Modal spectral Tchebyshev Petrov–Galerkin stratagem for the time-fractional nonlinear Burgers’ equation

Document Type : Research Article

Authors

• Y.H. Youssri ¹
• A.G. Atta ²

¹ Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.

² Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt.

10.22067/ijnao.2023.83389.1292

Abstract

Herein, we construct an explicit modal numerical solver based on the spec-tral Petrov–Galerkin method via a specific combination of shifted Cheby-shev polynomial basis for handling the nonlinear time-fractional Burger-type partial differential equation in the Caputo sense. The process reduces the problem to a nonlinear system of algebraic equations. Solving this alge-braic equation system will yield the approximate solution’s unknown coef-ficients. Many relevant properties of Chebyshev polynomials are reported, some connection and linearization formulas are reported and proved, and all elements of the obtained matrices are evaluated neatly. Also, conver-gence and error analyses are established. Various illustrative examples demonstrate the applicability and accuracy of the proposed method and depict the absolute and estimated error figures. Besides, the current ap-proach’s high efficiency is proved by comparing it with other techniques in the literature.

Keywords

• Time-fractional Burgers’ equation
• Chebyshev polynomials
• Petrov–Galerkin method
• Convergence analysis

Main Subjects

• Numerical methods for differential equations and its applications

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