On a Non–Local Problem for a Multi–Term Fractional Diffusion-Wave Equation

Abstract

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions.

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

“…In an arbitrary domain Ω initial-boundary value problems for subdiffusion equations (the fractional part of the equation is a multi-term and initial conditions are non-local) with the Caputo derivatives has been investigated by M. Ruzhansky et al [24]. The authors proved the existence and uniqueness of the generalized solution to the problem.…”

“…A. Alimov and S. R. Umarov for discussions of these results. We wish also to thank B. Turmetov for making us aware of the paper of M. Ruzhansky et al [24].…”

“…Therefore, similar to Theorem 2 statement holds true for any operator with these properties. As an example we may consider differential operator with involution and the Bessel operator (see the paper of M. Ruzhansky et al [24]), or the initial-boundary value problem in N -dimensional torus:…”

Bitsadze–Samarskii type problem for the integro-differential diffusion–wave equation on the Heisenberg group

“…The condition (2.2) can be interpreted as a multi-point non-resonance condition. Note that a similar problem for the time-fractional multi-term diffusion-wave equation was investigated by the authors in [29].…”

Eigenfunctions of the time‐fractional diffusion‐wave operator

“…, m ∈ N. (2.3) Another special function that we use in our work is the multivariate Mittag-Leffler function (see Luchko and Gorenflo and Ruzhansky et al 27,28 ). Definition 2.1.…”