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

Chein-Shan Liu1
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: Dtαu + εuux = vuxx + ηDxβu, 0 < α, β ≤ 1, and ut + D*β(D*1-βu)2/2 = vuxx, 0 < β ≤ 1. The time-fractional derivative Dtαu and space-fractional derivative Dxβu 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.

Keywords

Fractional Burgers equation, Fictitious time integration method (FTIM), Caputo derivative, Riemann-Liouville derivative, Group-preserving scheme

Cite This Article

. and . , "The fictitious time integration method to solve the space- and time-fractional burgers equations," Computers, Materials & Continua, vol. 15, no.3, pp. 221–240, 2010.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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