Abstract
In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers’ equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers’ equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods.
Highlights
The aim of this study is to present the numerical solutions by aid of the Gegenbauer wavelet collocation method with an operational matrix of fractional integration and the Gegenbauer wavelet
For α = 1, α = 0.90 and α = 0.75, the physical behaviors of the absolute errors obtained using the Gegenbauer wavelet Galerkin method and the Gegenbauer wavelet collocation method at different times are depicted in Figures 1–3, respectively
The main goal of this paper is to build up for obtaining numerical solutions of the coupled system different times are depicted in Figures 1–3, respectively
Summary
Galerkin method for the following coupled system of Burgers’ equations with time-fractional derivative [1]:. To solve the coupled system of Burgers’ equations with time-fractional derivative numerically, there are various approaches which have been studied by many authors. Islam and Akbar [23] applied the generalized (G0 /G)-expansion method to obtain exact wave solutions of the space-time fractional-coupled Burgers equations. We focus on the numerical analysis of the coupled system of Burgers’ equations with time-fractional derivative using the Gegenbauer wavelet collocation method with the operational matrix of fractional integration and the Gegenbauer wavelet.
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