Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative

By Trifce Sandev, Ralf Metzler, Ž. Tomovski. Journal of Physics A: Mathematical and Theoretical, 2011

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In this paper, the solution of a fractional diffusion equation with a Hilfer-generalized Riemann–Liouville time fractional derivative is obtained in terms of Mittag–Leffler-type functions and Fox's H-function. The considered equation represents a quite general extension of the classical diffusion (heat conduction) equation. The methods of separation of variables, Laplace transform, and analysis of the Sturm–Liouville problem are used to solve the fractional diffusion equation defined in a bounded domain. By using the Fourier–Laplace transform method, it is shown that the fundamental solution of the fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative defined in the infinite domain can be expressed via Fox's H-function. It is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the Caputo and Riemann–Liouville sense are special cases of those diffusion equations with the Hilfer-generalized Riemann–Liouville time fractional derivative. The asymptotic behaviour of the solutions are found for large values of the spatial variable. The fractional moments of the fundamental solution of the fractional diffusion equation are obtained. The obtained results are relevant in the context of glass relaxation and aquifer problems.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1ef7191b2148b67c4810c364810c1fbcae73b8c7

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Citations in the corpus

Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator (2012, background): “Applications of D a+ are given in [7,2,12,13].”
Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions (2017, background): “For more details of the theory and applications of Riemann−Liouville fractional differential equations, one can refer to [14]–[15], [24].”
Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives (2013, background): “More recently, new results for fractional differential equations involving GRLFDs have been obtained in [2, 18, 22, 23].”
A new fractional operator of variable order: Application in the description of anomalous diffusion (2016, methodology): “of the AD was reported in [10], and the AD equation with a generalized FOD of Riemann–Liouville type was proposed in [11].”
Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions (2017, methodology): “For more details of the theory and applications of Riemann–Liouville fractional differential equations and inclusions, we can refer to [15, 21, 31, 32].”
NONLOCAL PROBLEM FOR A MIXED TYPE FOURTH-ORDER DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL OPERATOR (2020, background): “In [23], boundary value problems for a fractional diffusion equation with the Hilfer fractional derivative in finite and infinite domains were studied.”
Existence results of Hilfer integro-differential equations with fractional order (2020, methodology): “The couple of parameter family of fractional derivative with ζ, η permits one to interpolate between the R-L and the Caputo derivative discussed in Hilfer [8, 10, 11, 22, 31] fractional derivative.”