A time-nonlocal inverse problem for a hyperbolic equation with an integral overdetermination condition

Yashar Mehraliyev,Elvin Azizbayov

doi:10.14232/ejqtde.2021.1.28

Abstract

This article is concerned with the study of the unique solvability of a timenonlocal inverse boundary value problem for second-order hyperbolic equation with an integral overdetermination condition. To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary system with trivial data and prove its equivalence (in a certain sense) to the original problem. Then using the Banach fixed point principle, the existence and uniqueness of a solution to this system is shown. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the inverse coefficient problem is proved for the smaller value of time.

Highlights

It is often required to recover the coefficients in an ordinary or partial differential equation from the final overspecified data
Substituting the expression of (3.4) into (3.1), we find the component u(x, t) of the classical solution to problem (2.1)–(2.3), (2.14) to be u(x, t) = ∑
Considered problem was reduced to an auxiliary problem in a certain sense and using the contraction mappings principle a unique existence conditions for a solution of equivalent problem are established

Summary

Introduction

It is often required to recover the coefficients in an ordinary or partial differential equation from the final overspecified data Problems of these types are called inverse problems of mathematical physics and are one of the most complicated and practically important problems. The theory of inverse problems is widely used to solve practical problems in almost all fields of science, in particular, in physics, medicine, ecology, and economics. As well as the direct nonlocal boundary value problems for hyperbolic equations with integral conditions (with respect to time variable) are considered in the papers [12, 22] and the references therein. A distinctive feature of this article is the consideration the inverse boundary value problem for a hyperbolic equation with both spatial and time nonlocal conditions

Mathematical formulation

Existence and uniqueness of the solution of the inverse problem

Conclusion