Approximate solutions to fractional subdiffusion equations |
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Authors: | J Hristov |
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Institution: | (1) Department of Nuclear and Energy Engineering, Cheju National University, Cheju, 690-756, Korea;(2) Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan, 701 Taiwan, People’s Republic of China; |
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Abstract: | The work presents integral solutions of the fractional subdiffusion equation by an integral method, as an alternative approach
to the solutions employing hypergeometric functions. The integral solution suggests a preliminary defined profile with unknown
coefficients and the concept of penetration (boundary layer). The prescribed profile satisfies the boundary conditions imposed
by the boundary layer that allows its coefficients to be expressed through its depth as unique parameter. The integral approach
to the fractional subdiffusion equation suggests a replacement of the real distribution function by the approximate profile.
The solution was performed with Riemann-Liouville time-fractional derivative since the integral approach avoids the definition
of the initial value of the time-derivative required by the Laplace transformed equations and leading to a transition to Caputo
derivatives. The method is demonstrated by solutions to two simple fractional subdiffusion equations (Dirichlet problems):
1) Time-Fractional Diffusion Equation, and 2) Time-Fractional Drift Equation, both of them having fundamental solutions expressed
through the M-Wright function. The solutions demonstrate some basic issues of the suggested integral approach, among them:
a) Choice of the profile, b) Integration problem emerging when the distribution (profile) is replaced by a prescribed one
with unknown coefficients; c) Optimization of the profile in view to minimize the average error of approximations; d) Numerical
results allowing comparisons to the known solutions expressed to the M-Wright function and error estimations. |
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