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

Abstract

The 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.

Applications of Two Methods in Exact Wave Solutions in the Space-Time Fractional Drinfeld–Sokolov–Wilson System

“…Step 5. To determine the positive integer m, we substitute (8) along with ( 12) into (9) and balance between the highest order derivatives and the highest order nonlinear terms appearing in (9). Furthermore, if the degree of v(ξ) is defined as deg[v (ξ)] � m, the degree of the other expressions are as follows:…”

“…In recent years, a variety of fractional partial di erential equations have been used in many studies, which has increased the incentive to use and develop numerical and approximate methods for such problems. Several numerical methods have been used to solve fractional di erential equations for example the PicardChelyshkov polynomial method (PCPM), the Chelyshkov polynomial method (CPM) that further coupled with a nite di erence scheme (FDM) and named as a semispectral method (SSM), and a hybrid method based on operational matrices of derivative has been provided by Muhammad Hamid et al [9][10][11]. In mathematically-oriented scienti c elds, obtaining exact solutions of nonlinear fractional di erential equations (NLFDEs) has become one of the most exciting and many active areas of research investigation.…”

Fractional order stagnation point flow of the hybrid nanofluid towards a stretching sheet

“…Usman et al 31 developed a unique computational approach for computing stable solutions in multi-dimensions of time-fractional viscous Burger's models equation. Hamid et al 32 generalized the premise of classical Chelyshkov polynomials to functions with more than one variable, while providing evidence for theorems and definitions. Keeping in view, the applicability of fractional derivative and the versatility of silver and titanium dioxide hybrid nanofluid using blood as base fluid, we have modeled the current problem.…”

Travelling Wave Solutions of Nonlinear Evolution Problem Arising in Mathematical Physics through (G′/G)-Expansion Scheme

“…Similarly, the same model can be used for two and three dimensional problems where the function U will be, respectively, expressed as U (x, y, t) and U (x, y, z, t) while details are given in references [7,8].…”

“…e Klein-Gordon equation of quadratic and cubic nonlinearity will be constructed, respectively, by taking H (U) as U − U 2 and U − U 3 . e readers can see the cited references [5,7,9] to understand the detailed physical aspects of the models and historical overview. However, solutions of these PDEs are of great significance and importance in physical mechanisms as mentioned above.…”