On some linear two-point inverse problem for a multidimensional heat conduction equation with semi-nonlocal boundary conditions

Abstract

It is known that V.A. Ilyin and E.I. Moiseev studied generalized nonlocal boundary value problems for the Sturm-Liouville equation, the nonlocal boundary conditions specified at the interior points of the interval under consideration. For such problems, uniqueness and existence theorems for a solution to the problem were proven. There are many difficulties in studying these generalized nonlocal boundary value problems for partial differential equations, especially in obtaining a priori estimates. Therefore, it is necessary to use new methods for solving generalized nonlocal problems (forward problems). As we know, it is not difficult to establish a connection between forward and inverse problems. Therefore, when solving generalized nonlocal boundary value problems for partial differential equations, reducing them to multipoint inverse problems is necessary. The first results in the direction belong to S.Z. Dzhamalov. In his works, he proposed and investigated multipoint inverse problems for some equations of mathematical physics. In this article, the authors studied the correctness of one linear two-point inverse problem for the multidimensional heat conduction equation. Using the methods of a priori estimates, Galerkin’s method, a sequence of approximations and contracting mappings, the unique solvability of the generalized solution of the linear two-point inverse problem for the multidimensional heat equation was proved.

Keywords

• multidimensional heat conduction equation
• linear two-point inverse problem
• unique solvability of a generalized solution
• methods of a priori estimates
• Galerkin’s method
• sequences of approximations and contracting mappings