@Article{cmc.2010.015.221,
AUTHOR = {Chein-Shan Liu},
TITLE = {The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations},
JOURNAL = {Computers, Materials \& Continua},
VOLUME = {15},
YEAR = {2010},
NUMBER = {3},
PAGES = {221--240},
URL = {http://www.techscience.com/cmc/v15n3/22538},
ISSN = {1546-2226},
ABSTRACT = {We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: <i>D<sub>t</sub><sup>α</sup>u + εuu<sub>x</sub> = vu<sub>xx</sub> + ηD<sub>x</sub><sup>β</sup>u, 0 < α, β ≤ 1</i>, and <i>u<sub>t</sub> + D<sub>*</sub><sup>β</sup>(D<sub>*</sub><sup>1-β</sup>u)<sup>2</sup>/2 = vu<sub>xx</sub>, 0 < β ≤ 1</i>. The time-fractional derivative <i>D<sub>t</sub><sup>α</sup>u</i> and space-fractional derivative <i>D<sub>x</sub><sup>β</sup>u</i> are defined in the Caputo sense, while <i>D<sub>*</sub><sup>β</sup>u</i> is the Riemann-Liouville space-fractional derivative. A fictitious time <i>τ</i> is used to transform the dependent variable <i>u(x,t)</i> into a new one by <i>(1+τ)<sup>γ</sup>u(x,t) =: v(x,t,τ)</i>, where <i>0 < γ ≤ 1</i> is a parameter, such that the original equation is written as a new functional-differential type partial differential equation in the space of <i>(x,t,τ)</i>. When the group-preserving scheme is used to integrate these equations under a semi-discretization of <i>u(x,t,τ)</i> at the spatial-temporal grid points, we can achieve rather accurate solutions.},
DOI = {10.3970/cmc.2010.015.221}
}