Transparent Boundary Conditions for a Diffusion Problem Modified by Hilfer Derivative
Vol. 21 (2014), No. 1, Page 129–152.
Ghanam, Ryad A. ; Malik, Nadeem A. ; Tatar, Nasser-eddine
Transparent Boundary Conditions for a Diffusion Problem Modified by Hilfer Derivative
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Abstract:
We consider a homogeneous fractional diffusion problem in an infinite reservoir sometimes called a ``modified" diffusion equation. The equation involves a (nonlocal in time) memory term in the form of a time fractional derivative (of the Laplacian). For the sake of reducing the computational domain to a bounded one we establish appropriate ``artificial" boundary conditions. This is to avoid the effect of reflected waves in case of a ``solid" standard boundary. Then, an equivalent problem is studied in this bounded domain. To this end we use the Laplace-Fourier transform, the two-parameter Mittag-Leffler function and some properties of fractional derivatives.
Keywords: Caputo fractional derivative, Hilfer fractional derivative, fractional diffusion problem, Mittag-Leffler function, artificial boundary condition, reduced equivalent problem.
Mathematics Subject Classification (2010): 35K20, 26A33, 42A38.
Mathematical Reviews Number: MR3235552
Received: 2014-01-06
