; Szameit, Alexander; Solano, Enrique

2017-01-01

We propose the realization of photonic circuits whose dynamics is governed by advanced-retarded differential equations. Beyond their mathematical interest, these photonic configurations enable the implementation of quantum feedback and feedforward without requiring any intermediate measurement. We show how this protocol can be applied to implement interesting delay effects in the quantum regime, as well as in the classical limit. Our results elucidate the potential of the protocol as a promising route towards integrated quantum control systems on a chip. PMID:28230090

3. Fast Numerical Methods for Stochastic Partial Differential Equations

DTIC Science & Technology

2016-04-15

uncertainty quantification. In the last decade much progress has been made in the construction of numerical algorithms to efficiently solve SPDES with...applicable SPDES with efficient numerical methods. This project is intended to address the numerical analysis as well as algorithm aspects of SPDES. Three...differential equations. Our work contains algorithm constructions, rigorous error analysis, and extensive numerical experiments to demonstrate our algorithm

4. On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

Glatt-Holtz, Nathan; Mattingly, Jonathan C.; Richards, Geordie

2017-02-01

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.

5. On qualitative properties in Volterra integro-differential equations

Tunç, Cemil

2017-01-01

In this research, we utilize a Lyapunov function and obtain sufficient conditions for the asymptotically stability and boundedness of solutions to the Volterra integro-differential equations of the form d/dt [ x (t)-∫0tD(t,s)x (s)ds ] =-A (t )x (t )+ ∫0tC(t,s) x (s )ds +e (t ,x ). The obtained results revise, correct and improve the results obtained in literature.

6. A Global Optimization Algorithm Using Stochastic Differential Equations.

DTIC Science & Technology

1985-02-01

Bari (Italy).2Istituto di Fisica , 2 UniversitA di Roma "Tor Vergata", Via Orazio Raimondo, 00173 (La Romanina) Roma (Italy). 3Istituto di Matematica ...accompanying Algorithm. lDipartininto di Matematica , Universita di Bari, 70125 Bar (Italy). Istituto di Fisica , 2a UniversitA di Roim ’"Tor Vergata", Via...Optimization, Stochastic Differential Equations Work Unit Number 5 (Optimization and Large Scale Systems) 6Dipartimento di Matematica , Universita di Bari, 70125

7. A Partial Differential Equation for the Rank One Convex Envelope

2017-02-01

A partial differential equation (PDE) for the rank one convex envelope is introduced. The existence and uniqueness of viscosity solutions to the PDE is established. Elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the PDE is proven. Computational results are presented and laminates are computed from the envelopes. Results include the Kohn-Strang example, the classical four gradient example, and an example with eight gradients which produces nontrivial laminates.

8. Computing spacetime curvature via differential-algebraic equations

SciTech Connect

Ashby, S.F.; Lee, S.L.; Petzold, L.R.; Saylor, P.E.; Seidel, E.

1996-01-01

The equations that govern the behavior of physical systems can often solved numerically using a method of lines approach and differential-algebraic equation (DAE) solvers. For example, such an approach can be used to solve the Einstein field equations of general relativity, and thereby simulate significant astrophysical events. In this paper, we describe some preliminary work in which two model problems in general relativity are formulated, spatially discretized, and then numerically solved as a DAE. In particular, we seek to reproduce the solution to the spherically symmetric Schwarzschild spacetime. This is an important testbed calculation in numerical relativity since the solution is the steady-state for the collision of two (or more) non-rotating black holes. Moreover, analytic late-time properties of the Schwarzschild spacetime are well known and can be used the accuracy of the simulation.

9. Oscillation properties of some functional fourth order hyperbolic differential equations

Petrova, Z.

2012-11-01

In this paper, we apply our recent results for fourth order functional ordinary differential equations and inequalities and obtain sufficient conditions for oscillation of all sufficiently smooth solutions of the following equation ∑ i+j = 2;4ai,j∂i+ju(x,y)/∂xi∂yj+ ∑ i = 1nbi(x,y)u(x-σi,y-τi)+c(x,y,u) = f(x,y), where x>0,y>0,ai,j∈R,σi≥0 and τi ≥ 0 are constants for all the indices. Also, we suppose that n∈N,bi(x,y)∈C(R+2;R+), ∀i = 1-n;c(x,y,u)∈C(R+2,R;R) and f(x,y)∈C(R+2;R). In particular, we establish sufficient conditions for the distribution of zeros this equation.

10. Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

SciTech Connect

2015-02-15

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.

11. Modelling biochemical reaction systems by stochastic differential equations with reflection.

PubMed

Niu, Yuanling; Burrage, Kevin; Chen, Luonan

2016-05-07

In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach.

12. New exact solutions of fractional Davey-Stewartson equation with power-law nonlinearity and new integrable Davey-Stewartson-type equation

Saha Ray, S.

2016-09-01

In this article, the Jacobi elliptic function method viz. the mixed dn-sn method has been presented for finding the travelling wave solutions of the Davey-Stewartson equations. As a result, some solitary wave solutions and doubly periodic solutions are obtained in terms of Jacobi elliptic functions. Moreover, solitary wave solutions are obtained as simple limits of doubly periodic functions. These solutions can be useful to explain some physical phenomena, viz. evolution of a three-dimensional wave packet on water of finite depth. The proposed Jacobi elliptic function method is efficient, powerful and can be used in order to establish newer exact solutions for other kinds of nonlinear fractional partial differential equations arising in mathematical physics.

13. [Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (2)].

PubMed

Murase, Kenya

2015-01-01

In this issue, symbolic methods for solving differential equations were firstly introduced. Of the symbolic methods, Laplace transform method was also introduced together with some examples, in which this method was applied to solving the differential equations derived from a two-compartment kinetic model and an equivalent circuit model for membrane potential. Second, series expansion methods for solving differential equations were introduced together with some examples, in which these methods were used to solve Bessel's and Legendre's differential equations. In the next issue, simultaneous differential equations and various methods for solving these differential equations will be introduced together with some examples in medical physics.

14. Fractional Step and Pseudocompressibility Methods for the Solution of Incompressible Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Kiris, Cetin; Kwak, Dochan

1995-01-01

The fractional step and the pseudocompressibility methods for the solution of the incompressible Navier-Stokes equations are outlined. The fractional step method is based on finite-volume formulation and uses the pressure and the volume fluxes across the faces of each cell as dependent variables. The momentum equations are solved implicitly and the Poisson equation for the pressure is solved by using the multigrid method. The pseudocompressibility approach uses an implicit-higher-order-upwind differencing scheme for the convective terms together with the Gauss-Seidel line relaxation method. The dependent variables in the pseudocompressibility approach are the pressure and the cartesian velocity components in unstaggered mesh orientation. The 90-degree square duct flow, the wing-tip vortex wake flow and unsteady turbulent flows over an oscillating NACA 0015 airfoil are computed using both the fractional step and the pseudocompressibility methods. The results obtained from two different schemes are compared against experimental measurements.

15. Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm

El-Ajou, Ahmad; Arqub, Omar Abu; Momani, Shaher

2015-07-01

In this paper, explicit and approximate solutions of the nonlinear fractional KdV-Burgers equation with time-space-fractional derivatives are presented and discussed. The solutions of our equation are calculated in the form of rabidly convergent series with easily computable components. The utilized method is a numerical technique based on the generalized Taylor series formula which constructs an analytical solution in the form of a convergent series. Five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method. Graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution. The results reveal that the method is very effective and simple in determination of solution of the fractional KdV-Burgers equation.

16. Existence and Stability Results for Second-Order Stochastic Equations Driven by Fractional Brownian Motion

Revathi, P.; Sakthivel, R.; Song, D.-Y.; Ren, Yong; Zhang, Pei

2013-09-01

Fractional Brownian motion has been widely used to model a number of phenomena in diverse fields of science and engineering. In this article, we investigate the existence, uniqueness and stability of mild solutions for a class of second-order nonautonomous neutral stochastic evolution equations with infinite delay driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1) in Hilbert spaces. More precisely, using semigroup theory and successive approximation approach, we establish a set of sufficient conditions for obtaining the required result under the assumption that coefficients satisfy non-Lipschitz condition with Lipschitz condition being considered as a special case. Further, the result is deduced to study the second-order autonomous neutral stochastic equations with fBm. The results generalize and improve some known results. Finally, as an application, stochastic wave equation with infinite delay driven by fractional Brownian motion is provided to illustrate the obtained theory.

17. A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains

Bonforte, Matteo; Vázquez, Juan Luis

2015-10-01

We investigate quantitative properties of the nonnegative solutions to the nonlinear fractional diffusion equation, , posed in a bounded domain, , with m > 1 for t > 0. As we use one of the most common definitions of the fractional Laplacian , 0 < s < 1, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. In addition, we obtain similar estimates for fractional semilinear elliptic equations. Either the standard Laplacian case s = 1 or the linear case m = 1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.

18. ASP: Automated symbolic computation of approximate symmetries of differential equations

Jefferson, G. F.; Carminati, J.

2013-03-01

A recent paper (Pakdemirli et al. (2004) [12]) compared three methods of determining approximate symmetries of differential equations. Two of these methods are well known and involve either a perturbation of the classical Lie symmetry generator of the differential system (Baikov, Gazizov and Ibragimov (1988) [7], Ibragimov (1996) [6]) or a perturbation of the dependent variable/s and subsequent determination of the classical Lie point symmetries of the resulting coupled system (Fushchych and Shtelen (1989) [11]), both up to a specified order in the perturbation parameter. The third method, proposed by Pakdemirli, Yürüsoy and Dolapçi (2004) [12], simplifies the calculations required by Fushchych and Shtelen's method through the assignment of arbitrary functions to the non-linear components prior to computing symmetries. All three methods have been implemented in the new MAPLE package ASP (Automated Symmetry Package) which is an add-on to the MAPLE symmetry package DESOLVII (Vu, Jefferson and Carminati (2012) [25]). To our knowledge, this is the first computer package to automate all three methods of determining approximate symmetries for differential systems. Extensions to the theory have also been suggested for the third method and which generalise the first method to systems of differential equations. Finally, a number of approximate symmetries and corresponding solutions are compared with results in the literature.

