An inverse problem for a 2D parabolic equation with nonlocal overdetermination condition

Abstract

We consider an inverse problem of identifying the time-dependent coefficient a(t) in a twodimensional parabolic equation:with the initial condition, Neumann boundary data and the nonlocal overdetermination conditionwhere y 0 is a fixed number from [0, l].The conditions of existence and uniqueness of the classical solution to this problem are established. For this purpose the Green function method, Schauder fixed point theorem and the theory of Volterra intergral equations are utilized.

Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition

“….) described by ( 17) into (14), to determine the first component of the solution (1)-( 4), (7), we obtain…”

“…Problems of the solvability of inverse problems for a two-dimensional heat equation is extensively studied by many authors, see, for example, Ismailov [5], Ivanchov [8,9], Kabanikhin [13], Kinash [14], Zaynullov [19], and others. But the statement of the problem and the proof techniques used in this study are different from representations in these papers.…”

Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor

“…and w 1 is defined by (10). Analogously, in order to get an operator equation with respect to a 2 (t), we differentiate (9) with respect to y and use the notation w 2 := u y (x, y, t) to obtain…”

“…The inverse problem that we formulate in Section 2 and propose to study combines the features of multi-dimensions, multiple coefficient identification, [6], as well as non-local overdetermination data, [2,7,8,10]. All these papers are unified by the approach utilized to prove the existence of solution: the inverse problem is reformulated as a fixed point problem for a certain nonlinear compact operator, so that the Schauder theorem can be applied to it.…”

Recovery of the unknown coefficients in a two‐dimensional hyperbolic equation

“…In the article published by Azizbayov and Mehraliyev, 16 a nonlocal inverse boundary value problem for a two-dimensional parabolic equation is studied almost by the same technique. Nevertheless, the inverse problems for two-dimensional partial differential equations associated with the recovery of the coefficients have been studied scarce (see literature [17][18][19][20][21] ) and need more thorough consideration.…”