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

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