Abstract
The space-time-fractional diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplacian is considered in the case of axial symmetry. Mass absorption (mass release) is described by a source term proportional to concentration. The integral transform technique is used. Different particular cases of the solution are studied. The numerical results are illustrated graphically.
Highlights
The conventional theory of diffusion is based on the classical Fick law, which relates the matter flux j to the concentration gradient j = −k grad c, (1)where k is the diffusion conductivity.In combination with the balance equation for mass, the classical Fick law leads to the standard diffusion equation ∂c = a∆c (2)∂t with a being the diffusivity coefficient.Mass transport in a medium with a first order chemical reaction is described by an additional linear source term in the diffusion equation [1]: = a ∆ c − bc, ∂t (3)where the values of the coefficient b > 0 and b < 0 correspond to mass absorption and mass release, respectively
If the time nonlocality is accompanied with the spatial nonlocalty, the general space-time-fractional heat conduction equation is obtained:
We have considered the fundamental solutions to the Cauchy problem and to the source problem for the space-time fractional diffusion equation with the linear source term
Summary
The time-nonlocal dependence between the matter flux j and the concentration gradient grad c with the long-tail power kernel (see [8,9,10,11]) can be interpreted in terms of fractional integrals and derivatives and results in the time-fractional diffusion equation. It is obvious that Equation (8) is a fractional generalization of the standard formula for the Fourier transform of the Laplace operator corresponding to β = 2:. Equation (8) for the Fourier transform of the fractional Laplace operator in the case of axial symmetry simplifies:. If the time nonlocality is accompanied with the spatial nonlocalty, the general space-time-fractional heat conduction equation is obtained (see [18,19,22,23]):.
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