Solvability of the Inverse Problem of Finding Thermal Conductivity

Abstract

We study the inverse problem of finding the coefficient of thermal conductivity of the heat equation (along with the solution). As the overdetermination condition we take the values of the solution at the final time. Existence of a regular solution is proven.The process of heat propagation in a rod is described by the following general heat equation:in which the function u(x, t) characterizes the temperature at a point x at time t; ρ, p, and q characterize the properties of the substance; and the function f (x, t) is determined by the exterior sources (see [1]). If the properties of the substance or/and the exterior medium are unknown a priori then alongside the solution u(x, t) one or several coefficients of the indicated equation or/and the function f (x, t) may turn out to be unknown. In mathematics, inverse problems we call the problems of the theory of partial differential equations in which both the solution to the equation and one or several its coefficients or/and the right-hand side are unknown. As a rule, these problems together with some boundary conditions appropriate for the corresponding direct problem (the problem with known coefficients and right-hand side) have some additional information connected with the presence of some unknown function (functions). If we confine study to the models in which the unknown coefficients are independent of t or the righthand side f (x, t) contains an unknown component independent of t (these models are rather adequate for description of the real processes, see [1]) then in the case of the heat equation the conditions of the direct problem are those of some initial-boundary value problem for a parabolic equation. As a rule, the additional information is either some information of the medium state (i.e., temperature) at a certain time or some information of the average temperature. In the first case the inverse problem is said to be a problem with final overdetermination; in the second we have a problem with integral overdetermination.

The Application of Boundary Element Method for The Solution of The Inverse Heat Conduction

“…[10] Problem solved for diff erent thermal conductivity. [11] The rest of this paper is arranged as follows. The second part explains the problem to be solved, the third part uses the boundary element method to analyze the problem, and the partial diff erential equation is changed to the integral form for discretization.…”

“…) we now use the investgated condition(11) at the point (x0 , t˜i ) for i = 1, 2, • • • , n, and we can obtain N equations.Then, constrain(11) shouid also be considered by applying the integral equation at the point (0, ˜i ) and (1, t˜i ) , we can acquire 2N equations.…”

Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

“…Inverse problems of finding coefficients in parabolic equations have been extensively studied (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]). These problems are nonlinear, since the unknown functions enter into the equation in the form of products.…”

On an over-determined problem of free boundary of a degenerate parabolic equation

“…We know that there are many works devoted to different kinds of such ill-posed problems on parabolic equations in the recent years (see [9], [7], [8], [14]). But most of them discussed the solvability of these problems and few of them are concerned with free boundary problems.…”