Existence and uniqueness of the solution for a time-fractional diffusion equation

Abstract

In the paper existence and uniqueness of the solution for a time-fractional diffusion equation on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a fundamental solution of the problem is known, we may seek the solution as the double layer potential. This approach leads to a Volterra integral equation of the second kind associated with a compact operator. Then classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary datum is continuous and satisfies a compatibility condition. This proves that the original problem has a unique solution and the solution is given by the double layer potential.MSC 2010 : Primary 45K05; Secondary 26A33, 45D05, 33C60

Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

“…For other methods and results in the theory of initial-boundary-value problems for the PDEs of fractional order we refer e.g. to [1], [6], [8], [21], [23], [25] to mention only few of many recent publications.…”

On a class of time-fractional differential equations

“…For example, it is known that the fractional diffusion equations Solutions of fractional diffusion equations have been considered by several authors such as Wyss et al [36,38] and Fujita [13,14], see also the very recent papers [18,28]. In [31], Orsingher and Beghin give the fundamental solution to the time-fractional telegraph equation ∂ 2α ∂t 2α u(t, x) + 2λ ∂ α ∂t α u(t, x) = c 2 ∂ 2 ∂x 2 u(t, x) with some proper initial data.…”

Thermal blow-up in a subdiffusive medium due to a nonlinear boundary flux

“…The method of conversion uses an appropriate Green's function associated with (1.1) to achieve an integral representation of (1.1)-(1.5) while imposing either (1.6) or (1.7). The utility of this approach has been demonstrated in [3] and references therein.…”