19. Strong solutions for differential equations in abstract spaces

Teixeira, Eduardo V.

Let (E,F) be a locally convex space. We denote the bounded elements of E by Eb:={x∈E:∥x∥F=supρ∈F ρ(x)<∞}. In this paper, we prove that if B is relatively compact with respect to the F topology and f:I×Eb→Eb is a measurable family of F-continuous maps then for each x0∈Eb there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥F norm) local solution to the initial valued problem ut(t)=f(t,u(t)), u(t0)=x0. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator.

20. Integro-differential equation for Bose-Einstein condensates

SciTech Connect

Adam, R. M.; Sofianos, S. A.

2010-11-15

We use the assumption that the potential for the A-boson system can be written as a sum of pairwise acting forces to decompose the wave function into Faddeev components that fulfill a Faddeev type equation. Expanding these components in terms of potential harmonic (PH) polynomials and projecting on the potential basis for a specific pair of particles results in a two-variable integro-differential equations suitable for A-boson bound-state studies. The solution of the equation requires the evaluation of Jacobi polynomials P{sub K}{sup {alpha},{beta}}(x) and of the weight function W(z) which give severe numerical problems for very large A. However, using appropriate limits for A{yields}{infinity} we obtain a variant equation which depends only on the input two-body interaction, and the kernel in the integral part has a simple analytic form. This equation can be readily applied to a variety of bosonic systems such as microclusters of noble gasses. We employ it to obtain results for A(set-membership sign)(10-100) {sup 87}Rb atoms interacting via interatomic interactions and confined by an externally applied trapping potential V{sub trap}(r). Our results are in excellent agreement with those previously obtained using the potential harmonic expansion method (PHEM) and the diffusion Monte Carlo (DMC) method.

1. Computations of Wall Distances Based on Differential Equations

NASA Technical Reports Server (NTRS)

Tucker, Paul G.; Rumsey, Chris L.; Spalart, Philippe R.; Bartels, Robert E.; Biedron, Robert T.

2004-01-01

The use of differential equations such as Eikonal, Hamilton-Jacobi and Poisson for the economical calculation of the nearest wall distance d, which is needed by some turbulence models, is explored. Modifications that could palliate some turbulence-modeling anomalies are also discussed. Economy is of especial value for deforming/adaptive grid problems. For these, ideally, d is repeatedly computed. It is shown that the Eikonal and Hamilton-Jacobi equations can be easy to implement when written in implicit (or iterated) advection and advection-diffusion equation analogous forms, respectively. These, like the Poisson Laplacian term, are commonly occurring in CFD solvers, allowing the re-use of efficient algorithms and code components. The use of the NASA CFL3D CFD program to solve the implicit Eikonal and Hamilton-Jacobi equations is explored. The re-formulated d equations are easy to implement, and are found to have robust convergence. For accurate Eikonal solutions, upwind metric differences are required. The Poisson approach is also found effective, and easiest to implement. Modified distances are not found to affect global outputs such as lift and drag significantly, at least in common situations such as airfoil flows.

2. Integro-differential equation for Bose-Einstein condensates

Adam, R. M.; Sofianos, S. A.

2010-11-01

We use the assumption that the potential for the A-boson system can be written as a sum of pairwise acting forces to decompose the wave function into Faddeev components that fulfill a Faddeev type equation. Expanding these components in terms of potential harmonic (PH) polynomials and projecting on the potential basis for a specific pair of particles results in a two-variable integro-differential equations suitable for A-boson bound-state studies. The solution of the equation requires the evaluation of Jacobi polynomials PKα,β(x) and of the weight function W(z) which give severe numerical problems for very large A. However, using appropriate limits for A→∞ we obtain a variant equation which depends only on the input two-body interaction, and the kernel in the integral part has a simple analytic form. This equation can be readily applied to a variety of bosonic systems such as microclusters of noble gasses. We employ it to obtain results for A∈(10-100) Rb87 atoms interacting via interatomic interactions and confined by an externally applied trapping potential Vtrap(r). Our results are in excellent agreement with those previously obtained using the potential harmonic expansion method (PHEM) and the diffusion Monte Carlo (DMC) method.

3. Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation

PubMed Central

Wang, Gang wei; Xu, Tian zhou; Feng, Tao

2014-01-01

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional fifth-order KdV equation. A systematic research to derive Lie point symmetries to time fractional fifth-order KdV equation is performed. In the sense of point symmetry, all of the vector fields and the symmetry reductions of the fractional fifth-order KdV equation are obtained. At last, by virtue of the sub-equation method, some exact solutions to the fractional fifth-order KdV equation are provided. PMID:24523885

4. On three explicit difference schemes for fractional diffusion and diffusion-wave equations

Quintana Murillo, Joaquín; Bravo Yuste, Santos

2009-10-01

Three explicit difference schemes for solving fractional diffusion and fractional diffusion-wave equations are studied. We consider these equations in both the Riemann-Liouville and the Caputo forms. We find that the Gorenflo et al (2000 J. Comput. Appl. Math. 118 175) and the Yuste-Acedo (2005 SIAM J. Numer. Anal. 42 1862) methods when applied to fractional diffusion equations are equivalent when BDF1 coefficients are used to discretize the fractional derivative operators, but that this is not the case for fractional diffusion-wave equations. The accuracy and stability of the three methods are studied. Surprisingly, the third method, that of Ciesielski-Leszczynski (2003 Proc. 15th Conf. on Computer Methods in Mechanics), although closely related to the Gorenflo et al method, is the least accurate, especially for short times. The stability analysis is carried out by means of a procedure close to the standard von Neumann method. We find that the stability bounds of the three methods are the same.

5. Sinc-Chebyshev Collocation Method for a Class of Fractional Diffusion-Wave Equations

PubMed Central

Mao, Zhi; Xiao, Aiguo; Yu, Zuguo; Shi, Long

2014-01-01

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed. PMID:24977177

6. Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

Abdelkawy, M. A.; Ahmed, Engy A.; Alqahtani, Rubayyi T.

2016-01-01

We introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs). We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

7. Stochastic Calculus and Differential Equations for Physics and Finance

McCauley, Joseph L.

2013-02-01

1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker-Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and Chapman-Kolmogorov; 8. Non Markov Ito processes; 9. Black-Scholes, martingales, and Feynman-Katz; 10. Stochastic calculus with martingales; 11. Statistical physics and finance, a brief history of both; 12. Introduction to new financial economics; 13. Statistical ensembles and time series analysis; 14. Econometrics; 15. Semimartingales; References; Index.

8. Minimal parameter solution of the orthogonal matrix differential equation

NASA Technical Reports Server (NTRS)

Bar-Itzhack, Itzhack Y.; Markley, F. Landis

1990-01-01

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

9. Minimal parameter solution of the orthogonal matrix differential equation

NASA Technical Reports Server (NTRS)

Bar-Itzhack, Itzhack Y.; Markley, F. Landis

1988-01-01

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed employing the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

10. Differential equation based method for accurate approximations in optimization

NASA Technical Reports Server (NTRS)

Pritchard, Jocelyn I.; Adelman, Howard M.

1990-01-01

This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.

11. Axially symmetric equations for differential pulsar rotation with superfluid entrainment

Antonelli, M.; Pizzochero, P. M.

2017-01-01

In this article we present an analytical two-component model for pulsar rotational dynamics. Under the assumption of axial symmetry, implemented by a paraxial array of straight vortices that thread the entire neutron superfluid, we are able to project exactly the 3D hydrodynamical problem to a 1D cylindrical one. In the presence of density-dependent entrainment the superfluid rotation is non-columnar: we circumvent this by using an auxiliary dynamical variable directly related to the areal density of vortices. The main result is a system of differential equations that take consistently into account the stratified spherical structure of the star, the dynamical effects of non-uniform entrainment, the differential rotation of the superfluid component and its coupling to the normal crust. These equations represent a mathematical framework in which to test quantitatively the macroscopic consequences of the presence of a stable vortex array, a working hypothesis widely used in glitch models. Even without solving the equations explicitly, we are able to draw some general quantitative conclusions; in particular, we show that the reservoir of angular momentum (corresponding to recent values of the pinning forces) is enough to reproduce the largest glitch observed in the Vela pulsar, provided its mass is not too large.

12. Solutions for the Mikhailov-Shabat-Yamilov Difference-Differential Equations and Generalized Solutions for the Volterra and the Toda Lattice Equations

Narita, K.

1998-03-01

We present two types of mixed 1-rational N-soliton solutions and two types of special solutions for four types of Volterra-related difference-differential equations arising in Mikhailov, Shabat and Yamilov's lists. We also find new expressions of mixed 1-rational N-soliton solutions for the Volterra and the Toda lattice equations based on the invariance of Gibbon and Tabor's equation (J. Math. Phys. 26 (1985), 1956) under the fractional linear transformation. By taking appropriate limits of wave numbers, we find some new rational solutions for the Volterra and the Toda lattice equations. We also present elliptic function solutions for the Volterra and the Toda lattice equations different from known ones based on the same formulation.

13. Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker-Planck and Fractional Equations

Boon, Jean Pierre; Lutsko, James F.

2017-03-01

The temporal Fokker-Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker-Planck equation for the first passage distribution function f_j(r,t) of a particle moving on a substrate with time delays τ _j. Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability P_j is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, P_j ∝ f_j^{ν - 1}, the generalized Fokker-Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, P_j ∝ τ _j^{-1-α } (with 0< α < 2), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.

