Open Access

ARTICLE

The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations

Chein-Shan Liu¹

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw

Computers, Materials & Continua 2010, 15(3), 221-240. https://doi.org/10.3970/cmc.2010.015.221

Abstract

We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: D_t^αu + εuu_x = vu_xx + ηD_x^βu, 0 < α, β ≤ 1, and u_t + D_*^β(D_*^1-βu)²/2 = vu_xx, 0 < β ≤ 1. The time-fractional derivative D_t^αu and space-fractional derivative D_x^βu are defined in the Caputo sense, while D_*^βu is the Riemann-Liouville space-fractional derivative. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ)^γu(x,t) =: v(x,t,τ), where 0 < γ ≤ 1 is a parameter, such that the original equation is written as a new functional-differential type partial differential equation in the space of (x,t,τ). When the group-preserving scheme is used to integrate these equations under a semi-discretization of u(x,t,τ) at the spatial-temporal grid points, we can achieve rather accurate solutions.

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Keywords

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