On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Ravshan Ashurov,Yusuf Fayziev

doi:10.3390/fractalfract6010041

Abstract

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

Highlights

Let H be a separable Hilbert space with the scalar product (·, ·) and the norm || · ||and A : H → H be an arbitrary unbounded positive selfadjoint operator in H
To the best of our knowledge, such an inverse problem was discussed only in the paper [17]. The authors considered this problem for the subdiffusion equation including the Caputo fractional derivative, the elliptical part of which is a two-variable differential expression with constant coefficients
The inverse problems of determining the right-hand side of various subdiffusion equations have been considered by a number of authors

Summary

Introduction

Let H be a separable Hilbert space with the scalar product (·, ·) and the norm || · ||. In contrast to the backward problems, that the solutions of problems (2) and (3) continuously depend on the right-hand side of the equation and on the function φ. The inverse problems of determining the right-hand side of the equation and function φ in the boundary conditions are investigated. To the best of our knowledge, such an inverse problem was discussed only in the paper [17] The authors considered this problem for the subdiffusion equation including the Caputo fractional derivative, the elliptical part of which is a two-variable differential expression with constant coefficients. In this case, we assume that the unknown function f does not depend on t.

Preliminaries

Inverse Problem of Determining the Heat Source Density

The Inverse Problem of Determining the Boundary Function φ

Conclusions

Methods