Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

Abstract

This article contributes to a balanced space–time spectral collocation method for solving nonlinear time‐fractional Burgers equations with given initial‐boundary conditions. Most of existing approximate methods for solving partial differential equations are unbalanced, since they have used a low order scheme such as finite difference methods for integrating the temporal variable and a high order numerical framework such as spectral Galerkin (or meshless) method for discretization of space variables. So in the current paper, our suggested scheme is balanced in both time and space variables. Due to the non‐smoothness of solutions of time‐fractional Burgers equations, we apply efficient basis functions as the fractional Lagrange functions for interpolating time variable. By collocating the main equation and the initial‐boundary conditions together with the implementation of the corresponding operational matrices of spatial and fractional temporal variables, the assumed model is transformed into the associated system of nonlinear algebraic equations, which can be solved via efficient iterative solvers such as the Levenberg–Marquardt method. Also, we fully analyze the convergence of method. Moreover, we consider several test problems for examining the suggested scheme that confirms its high accuracy and low computational cost with respect to recent numerical methods in the literature.

Analyzing Similarity Solution of Modified Fisher Equation

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Third‐kind Chebyshev cardinal functions for variable‐order time fractional RLW‐Burgers equation

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