The numerical solution of the boundary inverse problem for a parabolic equation

Abstract

Boundary inverse problems occupy an important place among the inverse problems of mathematical physics. They are connected with the problems of diagnosis, when additional measurements on one of the borders or inside the computational domain are necessary to restore the boundary regime in the other border, inaccessible to direct measurements. The boundary inverse problems belong to a class of conditionally correct problems, and therefore, their numerical solution requires the development of special computational algorithms.The paper deals with the solution of the boundary inverse problem for one-dimensional second-order parabolic equations, consisting in the restoration of boundary regime according to measurements inside the computational domain. For the numerical solution of the inverse problem it is proposed to use an analogue of a computational algorithm, proposed and developed to meet the challenges of identification of the right side of the parabolic equations in the works P.N.Vabishchevich and his students based on a special decomposition of solving the problem at each temporal layer. We present and discuss the results of a computational experiment conducted on model problems with quasi-solutions, including with random errors in the input data.

Numerical determination of the right-hand side of parabolic systems with point measurements

“…Our paper is devoted to implementation of the decomposition algorithm proposed for the heat equation in [9,11] to parabolic systems. Numerical experiments confirm the efficiency of the algorithm.…”

“…Much attention is given to the problem of determining the right-hand side and the coefficients of second order parabolic equations. Particularly, inverse problems are considered in which the right-hand side depends only on the time variables [9,10,11].…”

“…The numerical algorithm is based on a special decomposition of the numerical solution first proposed for the heat equation in [9,11]. Numerical experiments show the capacity of the numerical algorithm for parabolic systems.…”

Calculation of Tissue Oxygenation via an Inverse Boundary Problem for Transcutaneous Oxygenation Wearable Applications

“…The method, described in detail in ref , consists of discretizing the problem stated in eqs – using the finite difference method, and including a diffusivity D that is x dependent. Here, we have modified this algorithm, which used a Neumann left boundary condition, to instead include Dirichlet boundary conditions, as the problem under study (transcutaneous oxygenation) requires specifying the value of the concentration at the boundary condition, rather than its derivative.…”

Numerical Method for Solving Nonhomogeneous Backward Heat Conduction Problem

“…In many engineering and application areas we need to reconstruct the unknown initial heat energy or temperature from the final measured one; it is a typical inverse problem which is related to the initial boundary value problems in heat conduction. As we all know, there are many kinds of inverse problems in mathematical physics, such as boundary inverse problems [6,7], coefficient inverse problems [8,9], and evolutionary inverse problems (or time inverse problem) [10,11]. The backward heat conduction problem is a time inverse problem, in which the initial conditions are unknown; instead the final data are observable.…”