Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations

Abstract

The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented.

Simple Closed-Form Formulas for Conditional Moments of Inhomogeneous Nonlinear Drift Constant Elasticity of Variance Process

“…One main concern when we investigate the conditional moments described by the IND-CEV process is that the integral terms (6) in Theorem 2 cannot be directly evaluated. Thus, a very accurate numerical integration scheme is applied via the Chebyshev integration method; see [25][26][27][28] for more details.…”

“…One major concern is that our proposed formulas in Theorem 2 and Corollaries 1 and 2 are not in closed form when integral terms cannot be analytically computed. In this case, a numerical method can be applied to calculate the coefficients numerically; see [28,29].…”

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

“…Moreover, in [10], the authors solved a class of non-linear variable-order fractional reaction-diffusion equation based on using the shifted Chebyshev polynomials of the fifth kind. For some other articles concerned with the different kinds of Chebyshev polynomials, see for example [11][12][13][14].…”

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

“…This modified FIM also provides much higher accuracy than the FDM and those original FIMs with small computational nodes. Recently, the modified FIM was widely utilized to apply with many applications, see [20][21][22][23]. Also, it was demonstrated that results obtained by the modified FIM achieve significant improvement in terms of accuracy more than several existing methods.…”