Abstract
Solutions to time-fractional diffusion-wave equation with a source term in spherical coordinates are obtained for an infinite medium with a spherical cavity. The solutions are found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate Ï•, the Legendre transform with respect to the spatial coordinate μ, and the Weber transform of the order n+1/2 with respect to the radial coordinate r. In the central symmetric case with one spatial coordinate r the obtained resultscoincide with those studied earlier.
Highlights
Fractional order partial differential equation, in particular, the time-fractional diffusionwave equation are of great interest in studies of important physical phenomena in amorphous, colloid, glassy and porous materials, in fractals, percolation clusters, random and disordered media, in comb structures, dielectrics, semiconductors, polymers, biological systems, in geology, geophysics, medicine, economy, finance, etc
The major utility of this type fractional derivative is caused by the treatment of differential equations of fractional order for physical applications, where the initial conditions are usually expressed in terms of a given function and its derivatives of integer order, even if the governing equation is of fractional order [19, 25]
The non-axisymmetric solutions to the source, Cauchy, and Dirichlet problems for timefractional diffusion-wave equation have been found for a medium with a spherical cavity
Summary
Fractional order partial differential equation, in particular, the time-fractional diffusionwave equation are of great interest in studies of important physical phenomena in amorphous, colloid, glassy and porous materials, in fractals, percolation clusters, random and disordered media, in comb structures, dielectrics, semiconductors, polymers, biological systems, in geology, geophysics, medicine, economy, finance, etc. (see, for example, Bagley and Torvik [1], Carpinteri and Cornetti [2], Magin [3], Mainardi [4, 5], Metzler and Klafter [6, 7], Povstenko [8], Rabotnov [9, 10], Rossikhin and Shitikova [11], Uchaikin [12], West et al [13], Zaslavsky [14] and references therein). The asymptotic behavior for the solution of fractional differential equations in the nonlinear case was studied by Baleanu et al [22]. Consider the time-fractional diffusion-wave Eq(6) with a source term in spherical coordinates r, θ, and φ: The Caputo fractional derivative is a regularization in the time origin for the Riemann– Liouville fractional derivative by incorporating the relevant initial conditions [24]. The major utility of this type fractional derivative is caused by the treatment of differential equations of fractional order for physical applications, where the initial conditions are usually expressed in terms of a given function and its derivatives of integer (not fractional) order, even if the governing equation is of fractional order [19, 25].
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