On The Eventual Periodicity of Fractional Order Dispersive Wave Equations Using RBFs and Transform

TL;DR

This paper introduces a numerical method combining RBF-FD and Laplace transform to solve fractional dispersive wave equations, demonstrating its efficiency and accuracy in analyzing eventual periodicity and handling nonlinear PDEs.

Contribution

The paper presents a novel numerical scheme that integrates RBF-FD with Laplace transform for fractional PDEs, improving efficiency and stability over traditional methods.

Findings

The method accurately captures the eventual periodicity of fractional dispersive wave equations.

It efficiently solves large-scale nonlinear fractional PDEs with sparse matrices.

Numerical examples confirm high-order accuracy and stability of the approach.

Abstract

In this research work, let us 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 us 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 results