Propagation of nonlinear shock waves for the generalised Oskolkov equation and its dynamic motions in the presence of an external periodic perturbation

NASA Astrophysics Data System (ADS)

Ak, Turgut; Aydemir, Tugba; Saha, Asit; Kara, Abdul Hamid

2018-06-01

Propagation of nonlinear shock waves for the generalised Oskolkov equation and dynamic motions of the perturbed Oskolkov equation are investigated. Employing the unified method, a collection of exact shock wave solutions for the generalised Oskolkov equations is presented. Collocation finite element method is applied to the generalised Oskolkov equation for checking the accuracy of the proposed method by two test problems including the motion of shock wave and evolution of waves with Gaussian and undular bore initial conditions. Considering an external periodic perturbation, the dynamic motions of the perturbed generalised Oskolkov equation are studied depending on the system parameters with the help of phase portrait and time series plot. The perturbed generalised Oskolkov equation exhibits period-3, quasiperiodic and chaotic motions for some special values of the system parameters, whereas the generalised Oskolkov equation presents shock waves in the absence of external periodic perturbation.

Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd-Bullough-Mikhailov equation

NASA Astrophysics Data System (ADS)

Ilhan, O. A.; Bulut, H.; Sulaiman, T. A.; Baskonus, H. M.

2018-02-01

In this study, the modified exp ( - Φ (η )) -expansion function method is used in constructing some solitary wave solutions to the Oskolkov-Benjamin-Bona-Mahony-Burgers, one-dimensional Oskolkov equations and the Dodd-Bullough-Mikhailov equation. We successfully construct some singular solitons and singular periodic waves solutions with the hyperbolic, trigonometric and exponential function structures to these three nonlinear models. Under the choice of some suitable values of the parameters involved, we plot the 2D and 3D graphics to some of the obtained solutions in this study. All the obtained solutions in this study verify their corresponding equation. We perform all the computations in this study with the help of the Wolfram Mathematica software. The obtained solutions in this study may be helpful in explaining some practical physical problems.

On function classes related pertaining to strong approximation of double Fourier series

NASA Astrophysics Data System (ADS)

Baituyakova, Zhuldyz

2015-09-01

The investigation of embedding of function classes began a long time ago. After Alexits [1], Leindler [2], and Gogoladze[3] investigated estimates of strong approximation by Fourier series in 1965, G. Freud[4] raised the corresponding saturation problem in 1969. The list of the authors dealing with embedding problems partly is also very long. It suffices to mention some names: V. G. Krotov, W. Lenski, S. M. Mazhar, J. Nemeth, E. M. Nikisin, K. I. Oskolkov, G. Sunouchi, J. Szabados, R. Taberski and V. Totik. Study on this topic has since been carried on over a decade, but it seems that most of the results obtained are limited to the case of one dimension. In this paper, embedding results are considered which arise in the strong approximation by double Fourier series. We prove theorem on the interrelation between the classes Wr1,r2HS,M ω and H(λ, p, r1, r2, ω(δ1, δ2)), in the one-dimensional case proved by L. Leindler.