On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

Keywords: RBFs Methods, RBF-FD Method, Time-Fractional KdV and Burgers Equations, Laplace Transform

Abstract

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain.

Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense.

Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Methods

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Author Biographies

Hameed Ullah Jan, University of Engineering and Technology Peshawar

Department of Basic Sciences and Islamiat

Marjan Uddin, University of Engineering and Technology Peshawar

Department of Basic Sciences and Islamiat

Irshad Ali Shah, University of Science and Technology Bannu

Department of Mathematics

Salam Ullah Khan, University of Science and Technology Bannu

Department of Computer Science

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Published
2022-05-31
How to Cite
Jan, H. U., Uddin, M., Shah, I. A., & Khan, S. U. (2022). On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform. EUREKA: Physics and Engineering, (3), 133-148. https://doi.org/10.21303/2461-4262.2022.002394
Section
Mathematics