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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
Author(s) -
Jacobs Byron A.
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22046
Subject(s) - mathematics , laplace transform , mathematical analysis , fractional calculus , neumann boundary condition , partial differential equation , discretization , laplace's equation , dirichlet boundary condition , boundary value problem
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 O ( △ x 4 ) 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