14. Paraconformal structures, ordinary differential equations and totally geodesic manifolds

Kryński, Wojciech

2016-05-01

We construct point invariants of ordinary differential equations of arbitrary order that generalise the Tresse and Cartan invariants of equations of order two and three, respectively. The vanishing of the invariants is equivalent to the existence of a totally geodesic paraconformal structure which consists of a paraconformal structure, an adapted GL(2 , R) -connection and a two-parameter family of totally geodesic hypersurfaces on the solution space. The structures coincide with the projective structures in dimension 2 and with the Einstein-Weyl structures of Lorentzian signature in dimension 3. We show that the totally geodesic paraconformal structures in higher dimensions can be described by a natural analogue of the Hitchin twistor construction. We present a general example of Veronese webs that generalise the hyper-CR Einstein-Weyl structures in dimension 3. The Veronese webs are described by a hierarchy of integrable systems.

15. Modeling tree crown dynamics with 3D partial differential equations

PubMed Central

Beyer, Robert; Letort, Véronique; Cournède, Paul-Henry

2014-01-01

We characterize a tree's spatial foliage distribution by the local leaf area density. Considering this spatially continuous variable allows to describe the spatiotemporal evolution of the tree crown by means of 3D partial differential equations. These offer a framework to rigorously take locally and adaptively acting effects into account, notably the growth toward light. Biomass production through photosynthesis and the allocation to foliage and wood are readily included in this model framework. The system of equations stands out due to its inherent dynamic property of self-organization and spontaneous adaptation, generating complex behavior from even only a few parameters. The density-based approach yields spatially structured tree crowns without relying on detailed geometry. We present the methodological fundamentals of such a modeling approach and discuss further prospects and applications. PMID:25101095

16. Bringing partial differential equations to life for students

José Cano, María; Chacón-Vera, Eliseo; Esquembre, Francisco

2015-05-01

Teaching partial differential equations (PDEs) carries inherent difficulties that an interactive visualization might help overcome in an active learning process. However, the generation of this kind of teaching material implies serious difficulties, mainly in terms of coding efforts. This work describes how to use an authoring tool, Easy Java Simulations, to build interactive simulations using FreeFem++ (Hecht F 2012 J. Numer. Math. 20 251) as a PDE solver engine. It makes possible to build simulations where students can change parameters, the geometry and the equations themselves getting an immediate feedback. But it is also possible for them to edit the simulations to set deeper changes. The process is ilustrated with some basic examples. These simulations show PDEs in a pedagogic manner and can be tuned by no experts in the field, teachers or students. Finally, we report a classroom experience and a survey from the third year students in the Degree of Mathematics at the University of Murcia.

17. Algorithm Refinement for Stochastic Partial Differential Equations. I. Linear Diffusion

Alexander, Francis J.; Garcia, Alejandro L.; Tartakovsky, Daniel M.

2002-10-01

A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. Results from a variety of numerical experiments are presented for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a nonstochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except in particle regions away from the interface. Extensions of the methodology to fluid mechanics applications are discussed.

18. Einstein-Weyl spaces and third-order differential equations

Tod, K. P.

2000-08-01

The three-dimensional null-surface formalism of Tanimoto [M. Tanimoto, "On the null surface formalism," Report No. gr-qc/9703003 (1997)] and Forni et al. [Forni et al., "Null surfaces formation in 3D," J. Math Phys. (submitted)] are extended to describe Einstein-Weyl spaces, following Cartan [E. Cartan, "Les espaces généralisées et l'integration de certaines classes d'equations différentielles," C. R. Acad. Sci. 206, 1425-1429 (1938); "La geometria de las ecuaciones diferenciales de tercer order," Rev. Mat. Hispano-Am. 4, 1-31 (1941)]. In the resulting formalism, Einstein-Weyl spaces are obtained from a particular class of third-order differential equations. Some examples of the construction which include some new Einstein-Weyl spaces are given.

19. A fast finite volume method for conservative space-fractional diffusion equations in convex domains

Jia, Jinhong; Wang, Hong

2016-04-01

We develop a fast finite volume method for variable-coefficient, conservative space-fractional diffusion equations in convex domains via a volume-penalization approach. The method has an optimal storage and an almost linear computational complexity. The method retains second-order accuracy without requiring a Richardson extrapolation. Numerical results are presented to show the utility of the method.

20. On a fractional integral equation of periodic functions involving Weyl-Riesz operator in Banach algebras

Momani, Shaher; Ibrahim, Rabha W.

2008-03-01

In this paper, we study the existence of periodic solutions for a nonlinear integral equation of periodic functions involving Weyl-Riesz fractional integral operator under the mixed generalized Lipschitz, Carathéodory and monotonicity conditions. The fixed point theorems due to Dhage are the main tool in carrying out our proofs.

1. Naturally occurring continued fractions in the variation of Kepler's equation. [for guidance and trajectory optimization

NASA Technical Reports Server (NTRS)

Shepperd, Stanley W.

1988-01-01

A family of functions involving integrals of universal functions is introduced. These functions have some interesting mathematical properties including the fact that they may be expressed as Gaussian continued fractions. A unique method of performing the integration is demonstrated which indicates why these functions may be important in the variation of Kepler's equation.

2. Wave equation for generalized Zener model containing complex order fractional derivatives

Atanacković, Teodor M.; Janev, Marko; Konjik, Sanja; Pilipović, Stevan

2017-03-01

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed.

3. A unique transformation from ordinary differential equations to reaction networks.

PubMed

Soliman, Sylvain; Heiner, Monika

2010-12-22

Many models in Systems Biology are described as a system of Ordinary Differential Equations, which allows for transient, steady-state or bifurcation analysis when kinetic information is available. Complementary structure-related qualitative analysis techniques have become increasingly popular in recent years, like qualitative model checking or pathway analysis (elementary modes, invariants, flux balance analysis, graph-based analyses, chemical organization theory, etc.). They do not rely on kinetic information but require a well-defined structure as stochastic analysis techniques equally do. In this article, we look into the structure inference problem for a model described by a system of Ordinary Differential Equations and provide conditions for the uniqueness of its solution. We describe a method to extract a structured reaction network model, represented as a bipartite multigraph, for example, a continuous Petri net (CPN), from a system of Ordinary Differential Equations (ODEs). A CPN uniquely defines an ODE, and each ODE can be transformed into a CPN. However, it is not obvious under which conditions the transformation of an ODE into a CPN is unique, that is, when a given ODE defines exactly one CPN. We provide biochemically relevant sufficient conditions under which the derived structure is unique and counterexamples showing the necessity of each condition. Our method is implemented and available; we illustrate it on some signal transduction models from the BioModels database. A prototype implementation of the method is made available to modellers at http://contraintes.inria.fr/~soliman/ode2pn.html, and the data mentioned in the "Results" section at http://contraintes.inria.fr/~soliman/ode2pn_data/. Our results yield a new recommendation for the import/export feature of tools supporting the SBML exchange format.

4. Deformed cohomologies of symmetry pseudo-groups and coverings of differential equations

Morozov, Oleg I.

2017-03-01

The work establishes a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov-Zabolotskaya equation and the Boyer-Finley equation.

5. Method for Solving Physical Problems Described by Linear Differential Equations

Belyaev, B. A.; Tyurnev, V. V.

2017-01-01

A method for solving physical problems is suggested in which the general solution of a differential equation in partial derivatives is written in the form of decomposition in spherical harmonics with indefinite coefficients. Values of these coefficients are determined from a comparison of the decomposition with a solution obtained for any simplest particular case of the examined problem. The efficiency of the method is demonstrated on an example of calculation of electromagnetic fields generated by a current-carrying circular wire. The formulas obtained can be used to analyze paths in the near-field magnetic (magnetically inductive) communication systems working in moderately conductive media, for example, in sea water.

6. Partial differential equation models in the socio-economic sciences

PubMed Central

Burger, Martin; Caffarelli, Luis; Markowich, Peter A.

2014-01-01

Mathematical models based on partial differential equations (PDEs) have become an integral part of quantitative analysis in most branches of science and engineering, recently expanding also towards biomedicine and socio-economic sciences. The application of PDEs in the latter is a promising field, but widely quite open and leading to a variety of novel mathematical challenges. In this introductory article of the Theme Issue, we will provide an overview of the field and its recent boosting topics. Moreover, we will put the contributions to the Theme Issue in an appropriate perspective. PMID:25288814

7. Partial differential equation models in the socio-economic sciences.

PubMed

Burger, Martin; Caffarelli, Luis; Markowich, Peter A

2014-11-13

Mathematical models based on partial differential equations (PDEs) have become an integral part of quantitative analysis in most branches of science and engineering, recently expanding also towards biomedicine and socio-economic sciences. The application of PDEs in the latter is a promising field, but widely quite open and leading to a variety of novel mathematical challenges. In this introductory article of the Theme Issue, we will provide an overview of the field and its recent boosting topics. Moreover, we will put the contributions to the Theme Issue in an appropriate perspective.

8. Behavior near constant solutions of functional differential equations

NASA Technical Reports Server (NTRS)

Hale, J. K.

1974-01-01

Techniques have been developed to determine in a systematic way the local behavior near constant solutions. Local integral manifolds play a very important role in this development, as they have also for ordinary differential equations. An attempt is made to indicate a few more applications of these methods to some problems in bifurcation in the spirit of Sotomayor (to appear) and to a growth model of Cooke and Yorke (to appear). It is also shown how to prove a theorem on stability under constantly acting disturbances using these methods.

