Open AccessThe Numerical Solutions of a Two-Dimensional Space-Time Riesz-Caputo Fractional Diffusion Equation
Author(s) -
Necati Özdemir,
Derya Avcı,
Beyza Billur İskender
Publication year - 2011
Publication title -
an international journal of optimization and control theories and applications (ijocta)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.287
H-Index - 6
eISSN - 2146-5703
pISSN - 2146-0957
DOI - 10.11121/ijocta.01.2011.0028
Subject(s) - mathematics , laplace transform , anomalous diffusion , fractional calculus , mathematical analysis , space (punctuation) , representation (politics) , diffusion , fourier transform , riesz potential , order (exchange) , diffusion equation , function (biology) , differential equation , mittag leffler function , physics , computer science , knowledge management , innovation diffusion , economy , finance , evolutionary biology , politics , biology , political science , law , economics , thermodynamics , service (business) , operating system
This paper is concerned with the numerical solutions of a two dimensional space-time fractional differential equation used to model the dynamic properties of complex systems governed by anomalous diffusion. The space-time fractional anomalous diffusion equation is defined by replacing second order space and first order time derivatives with Riesz and Caputo operators, respectively. By using Laplace and Fourier transforms, a general representation of analytical solution is obtained as Mittag-Leffler function. Gr\"{u}nwald-Letnikov (GL) approximation is also used to find numerical solution of the problem. Finally, simulation results for two examples illustrate the comparison of the analytical and numerical solutions and also validity of the GL approach to this problem.
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