Initial-boundary Value Problem for a Time-fractional Subdiffusion Equation with an Arbitrary Elliptic Differential Operator

Abstract

An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. In the case of an initial-boundary value problem on N -dimensional torus, one can easily see that these conditions are not only sufficient, but also necessary.

Initial-boundary value problem for a time-fractional subdiffusion equation on the torus

“…A result similar to Theorem 1.2 was obtained in the recent paper [11] for a more general subdiffusion equation. But the conditions on the functions f (x, t) and ϕ(x) that guarantee the existence and uniqueness of the solution to problem (1.1)-(1.2) found in [11] are more stringent.…”

“…A result similar to Theorem 1.2 was obtained in the recent paper [11] for a more general subdiffusion equation. But the conditions on the functions f (x, t) and ϕ(x) that guarantee the existence and uniqueness of the solution to problem (1.1)-(1.2) found in [11] are more stringent. This is due to the fact that in the present paper we give a more precise estimate for the Mittag-Leffler function Eρ,ρ(−t), t > 0 (see also [12]).…”

“…The uniqueness of the solution can be proved by the standard technique based on completeness in L2(T N ) of the set of eigenfunctions {γe inx } (see, for example, [11]).…”

Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation

“…A result similar to the above, in the case when the fractional part of the equation ( 1) is the Caputo derivative, was obtained by M. Ruzhansky et al [11]. In the case when A is an arbitrary elliptic differential operator, this theorem was proved in [12].…”

Initial-boundary value problem for a subdiffusion equation with the Caputo derivative

“…The authors of papers [7] and [8] used the Fourier method to construct a classical solution of the subdiffusion equations with the Riemann-Liouville derivative and various elliptic operators.…”