Abstract
We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using the Green function, we give a solution of the first boundary-value problem in a multidimensional cylindrical domain. The fractional differentiation is given by the Dzhrbashyan–Nersesyan fractional differentiation operator. In particular, this covers the cases of equations with the Riemann–Liouville and Caputo derivatives.
Highlights
Consider the equation where ∂α ∂yα− ∆x u( x, y) = f ( x, y), (1)stands for a fractional derivative with respect to y of order α ∈ (0, 2), y > 0, and ∆x = n ∂2∑ ∂x2 j =1 j is the Laplace operator with respect to x = ( x1, x2, ..., xn ) ∈ S ⊂ Rn .The fractional diffusion (0 < α ≤ 1) and diffusion-wave (0 < α < 2) equations have attracted great attention in recent years
The sequence {γ0, γ1, ..., γm } or, in other words, the used form of fractional differentiation affects only the part of the solution that corresponds to the initial conditions
We find the regular part of the Green function as a solution of an integral equation generated by a jump relation for the double-layer potential of the considered equation
Summary
Stands for a fractional derivative with respect to y of order α ∈ (0, 2), y > 0, and. The Green functions of boundary-value problems for the Equation (1) have been constructed for one-dimensional case (n = 1) in [23], and —for problems in multidimensional rectangular domains with n > 1—in [32]. We note [31], in which the first boundary-value problem for the fractional diffusion equation (0 < α < 1) with the Caputo derivative has been solved using the layer potential technique. The results obtained here cover the cases of equations with the Riemann–Liouville and Caputo derivatives
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