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High‐order compact finite difference and laplace transform method for the solution of time‐fractional heat equations with dirchlet and neumann boundary conditions
Authors:Byron A Jacobs
Institution:1. School of Computer Science and Applied Mathematics, University of the Witswatersrand, Wits, Johannesburg, South Africa;2. DST‐NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE‐MaSS), University of the Witwatersrand, Wits, Johannesburg, South Africa
Abstract:The work presents a novel coupling of the Laplace Transform and the compact fourth‐order finite‐difference discretization scheme for the efficient and accurate solution of linear time‐fractional nonhomogeneous diffusion equations subject to both Dirichlet and Neumann boundary conditions. A translational transformation of the dependent variable ensures the Caputo derivative is aligned with the Riemann‐Louiville fractional derivative. The resulting scheme is computationally efficient and shown to be uniquely solvable in all cases, accurate and convergent to urn:x-wiley:0749159X:media:num22046:num22046-math-0001 in the spatial domain. The convergence rates in the temporal domain are contour dependent but exhibit geometric convergence. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1184–1199, 2016
Keywords:compact finite‐difference  fourth‐order accurate  fractional derivative  Laplace transform
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