9. An Exponential Finite Difference Technique for Solving Partial Differential Equations.

DTIC Science & Technology

1987-06-01

density , kg/N 3 (lbm/ft 3) 91.*,e separation variables (At dimensionless timelAX) 2 vi -W sNiv W- NiW.4%1 1. INTRODUCTION Partial differential equations...competing numerical analysis were run in double precision on either the IBM-3033 or the Cray X-MP mainframes. The computer codes developed for the...is increased. - R P~p~ 15 Effect of Initial and Boundary Conditions on the Exponential Finite Difference Method In this section the effect of

10. Neural network error correction for solving coupled ordinary differential equations

NASA Technical Reports Server (NTRS)

Shelton, R. O.; Darsey, J. A.; Sumpter, B. G.; Noid, D. W.

1992-01-01

A neural network is presented to learn errors generated by a numerical algorithm for solving coupled nonlinear differential equations. The method is based on using a neural network to correctly learn the error generated by, for example, Runge-Kutta on a model molecular dynamics (MD) problem. The neural network programs used in this study were developed by NASA. Comparisons are made for training the neural network using backpropagation and a new method which was found to converge with fewer iterations. The neural net programs, the MD model and the calculations are discussed.

11. Investigation of ODE integrators using interactive graphics. [Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Brown, R. L.

1978-01-01

Two FORTRAN programs using an interactive graphic terminal to generate accuracy and stability plots for given multistep ordinary differential equation (ODE) integrators are described. The first treats the fixed stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution, as well as the linear stability region. The second generates an analog to the stability region for classes of non-linear ODE's as well as accuracy plots. Both systems can compute method coefficients from a simple specification of the method. Example plots are given.

12. Differential Equations, Related Problems of Pade Approximations and Computer Applications

DTIC Science & Technology

1988-01-01

geometric sense, like the Picard-Fuchs equations satisfied by the variation of periods, possess strong arithmetic properties (global nilpotence ...result, and the (G, C)-function conditions, one needs the definition of the p-curvature. We consider a system of matrix first order linear differential...the system (1.1) in the matrix form df f /dx = Aff ; A E M (Q(x)), one can introduce the p-curvature operators Ip, associated with the system (1.1). The

13. Analytical solutions for systems of partial differential-algebraic equations.

PubMed

Benhammouda, Brahim; Vazquez-Leal, Hector

2014-01-01

This work presents the application of the power series method (PSM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-one and index-three are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a useful strategy to find exact solutions. The main advantage of the proposed methodology is that the procedure is based on a few straightforward steps and it does not generate secular terms or depends of a perturbation parameter.

14. Renormalization group and perfect operators for stochastic differential equations.

PubMed

Hou, Q; Goldenfeld, N; McKane, A

2001-03-01

We develop renormalization group (RG) methods for solving partial and stochastic differential equations on coarse meshes. RG transformations are used to calculate the precise effect of small-scale dynamics on the dynamics at the mesh size. The fixed point of these transformations yields a perfect operator: an exact representation of physical observables on the mesh scale with minimal lattice artifacts. We apply the formalism to simple nonlinear models of critical dynamics, and show how the method leads to an improvement in the computational performance of Monte Carlo methods.

15. Estimating varying coefficients for partial differential equation models.

PubMed

Zhang, Xinyu; Cao, Jiguo; Carroll, Raymond J

2017-01-11

Partial differential equations (PDEs) are used to model complex dynamical systems in multiple dimensions, and their parameters often have important scientific interpretations. In some applications, PDE parameters are not constant but can change depending on the values of covariates, a feature that we call varying coefficients. We propose a parameter cascading method to estimate varying coefficients in PDE models from noisy data. Our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed. The performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a PDE model arising from LIDAR data.

16. Approximate Solvability of Forward-Backward Stochastic Differential Equations

SciTech Connect

Ma, J. Yong, J.

2002-07-01

The solvability of forward-backward stochastic differential equations (FBSDEs for short) has been studied extensively in recent years. To guarantee the existence and uniqueness of adapted solutions, many different conditions, some quite restrictive, have been imposed. In this paper we propose a new notion: the approximate solvability of FBSDEs, based on the method of optimal control introduced in our primary work [15]. The approximate solvability of a class of FBSDEs is shown under mild conditions; and a general scheme for constructing approximate adapted solutions is proposed.

17. State-Constrained Optimal Control Problems of Impulsive Differential Equations

SciTech Connect

Forcadel, Nicolas; Rao Zhiping Zidani, Hasnaa

2013-08-01

The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption.

18. Some studies of the numerical solution of ordinary differential equations

Mehdiyeva, G.; Ibrahimov, V.; Imanova, M.

2012-08-01

With the numerical solution of ordinary differential equations(ODE), scientists engaged in the Middle Ages, beginning with the work of Clairaut. The domain of the numerical methods involved in many famous mathematicians - Euler, Runge, Kutta, Adams, Laplace, and others. They have constructed methods with different properties. In this paper we consider the construction of numerical methods with high accuracy and to this end is proposed to use multi-step multi-derivative and hybrid methods. As well as specific methods are constructed with a certain accuracy.

19. Application of a partial differential equation in image processing

Wang, Huijuan; Yin, Zengqian; Wan, Jingyu; Pang, Juan

2008-03-01

Edge detection is realized by using a FitzHugh Nagumo model which is one type of partial differential equations. This model has three types of dynamic, excitable, Turing/Hopf bifurcation, and bistable. In the excitable region the model can realize the edge detection. In the simulation only one image is processed in order to confirm the effect of control parameters on the edge detection. A satisfying effect of edge detection can be obtained by choosing appropriate control parameters. By comparing with other operators it is found that the FitzHugh Nagumo model is superior to Canny operator, Prewitt operator, Roberts operator, and Sobel operator on edge detection.

20. Infinite time interval backward stochastic differential equations with continuous coefficients.

PubMed

Zong, Zhaojun; Hu, Feng

2016-01-01

In this paper, we study the existence theorem for [Formula: see text] [Formula: see text] solutions to a class of 1-dimensional infinite time interval backward stochastic differential equations (BSDEs) under the conditions that the coefficients are continuous and have linear growths. We also obtain the existence of a minimal solution. Furthermore, we study the existence and uniqueness theorem for [Formula: see text] [Formula: see text] solutions of infinite time interval BSDEs with non-uniformly Lipschitz coefficients. It should be pointed out that the assumptions of this result is weaker than that of Theorem 3.1 in Zong (Turkish J Math 37:704-718, 2013).

1. Informed Conjecturing of Solutions for Differential Equations in a Modeling Context

ERIC Educational Resources Information Center

Winkel, Brian

2015-01-01

We examine two differential equations. (i) first-order exponential growth or decay; and (ii) second order, linear, constant coefficient differential equations, and show the advantage of learning differential equations in a modeling context for informed conjectures of their solution. We follow with a discussion of the complete analysis afforded by…

2. Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation.

PubMed

Zhang, Yiqi; Liu, Xing; Belić, Milivoj R; Zhong, Weiping; Zhang, Yanpeng; Xiao, Min

2015-10-30

The dynamics of wave packets in the fractional Schrödinger equation is still an open problem. The difficulty stems from the fact that the fractional Laplacian derivative is essentially a nonlocal operator. We investigate analytically and numerically the propagation of optical beams in the fractional Schrödinger equation with a harmonic potential. We find that the propagation of one- and two-dimensional input chirped Gaussian beams is not harmonic. In one dimension, the beam propagates along a zigzag trajectory in real space, which corresponds to a modulated anharmonic oscillation in momentum space. In two dimensions, the input Gaussian beam evolves into a breathing ring structure in both real and momentum spaces, which forms a filamented funnel-like aperiodic structure. The beams remain localized in propagation, but with increasing distance display an increasingly irregular behavior, unless both the linear chirp and the transverse displacement of the incident beam are zero.

3. Numerical solution of fractional-in-space nonlinear Schrödinger equation with the Riesz fractional derivative

2016-09-01

In this paper, dynamics of time-dependent fractional-in-space nonlinear Schrödinger equation with harmonic potential V(x),x in R in one, two and three dimensions have been considered. We approximate the Riesz fractional derivative with the Fourier pseudo-spectral method and advance the resulting equation in time with both Strang splitting and exponential time-differencing methods. The Riesz derivative introduced in this paper is found to be so convenient to be applied in models that are connected with applied science, physics, and engineering. We must also report that the Riesz derivative introduced in this work will serve as a complementary operator to the commonly used Caputo or Riemann-Liouville derivatives in the higher-dimensional case. In the numerical experiments, one expects the travelling wave to evolve from such an initial function on an infinite computational domain (-∞, &infty); , which we truncate at some large, but finite values L. It is important that the value of L is chosen large enough to give enough room for the wave function to propagate. We observe a different distribution of complex wave functions for the focusing and defocusing cases.

4. Variations in the Solution of Linear First-Order Differential Equations. Classroom Notes

ERIC Educational Resources Information Center

Seaman, Brian; Osler, Thomas J.

2004-01-01

A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx)+p(x)v=q(x) by means of a substitution v=f(y). The student then creates a table of the new equations and…

5. Fractional diffusion equation for an n-dimensional correlated Lévy walk.

PubMed

Taylor-King, Jake P; Klages, Rainer; Fedotov, Sergei; Van Gorder, Robert A

2016-07-01

Lévy walks define a fundamental concept in random walk theory that allows one to model diffusive spreading faster than Brownian motion. They have many applications across different disciplines. However, so far the derivation of a diffusion equation for an n-dimensional correlated Lévy walk remained elusive. Starting from a fractional Klein-Kramers equation here we use a moment method combined with a Cattaneo approximation to derive a fractional diffusion equation for superdiffusive short-range auto-correlated Lévy walks in the large time limit, and we solve it. Our derivation discloses different dynamical mechanisms leading to correlated Lévy walk diffusion in terms of quantities that can be measured experimentally.

6. A space-time spectral collocation algorithm for the variable order fractional wave equation.

PubMed

Bhrawy, A H; Doha, E H; Alzaidy, J F; Abdelkawy, M A

2016-01-01

The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space-time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi-Gauss-Lobatto collocation scheme for the spatial discretization and the shifted Jacobi-Gauss-Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.

