Linearized stable spectral method to analyze two‐dimensional nonlinear evolutionary and reaction‐diffusion models

Abstract

AbstractThe work is devoted to the development of a new spectral method based on higher dimensional orthogonal polynomials. Firstly, the concept of traditional Chelyshkov polynomials is extended for the function of more than one variable while definitions and theorems are presented with proofs. The operational matrices of derivative have been constructed assisted by defined higher‐order polynomials and used to the development of a spectral method. The method is further coupled with a Picard‐iterative scheme to tackle the high nonlinearities and termed as the Picard–Chelyshkov polynomial method (PCPM). The convergence and error‐bound have been analyzed through theorems and their proofs in order to prove the mathematical authenticity of the method. The PCPM is applied for some two‐dimensional unsteady nonlinear fractional partial differential equations and efficient results have been attained. In addition, it is evident from the comparative analysis with existing literature that the proposed method is fair enough in terms of accuracy, efficiency, and cost to deal with the problem in higher dimensions whilst could be further modified for other classes of dynamical problems.