A stable computational approach to analyze semi‐relativistic behavior of fractional evolutionary problems
Muhammad Hamid, Muhammad Usman, Wei Wang, Zhenfu Tian
Abstract
Abstract The current work is related to the development of a hybrid semi‐spectral computational scheme based on multidimensional Chelyshkov polynomials (CPs). One dimensional traditional CPs have been efficiently converted to multidimensional Chelyshkov polynomials (MDCPs) and used to construct some novel operational matrices of fractional‐order derivatives. A method is developed via obtained polynomials as the Chelyshkov polynomial method (CPM) and further coupled with a finite difference scheme (FDM) and named as a semi‐spectral method (SSM). The method is used to evaluate the results of nonlinear evolutionary problems of fractional order where the space domain is collocated via CPM and the time variable is discretized through FDM. The stability analysis of the proposed scheme is proved to show its mathematical authenticity. Additionally, the reported study is evidence that the proposed methodology is reasonable in terms of accuracy, efficiency and low‐cost to tackle the nonlinear problem in higher dimensions while could be further modified for other class of dynamical problems.