7. Coupled latent differential equation with moderators: simulation and application.

PubMed

Hu, Yueqin; Boker, Steve; Neale, Michael; Klump, Kelly L

2014-03-01

Latent differential equations (LDE) use differential equations to analyze time series data. Because of the recent development of this technique, some issues critical to running an LDE model remain. In this article, the authors provide solutions to some of these issues and recommend a step-by-step procedure demonstrated on a set of empirical data, which models the interaction between ovarian hormone cycles and emotional eating. Results indicated that emotional eating is self-regulated. For instance, when people do more emotional eating than normal, they will subsequently tend to decrease their emotional eating behavior. In addition, a sudden increase will produce a stronger tendency to decrease than will a slow increase. We also found that emotional eating is coupled with the cycle of the ovarian hormone estradiol, and the peak of emotional eating occurs after the peak of estradiol. The self-reported average level of negative affect moderates the frequency of eating regulation and the coupling strength between eating and estradiol. Thus, people with a higher average level of negative affect tend to fluctuate faster in emotional eating, and their eating behavior is more strongly coupled with the hormone estradiol. Permutation tests on these empirical data supported the reliability of using LDE models to detect self-regulation and a coupling effect between two regulatory behaviors.

8. A hybrid Pade-Galerkin technique for differential equations

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1993-01-01

A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter epsilon associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade approximation in the form of a rational function in the parameter epsilon. In the third step, the various powers of epsilon which appear in the Pade approximation are replaced by new (unknown) parameters (delta(sub j)). These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade approximations fail to do so. The method is discussed and topics for future investigations are indicated.

9. Runge-Kutta Methods for Linear Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Zingg, David W.; Chisholm, Todd T.

1997-01-01

Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

10. A variational approach to parameter estimation in ordinary differential equations

PubMed Central

2012-01-01

Background Ordinary differential equations are widely-used in the field of systems biology and chemical engineering to model chemical reaction networks. Numerous techniques have been developed to estimate parameters like rate constants, initial conditions or steady state concentrations from time-resolved data. In contrast to this countable set of parameters, the estimation of entire courses of network components corresponds to an innumerable set of parameters. Results The approach presented in this work is able to deal with course estimation for extrinsic system inputs or intrinsic reactants, both not being constrained by the reaction network itself. Our method is based on variational calculus which is carried out analytically to derive an augmented system of differential equations including the unconstrained components as ordinary state variables. Finally, conventional parameter estimation is applied to the augmented system resulting in a combined estimation of courses and parameters. Conclusions The combined estimation approach takes the uncertainty in input courses correctly into account. This leads to precise parameter estimates and correct confidence intervals. In particular this implies that small motifs of large reaction networks can be analysed independently of the rest. By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields. PMID:22892133

11. Pseudospectral collocation methods for fourth order differential equations

NASA Technical Reports Server (NTRS)

Malek, Alaeddin; Phillips, Timothy N.

1994-01-01

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.

12. Accelerating numerical solution of stochastic differential equations with CUDA

Januszewski, M.; Kostur, M.

2010-01-01

Numerical integration of stochastic differential equations is commonly used in many branches of science. In this paper we present how to accelerate this kind of numerical calculations with popular NVIDIA Graphics Processing Units using the CUDA programming environment. We address general aspects of numerical programming on stream processors and illustrate them by two examples: the noisy phase dynamics in a Josephson junction and the noisy Kuramoto model. In presented cases the measured speedup can be as high as 675× compared to a typical CPU, which corresponds to several billion integration steps per second. This means that calculations which took weeks can now be completed in less than one hour. This brings stochastic simulation to a completely new level, opening for research a whole new range of problems which can now be solved interactively. Program summaryProgram title: SDE Catalogue identifier: AEFG_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFG_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Gnu GPL v3 No. of lines in distributed program, including test data, etc.: 978 No. of bytes in distributed program, including test data, etc.: 5905 Distribution format: tar.gz Programming language: CUDA C Computer: any system with a CUDA-compatible GPU Operating system: Linux RAM: 64 MB of GPU memory Classification: 4.3 External routines: The program requires the NVIDIA CUDA Toolkit Version 2.0 or newer and the GNU Scientific Library v1.0 or newer. Optionally gnuplot is recommended for quick visualization of the results. Nature of problem: Direct numerical integration of stochastic differential equations is a computationally intensive problem, due to the necessity of calculating multiple independent realizations of the system. We exploit the inherent parallelism of this problem and perform the calculations on GPUs using the CUDA programming environment. The GPU's ability to execute

13. Solving cochlear mechanics problems with higher-order differential equations.

PubMed

de Boer, E; van Bienema, E

1982-11-01

Since most "exact" solution methods for cochlear models are rather unwieldy, they do not lend themselves to easy and multi-purpose application. In this paper a new solution method is described that is more flexible in this respect. A three-dimensional cochlear model is considered. It can be described by an integral equation in terms of the wavenumber k. The kernel Q (k) of that equation is approximated by a rational function of k and this makes it possible to reformulate the problem as a differential equation. The latter can be solved by a straightforward and well-known method. Results of computations with this technique are presented in two forms: an overview of the entire cochlear wave pattern and a detailed representation of the response peak. The method is also used to determine whether a discernible reflected wave is produced in the cochlea or not. For this purpose the wavenumber spectrum of the cochlear wave is studied: it is found to be a one-sided function of k. With surprisingly simple means it is thus shown that no appreciable reflection occurs from the inhomogeneity that is characteristic in cochlear wave propagation. This holds true for values of damping constant delta as low as 0.01, a factor of 5 smaller than is commonly used in cochlear modeling.

14. Tunable fractional-order photonic differentiator using a distributed feedback semiconductor optical amplifier

Sun, Shuqian; Deng, Ye; Zhu, Ninghua; Li, Ming

2016-03-01

We propose a tunable fractional-order photonic differentiator based on a distributed feedback semiconductor optical amplifier (SOA) working in reflection mode. The phase shift at the resonant wavelength can be adjusted by controlling the current injected into the DFB-SOA, which can implement the fractional-order differentiation. A 60 ps Gaussian pulse is temporally differentiated with a tunable order range from 0.7 to 1.3.

15. The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion

Bessaih, H.; Ferrario, B.

2017-02-01

In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à la Leray through a smoothing kernel of order α in the nonlinear term and a β-fractional Laplacian; we consider the critical case α + β =5/4 and we assume 1/2 < β <5/4. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order α. We prove global well posedness when the initial velocity is in Hr and the initial temperature is in H r - β for r > max ⁡ (2 β , β + 1). This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions on the initial conditions.

16. Lie Symmetries, Conservation Laws and Explicit Solutions for Time Fractional Rosenau–Haynam Equation

Qin, Chun-Yan; Tian, Shou-Fu; Wang, Xiu-Bin; Zhang, Tian-Tian

2017-02-01

Under investigation in this paper is the invariance properties of the time fractional Rosenau-Haynam equation, which can be used to describe the formation of patterns in liquid drops. By using the Lie group analysis method, the vector fields and symmetry reductions of the equation are derived, respectively. Moreover, based on the power series theory, a kind of explicit power series solutions for the equation are well constructed with a detailed derivation. Finally, by using the new conservation theorem, two kinds of conservation laws of the equation are well constructed with a detailed derivation. Supported by the Fundamental Research Fund for Talents Cultivation Project of the China University of Mining and Technology under Grant No. YC150003

17. A Stochastic Differential Equation Model for the Spread of HIV amongst People Who Inject Drugs.

PubMed

Liang, Yanfeng; Greenhalgh, David; Mao, Xuerong

2016-01-01

We introduce stochasticity into the deterministic differential equation model for the spread of HIV amongst people who inject drugs (PWIDs) studied by Greenhalgh and Hay (1997). This was based on the original model constructed by Kaplan (1989) which analyses the behaviour of HIV/AIDS amongst a population of PWIDs. We derive a stochastic differential equation (SDE) for the fraction of PWIDs who are infected with HIV at time. The stochasticity is introduced using the well-known standard technique of parameter perturbation. We first prove that the resulting SDE for the fraction of infected PWIDs has a unique solution in (0, 1) provided that some infected PWIDs are initially present and next construct the conditions required for extinction and persistence. Furthermore, we show that there exists a stationary distribution for the persistence case. Simulations using realistic parameter values are then constructed to illustrate and support our theoretical results. Our results provide new insight into the spread of HIV amongst PWIDs. The results show that the introduction of stochastic noise into a model for the spread of HIV amongst PWIDs can cause the disease to die out in scenarios where deterministic models predict disease persistence.

18. A Stochastic Differential Equation Model for the Spread of HIV amongst People Who Inject Drugs

PubMed Central

Mao, Xuerong

2016-01-01

We introduce stochasticity into the deterministic differential equation model for the spread of HIV amongst people who inject drugs (PWIDs) studied by Greenhalgh and Hay (1997). This was based on the original model constructed by Kaplan (1989) which analyses the behaviour of HIV/AIDS amongst a population of PWIDs. We derive a stochastic differential equation (SDE) for the fraction of PWIDs who are infected with HIV at time. The stochasticity is introduced using the well-known standard technique of parameter perturbation. We first prove that the resulting SDE for the fraction of infected PWIDs has a unique solution in (0, 1) provided that some infected PWIDs are initially present and next construct the conditions required for extinction and persistence. Furthermore, we show that there exists a stationary distribution for the persistence case. Simulations using realistic parameter values are then constructed to illustrate and support our theoretical results. Our results provide new insight into the spread of HIV amongst PWIDs. The results show that the introduction of stochastic noise into a model for the spread of HIV amongst PWIDs can cause the disease to die out in scenarios where deterministic models predict disease persistence. PMID:27051461

19. Optimal control of systems with discontinuous differential equations.

SciTech Connect

Stewart, D. E.; Anitescu, M.; Mathematics and Computer Science; Univ. of Iowa

2010-02-01

In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right-hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such a right-hand side. Notwithstanding the impressive development of nonsmooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. We show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize. This means that the strategy of 'optimize the discretization' will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that Euler's method is used where the step size is 'sufficiently small' in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing that involves Coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity.

20. An algorithm for solving the fractional convection diffusion equation with nonlinear source term

Momani, Shaher

2007-10-01

In this paper an algorithm based on Adomian's decomposition method is developed to approximate the solution of the nonlinear fractional convection-diffusion equation {∂αu}/{∂tα}={∂2u}/{∂x2}-c{∂u}/{∂x}+Ψ(u)+f(x,t),00. The fractional derivative is considered in the Caputo sense. The approximate solutions are calculated in the form of a convergent series with easily computable components. The analysis is accompanied by numerical examples and the obtained results are found to be in good agreement with the exact solutions known for some special cases.

1. A finite volume method for two-sided fractional diffusion equations on non-uniform meshes

Simmons, Alex; Yang, Qianqian; Moroney, Timothy

2017-04-01

We derive a finite volume method for two-sided fractional diffusion equations with Riemann-Liouville derivatives in one spatial dimension. The method applies to non-uniform meshes, with arbitrary nodal spacing. The discretisation utilises the integral definition of the fractional derivatives, and we show that it leads to a diagonally dominant matrix representation, and a provably stable numerical scheme. Being a finite volume method, the numerical scheme is fully conservative, and the ability to locally refine the mesh can produce solutions with more accuracy for the same number of nodes compared to a uniform mesh, as we demonstrate numerically.

2. A new class of traveling solitons for cubic fractional nonlinear Schrödinger equations

Hong, Younghun; Sire, Yannick

2017-04-01

We consider the one-dimensional cubic fractional nonlinear Schrödinger equation i∂tu‑(‑Δ)σu + |u|2u=0, where σ \\in ≤ft(\\frac{1}{2},1\\right) and the operator {{(- Δ )}σ} is the fractional Laplacian of symbol |ξ {{|}2σ} . Despite the lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form u(t,x)=e‑it(|k|2σ‑ω2σ)Qω,k(x‑2tσ|k|2σ‑2k),k∈R, ω>0 by a rather involved variational argument.

3. Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation

Li, Gongsheng; Zhang, Dali; Jia, Xianzheng; Yamamoto, Masahiro

2013-06-01

This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D time-fractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem, and the Lipschitz continuity of the solution operator is established. A modified optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function is put forward for the discretization of the minimization problem. Numerical inversions are performed for the diffusion coefficient taking on different functional forms and the additional data having random noise. Several factors which have important influences on the realization of the algorithm are discussed, including the approximate space of the diffusion coefficient, the regularization parameter and the initial iteration. The inversion solutions are good approximations to the exact solutions with stability and adaptivity demonstrating that the optimal perturbation algorithm with the sigmoid-type regularization parameter is efficient for the simultaneous inversion.

4. Exact solutions of a modified fractional diffusion equation in the finite and semi-infinite domains

Guo, Gang; Li, Kun; Wang, Yuhui

2015-01-01

We investigate the solutions of a modified fractional diffusion equation which has a secondary fractional time derivative acting on a diffusion operator. We obtain analytical solutions for the modified equation in the finite and semi-infinite domains subject to absorbing boundary conditions. Most of the results have been derived by using the Laplace transform, the Fourier Cosine transform, the Mellin transform and the properties of Fox H function. We show that the semi-infinite solution can be expressed using an infinite series of Fox H functions similar to the infinite case, while the finite solution requires double infinite series including both Fox H functions and trigonometric functions instead of one infinite series. The characteristic crossover between more and less anomalous behaviour as well as the effect of absorbing boundary conditions are clearly demonstrated according to the analytical solutions.

5. An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit

Wang, Li; Yan, Bokai

2016-05-01

We present a new asymptotic-preserving scheme for the linear Boltzmann equation which, under appropriate scaling, leads to a fractional diffusion limit. Our scheme rests on novel micro-macro decomposition to the distribution function, which splits the original kinetic equation following a reshuffled Hilbert expansion. As opposed to classical diffusion limit, a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter. Our idea is, while solving the macro-micro part in a truncated velocity domain (truncation only depends on numerical accuracy), to incorporate an integrated tail over the velocity space that is beyond the truncation, and its major component can be precomputed once with any accuracy. Such an addition is essential to drive the solution to the correct asymptotic limit. Numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes.

6. A Legendre tau-Spectral Method for Solving Time-Fractional Heat Equation with Nonlocal Conditions

PubMed Central

Bhrawy, A. H.; Alghamdi, M. A.

2014-01-01

We develop the tau-spectral method to solve the time-fractional heat equation (T-FHE) with nonlocal condition. In order to achieve highly accurate solution of this problem, the operational matrix of fractional integration (described in the Riemann-Liouville sense) for shifted Legendre polynomials is investigated in conjunction with tau-spectral scheme and the Legendre operational polynomials are used as the base function. The main advantage in using the presented scheme is that it converts the T-FHE with nonlocal condition to a system of algebraic equations that simplifies the problem. For demonstrating the validity and applicability of the developed spectral scheme, two numerical examples are presented. The logarithmic graphs of the maximum absolute errors is presented to achieve the exponential convergence of the proposed method. Comparing between our spectral method and other methods ensures that our method is more accurate than those solved similar problem. PMID:25057507

7. The exotic conformal Galilei algebra and nonlinear partial differential equations

Cherniha, Roman; Henkel, Malte

2010-09-01

The conformal Galilei algebra (CGA) and the exotic conformal Galilei algebra (ECGA) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the CGA but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under CGA. It is further shown that there are systems of non-linear PDEs admitting ECGA with the realisation obtained very recently in [D. Martelli and Y. Tachikawa, arXiv:0903.5184v2 [hep-th] (2009)]. Moreover, wide classes of non-linear systems, invariant under two different 10-dimensional subalgebras of ECGA are explicitly constructed and an example with possible physical interpretation is presented.

8. Finitely approximable random sets and their evolution via differential equations

Ananyev, B. I.

2016-12-01

In this paper, random closed sets (RCS) in Euclidean space are considered along with their distributions and approximation. Distributions of RCS may be used for the calculation of expectation and other characteristics. Reachable sets on initial data and some ways of their approximate evolutionary description are investigated for stochastic differential equations (SDE) with initial state in some RCS. Markov property of random reachable sets is proved in the space of closed sets. For approximate calculus, the initial RCS is replaced by a finite set on the integer multidimensional grid and the multistage Markov chain is substituted for SDE. The Markov chain is constructed by methods of SDE numerical integration. Some examples are also given.

9. Spreading disease: integro-differential equations old and new.

PubMed

Medlock, Jan; Kot, Mark

2003-08-01

We investigate an integro-differential equation for a disease spread by the dispersal of infectious individuals and compare this to Mollison's [Adv. Appl. Probab. 4 (1972) 233; D. Mollison, The rate of spatial propagation of simple epidemics, in: Proc. 6th Berkeley Symp. on Math. Statist. and Prob., vol. 3, University of California Press, Berkeley, 1972, p. 579; J. R. Statist. Soc. B 39 (3) (1977) 283] model of a disease spread by non-local contacts. For symmetric kernels with moment generating functions, spreading infectives leads to faster traveling waves for low rates of transmission, but to slower traveling waves for high rates of transmission. We approximate the shape of the traveling waves for the two models using both piecewise linearization and a regular-perturbation scheme.

10. Workload Characterization of CFD Applications Using Partial Differential Equation Solvers

NASA Technical Reports Server (NTRS)

Waheed, Abdul; Yan, Jerry; Saini, Subhash (Technical Monitor)

1998-01-01

Workload characterization is used for modeling and evaluating of computing systems at different levels of detail. We present workload characterization for a class of Computational Fluid Dynamics (CFD) applications that solve Partial Differential Equations (PDEs). This workload characterization focuses on three high performance computing platforms: SGI Origin2000, EBM SP-2, a cluster of Intel Pentium Pro bases PCs. We execute extensive measurement-based experiments on these platforms to gather statistics of system resource usage, which results in workload characterization. Our workload characterization approach yields a coarse-grain resource utilization behavior that is being applied for performance modeling and evaluation of distributed high performance metacomputing systems. In addition, this study enhances our understanding of interactions between PDE solver workloads and high performance computing platforms and is useful for tuning these applications.

11. Cause and cure of sloppiness in ordinary differential equation models.

PubMed

Tönsing, Christian; Timmer, Jens; Kreutz, Clemens

2014-08-01

Data-based mathematical modeling of biochemical reaction networks, e.g., by nonlinear ordinary differential equation (ODE) models, has been successfully applied. In this context, parameter estimation and uncertainty analysis is a major task in order to assess the quality of the description of the system by the model. Recently, a broadened eigenvalue spectrum of the Hessian matrix of the objective function covering orders of magnitudes was observed and has been termed as sloppiness. In this work, we investigate the origin of sloppiness from structures in the sensitivity matrix arising from the properties of the model topology and the experimental design. Furthermore, we present strategies using optimal experimental design methods in order to circumvent the sloppiness issue and present nonsloppy designs for a benchmark model.

12. Linear Multistep Methods for Integrating Reversible Differential Equations

Evans, N. Wyn; Tremaine, Scott

1999-10-01

This paper studies multistep methods for the integration of reversible dynamical systems, with particular emphasis on the planar Kepler problem. It has previously been shown by Cano & Sanz-Serna that reversible linear multisteps for first-order differential equations are generally unstable. Here we report on a subset of these methods-the zero-growth methods-that evade these instabilities. We provide an algorithm for identifying these rare methods. We find and study all zero-growth, reversible multisteps with six or fewer steps. This select group includes two well-known second-order multisteps (the trapezoidal and explicit midpoint methods), as well as three new fourth-order multisteps-one of which is explicit. Variable time steps can be readily implemented without spoiling the reversibility. Tests on Keplerian orbits show that these new reversible multisteps work well on orbits with low or moderate eccentricity, although at least 100 steps per radian are required for stability.

13. Numerical solution of differential equations by artificial neural networks

NASA Technical Reports Server (NTRS)

1995-01-01

Conventionally programmed digital computers can process numbers with great speed and precision, but do not easily recognize patterns or imprecise or contradictory data. Instead of being programmed in the conventional sense, artificial neural networks (ANN's) are capable of self-learning through exposure to repeated examples. However, the training of an ANN can be a time consuming and unpredictable process. A general method is being developed by the author to mate the adaptability of the ANN with the speed and precision of the digital computer. This method has been successful in building feedforward networks that can approximate functions and their partial derivatives from examples in a single iteration. The general method also allows the formation of feedforward networks that can approximate the solution to nonlinear ordinary and partial differential equations to desired accuracy without the need of examples. It is believed that continued research will produce artificial neural networks that can be used with confidence in practical scientific computing and engineering applications.

14. Cause and cure of sloppiness in ordinary differential equation models

Tönsing, Christian; Timmer, Jens; Kreutz, Clemens

2014-08-01

Data-based mathematical modeling of biochemical reaction networks, e.g., by nonlinear ordinary differential equation (ODE) models, has been successfully applied. In this context, parameter estimation and uncertainty analysis is a major task in order to assess the quality of the description of the system by the model. Recently, a broadened eigenvalue spectrum of the Hessian matrix of the objective function covering orders of magnitudes was observed and has been termed as sloppiness. In this work, we investigate the origin of sloppiness from structures in the sensitivity matrix arising from the properties of the model topology and the experimental design. Furthermore, we present strategies using optimal experimental design methods in order to circumvent the sloppiness issue and present nonsloppy designs for a benchmark model.

15. Computationally efficient statistical differential equation modeling using homogenization

USGS Publications Warehouse

Hooten, Mevin B.; Garlick, Martha J.; Powell, James A.

2013-01-01

Statistical models using partial differential equations (PDEs) to describe dynamically evolving natural systems are appearing in the scientific literature with some regularity in recent years. Often such studies seek to characterize the dynamics of temporal or spatio-temporal phenomena such as invasive species, consumer-resource interactions, community evolution, and resource selection. Specifically, in the spatial setting, data are often available at varying spatial and temporal scales. Additionally, the necessary numerical integration of a PDE may be computationally infeasible over the spatial support of interest. We present an approach to impose computationally advantageous changes of support in statistical implementations of PDE models and demonstrate its utility through simulation using a form of PDE known as “ecological diffusion.” We also apply a statistical ecological diffusion model to a data set involving the spread of mountain pine beetle (Dendroctonus ponderosae) in Idaho, USA.

16. A partial differential equation model of metastasized prostatic cancer.

PubMed

Friedman, Avner; Jain, Harsh Vardhan

2013-06-01

Biochemically failing metastatic prostate cancer is typically treated with androgen ablation. However, due to the emergence of castration-resistant cells that can survive in low androgen concentrations, such therapy eventually fails. Here, we develop a partial differential equation model of the growth and response to treatment of prostate cancer that has metastasized to the bone. Existence and uniqueness results are derived for the resulting free boundary problem. In particular, existence and uniqueness of solutions for all time are proven for the radially symmetric case. Finally, numerical simulations of a tumor growing in 2-dimensions with radial symmetry are carried in order to evaluate the therapeutic potential of different treatment strategies. These simulations are able to reproduce a variety of clinically observed responses to treatment, and suggest treatment strategies that may result in tumor remission, underscoring our model's potential to make a significant contribution in the field of prostate cancer therapeutics.

17. Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion

Kamrani, Minoo; Jamshidi, Nahid

2017-03-01

This work was intended as an attempt to motivate the approximation of quasi linear evolution equations driven by infinite-dimensional fractional Brownian motion with Hurst parameter H >1/2 . The spatial approximation method is based on Galerkin and the temporal approximation is based on implicit Euler scheme. An error bound and the convergence of the numerical method are given. The numerical results show usefulness and accuracy of the method.

18. Investigation of Conformable Fractional Schrödinger Equation in Presence of Killingbeck and Hyperbolic Potentials

Chung, Won Sang; Zare, Soroush; Hassanabadi, Hassan

2017-03-01

In this article, conformable fractional form of Schrödinger equation has been presented. Then in this formalism two different and well-known potential have been come in. Wave function of these potential are obtained in terms of Heun function and energy eigen values of each case is determined as well. Supported by the National Research Foundation of Korea Grant Funded by the Korean Government under Grant No. NRF-2015R1D1A1A01057792

19. Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation

Aksoy, Esin; Guner, Ozkan; Bekir, Ahmet; Cevikel, Adem C.

2016-06-01

Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation are studied by the generalized Kudryashov method, the exp-function method and the (G'/G)-expansion method. The solutions obtained include the form of hyperbolic functions, trigonometric and rational functions. These methods are effective, simple, and many types of solutions can be obtained at the same time.

20. Dark Soliton Solutions of Space-Time Fractional Sharma–Tasso–Olver and Potential Kadomtsev–Petviashvili Equations

Guner, Ozkan; Korkmaz, Alper; Bekir, Ahmet

2017-02-01

Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the \\tanh ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space.

1. A delay differential equation model of follicle waves in women.

PubMed

Panza, Nicole M; Wright, Andrew A; Selgrade, James F

2016-01-01

This article presents a mathematical model for hormonal regulation of the menstrual cycle which predicts the occurrence of follicle waves in normally cycling women. Several follicles of ovulatory size that develop sequentially during one menstrual cycle are referred to as follicle waves. The model consists of 13 nonlinear, delay differential equations with 51 parameters. Model simulations exhibit a unique stable periodic cycle and this menstrual cycle accurately approximates blood levels of ovarian and pituitary hormones found in the biological literature. Numerical experiments illustrate that the number of follicle waves corresponds to the number of rises in pituitary follicle stimulating hormone. Modifications of the model equations result in simulations which predict the possibility of two ovulations at different times during the same menstrual cycle and, hence, the occurrence of dizygotic twins via a phenomenon referred to as superfecundation. Sensitive parameters are identified and bifurcations in model behaviour with respect to parameter changes are discussed. Studying follicle waves may be helpful for improving female fertility and for understanding some aspects of female reproductive ageing.

2. BOOK REVIEW: Partial Differential Equations in General Relativity

Choquet-Bruhat, Yvonne

2008-09-01

General relativity is a physical theory basic in the modeling of the universe at the large and small scales. Its mathematical formulation, the Einstein partial differential equations, are geometrically simple, but intricate for the analyst, involving both hyperbolic and elliptic PDE, with local and global problems. Many problems remain open though remarkable progress has been made recently towards their solutions. Alan Rendall's book states, in a down-to-earth form, fundamental results used to solve different types of equations. In each case he gives applications to special models as well as to general properties of Einsteinian spacetimes. A chapter on ODE contains, in particular, a detailed discussion of Bianchi spacetimes. A chapter entitled 'Elliptic systems' treats the Einstein constraints. A chapter entitled 'Hyperbolic systems' is followed by a chapter on the Cauchy problem and a chapter 'Global results' which contains recently proved theorems. A chapter is dedicated to the Einstein Vlasov system, of which the author is a specialist. On the whole, the book surveys, in a concise though precise way, many essential results of recent interest in mathematical general relativity, and it is very clearly written. Each chapter is followed by an up to date bibliography. In conclusion, this book will be a valuable asset to relativists who wish to learn clearly-stated mathematical results and to mathematicians who want to penetrate into the subtleties of general relativity, as a mathematical and physical theory.

3. A more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators (L).

PubMed

Prieur, Fabrice; Vilenskiy, Gregory; Holm, Sverre

2012-10-01

A corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented. The fundamental approach is based on fractional formulations of the stress-strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in Prieur and Holm ["Nonlinear acoustic wave equations with fractional loss operators," J. Acoust. Soc. Am. 130(3), 1125-1132 (2011)]. The loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation, but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged.

4. Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front

Chan, Hardy; Wei, Juncheng

2017-05-01

Using the method of sub-super-solution, we construct a solution of (- Δ) s u - cuz - f (u) = 0 on R3 of pyramidal shape. Here (- Δ) s is the fractional Laplacian of sub-critical order 1 / 2 < s < 1 and f is a bistable nonlinearity. Hence, the existence of a traveling wave solution for the parabolic fractional Allen-Cahn equation with pyramidal front is asserted. The maximum of planar traveling wave solutions in various directions gives a sub-solution. A super-solution is roughly defined as the one-dimensional profile composed with the signed distance to a rescaled mollified pyramid. In the main estimate we use an expansion of the fractional Laplacian in the Fermi coordinates.

5. Chaotic attractors in tumor growth and decay: a differential equation model.

PubMed

Harney, Michael; Yim, Wen-sau

2015-01-01

Tumorigenesis can be modeled as a system of chaotic nonlinear differential equations. A simulation of the system is realized by converting the differential equations to difference equations. The results of the simulation show that an increase in glucose in the presence of low oxygen levels decreases tumor growth.

6. OPTIMAL CONTROL THEORY APPLIED TO SYSTEMS DESCRIBED BY PARTIAL DIFFERENTIAL EQUATIONS. VOL. 1 OF FINAL REPORT.

DTIC Science & Technology

control theory to systems described by partial differential equations. The intent is not to advance the theory of partial differential equations per se. Thus all considerations will be restricted to the more familiar equations of the type which often occur in mathematical physics. Specifically, the distributed parameter systems under consideration are represented by a set of field

7. Differential equation dynamical system based assessment model in GNSS interoperability

Han, Tao; Lu, XiaoChun; Wang, Xue; Rao, YongNan; Zou, DeCai; Yang, JianFei; Wu, YangYang

2011-06-01

With the development of Global Navigation Satellite System (GNSS), the idea of GNSS interoperability is born and has become the focus of study in the field of satellite navigation. The popularity for GNSS to augment the interoperability with the existing ones necessitates the study of the assessment algorithm of this idea. In this paper, an assessment algorithm for interoperability comprehensive benefits based on the differential equation dynamical system is discussed. There are two important aspects in GNSS that interoperability will affect: one is the performance advancement; the other one is the cost of adopting interoperability. While researching the complex relationship between the performance and cost, we found this relationship is similar as what between prey and predator in biomathematics, so the Lotka-Volterra model used to depict the prey-predator relationship is a felicitous tool. After building a differential dynamical model, we analyze the existence and stability of the positive equilibrium in the model. Then a Cost-Effective Function of GNSS is constructed based on the positive equilibrium, which is employed to assess the interoperability, qualitatively and quantitatively. Finally, the paper demonstrates the significance of the model and its application by citing a numerical example.

8. A Variable Order Fractional Differential-Based Texture Enhancement Algorithm with Application in Medical Imaging.

PubMed

Yu, Qiang; Vegh, Viktor; Liu, Fawang; Turner, Ian

2015-01-01

Texture enhancement is one of the most important techniques in digital image processing and plays an essential role in medical imaging since textures discriminate information. Most image texture enhancement techniques use classical integral order differential mask operators or fractional differential mask operators using fixed fractional order. These masks can produce excessive enhancement of low spatial frequency content, insufficient enhancement of large spatial frequency content, and retention of high spatial frequency noise. To improve upon existing approaches of texture enhancement, we derive an improved Variable Order Fractional Centered Difference (VOFCD) scheme which dynamically adjusts the fractional differential order instead of fixing it. The new VOFCD technique is based on the second order Riesz fractional differential operator using a Lagrange 3-point interpolation formula, for both grey scale and colour image enhancement. We then use this method to enhance photographs and a set of medical images related to patients with stroke and Parkinson's disease. The experiments show that our improved fractional differential mask has a higher signal to noise ratio value than the other fractional differential mask operators. Based on the corresponding quantitative analysis we conclude that the new method offers a superior texture enhancement over existing methods.

9. A Variable Order Fractional Differential-Based Texture Enhancement Algorithm with Application in Medical Imaging

PubMed Central

Yu, Qiang; Vegh, Viktor

2015-01-01

Texture enhancement is one of the most important techniques in digital image processing and plays an essential role in medical imaging since textures discriminate information. Most image texture enhancement techniques use classical integral order differential mask operators or fractional differential mask operators using fixed fractional order. These masks can produce excessive enhancement of low spatial frequency content, insufficient enhancement of large spatial frequency content, and retention of high spatial frequency noise. To improve upon existing approaches of texture enhancement, we derive an improved Variable Order Fractional Centered Difference (VOFCD) scheme which dynamically adjusts the fractional differential order instead of fixing it. The new VOFCD technique is based on the second order Riesz fractional differential operator using a Lagrange 3-point interpolation formula, for both grey scale and colour image enhancement. We then use this method to enhance photographs and a set of medical images related to patients with stroke and Parkinson’s disease. The experiments show that our improved fractional differential mask has a higher signal to noise ratio value than the other fractional differential mask operators. Based on the corresponding quantitative analysis we conclude that the new method offers a superior texture enhancement over existing methods. PMID:26186221

10. Lie Symmetry Analysis and Conservation Laws of a Generalized Time Fractional Foam Drainage Equation

Wang, Li; Tian, Shou-Fu; Zhao, Zhen-Tao; Song, Xiao-Qiu

2016-07-01

In this paper, a generalized time fractional nonlinear foam drainage equation is investigated by means of the Lie group analysis method. Based on the Riemann—Liouville derivative, the Lie point symmetries and symmetry reductions of the equation are derived, respectively. Furthermore, conservation laws with two kinds of independent variables of the equation are performed by making use of the nonlinear self-adjointness method. Supported by the National Training Programs of Innovation and Entrepreneurship for Undergraduates under Grant No. 201410290039, the Fundamental Research Funds for the Central Universities under Grant Nos. 2015QNA53 and 2015XKQY14, the Fundamental Research Funds for Postdoctoral at the Key Laboratory of Gas and Fire Control for Coal Mines, the General Financial Grant from the China Postdoctoral Science Foundation under Grant No. 2015M570498, and Natural Sciences Foundation of China under Grant No. 11301527

11. Mathematical and numerical studies of nonstandard difference equation models of differential equations. Final technical report, September 1995--September 1997

SciTech Connect

Mickens, R.E.

1997-12-12

The major thrust of this proposal was to continue our investigations of so-called non-standard finite-difference schemes as formulated by other authors. These schemes do not follow the standard rules used to model continuous differential equations by discrete difference equations. The two major aspects of this procedure consist of generalizing the definition of the discrete derivative and using a nonlocal model (on the computational grid or lattice) for nonlinear terms that may occur in the differential equations. Our aim was to investigate the construction of nonstandard finite-difference schemes for several classes of ordinary and partial differential equations. These equations are simple enough to be tractable, yet, have enough complexity to be both mathematically and scientifically interesting. It should be noted that all of these equations differential equations model some physical phenomena under an appropriate set of experimental conditions. The major goal of the project was to better understand the process of constructing finite-difference models for differential equations. In particular, it demonstrates the value of using nonstandard finite-difference procedures. A secondary goal was to construct and study a variety of analytical techniques that can be used to investigate the mathematical properties of the obtained difference equations. These mathematical procedures are of interest in their own right and should be a valuable contribution to the mathematics research literature in difference equations. All of the results obtained from the research done under this project have been published in the relevant research/technical journals or submitted for publication. Our expectation is that these results will lead to improved finite difference schemes for the numerical integration of both ordinary and partial differential equations. Section G of the Appendix gives a concise summary of the major results obtained under funding by the grant.

12. Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients

Slavyanov, S. Yu.; Satco, D. A.; Ishkhanyan, A. M.; Rotinyan, T. A.

2016-12-01

We discuss several examples of generating apparent singular points as a result of differentiating particular homogeneous linear ordinary differential equations with polynomial coefficients and formulate two general conjectures on the generation and removal of apparent singularities in arbitrary Fuchsian differential equations with polynomial coefficients. We consider a model problem in polymer physics.

13. Analytic study on a state observer synchronizing a class of linear fractional differential systems

Zhou, Xian-Feng; Huang, Qun; Jiang, Wei; Liu, Song

2014-10-01

This paper is concerned with Theorem 2 in Matignon and d’André-Novel (1997) [1], which was sufficient and necessary criterion on a state observer for a class of linear fractional differential systems. Based on the stability theory, the dual principle and the pole assignment theory of the fractional differential system, we have proved the validity of sufficiency of Theorem 2 in details. A counterexample is provided to show that the condition of Theorem 2 is not necessary.

14. Construction of finite difference schemes having special properties for ordinary and partial differential equations

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1984-01-01

Work on the construction of finite difference models of differential equations having zero truncation errors is summarized. Both linear and nonlinear unidirectional wave equations are discussed. Results regarding the construction of zero truncation error schemes for the full wave equation and Burger's equation are also briefly reported.

15. Laplace and Z Transform Solutions of Differential and Difference Equations With the HP-41C.

ERIC Educational Resources Information Center

Harden, Richard C.; Simons, Fred O., Jr.

1983-01-01

A previously developed program for the HP-41C programmable calculator is extended to handle models of differential and difference equations with multiple eigenvalues. How to obtain difference equation solutions via the Z transform is described. (MNS)

16. The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

Yan, Zhenya

2003-04-01

In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.

17. Finite Difference Methods for Time-Dependent, Linear Differential Algebraic Equations

DTIC Science & Technology

1993-10-27

Time-Dependent, Linear Differential Algebraic Equations ’ BY PATRICK J. RABIER AND WERNER C. RHEINBOLDT 2 T r e n - sa le; its tot puba"- c. 2 ed...1993 Finite Difference Methods for Time-Dependent, I Linear Differential Algebraic Equations ’ BY PATRICK J. RABIER AND WERNER C. RHEINBOLDT2...LINEAR DIFFERENTIAL ALGEBRAIC EQUATIONS 1 BY PATRICK J. RABIER AND WERNER C. RHEINBOLDT 2 ABSTRACT. Recently the authors developed a global reduction

18. Estimation of Delays and Other Parameters in Nonlinear Functional Differential Equations.

DTIC Science & Technology

1981-12-01

FSTIMATION OF DELAYS AND OTHER PARAMETERS IN NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS by K. T. Banks and P. L. Daniel December 1981 LCDS Report #82...ESTIMATION OF DELAYS AND OTHER PARAMETERS IN NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS H. T. Banks and P. L. Daniel ABSTRACT We discuss a spline...based approximation scheme for nonlinear nonautonomous delay differential equations . Convergence results (using dissipative type estimates on the

19. Fractionation of gold in a differentiated tholeiitic dolerite

USGS Publications Warehouse

Rowe, J.J.

1969-01-01

Gold content was determined, by neutron-activation analysis, in samples from a drill core through the Great Lake sheet, Tasmania, a differentiated tholeiitic dolerite. The gold content of parts of the core seems to be related to the mafic index. The variation of gold content with depth and mafic index is similar to that of copper, indicating that gold and copper may have been concomitantly crystallized from the magma. ?? 1969.

20. Time-fractional wave-diffusion equation in an inhomogeneous half-space