On the problem of recovering boundary conditions in the third boundary value problem for parabolic equation

Abstract

Background. In recent decades the theory for solution of inverse and ill-posed problems has become one of the most important and fast-growing branch of modern mathematics. A relevancy of this theory is due to not only significant growth in the number of applications of inverse and ill-posed problems in different fields of physical and technical sciences but also rapid development of computer technology. It is known that most of inverse problems of mathematical physics belong to the class if ill-posed problems, and the most important property of these problems is their instability to small perturbations of the initial data of the problem. This property induces the need for the development of regularization methods of special type. The boundary value problems constitute one of the most significant classes of inverse problems. An inverse problem is termed boundary if it is required to recover the function that is the part of a boundary value. Such problems arise when direct measuring of heat field characteristics at a domain boundary is difficult or impossible at all. Constructing numerical methods for solution of these problems is very crucial due to a vast number of their physical and technical applications. Materials and methods. In this paper we propose a numerical method for simultaneous recovering of boundary value coefficients in the third boundary problem for a heat equation. At the core of the method there is continuous method for solution of operator equations in Banach spaces. The main idea of the method is composing and solving the auxiliary system of differential equations of special type relative to the unknown coefficients of the basic problem. This system is to be solved numerically with the help of some method for approximate solution of differential equations. Simultaneous recovering of the coefficients by means of the proposed method additionally requires knowledge of values of a solution of the basic parabolic equation at two different points. Results. A numerical method for solution of the problem of recovering boundary value coefficients in the third boundary value problem for one-dimensional heat equation is constructed. We show applicability of continuous operator method to solution of inverse boundary value problems for parabolic equations. Convergence of the method is proven with the help of the theory of stability of ordinary differential equations. Successful solution of the model example shows effectiveness of the proposed method. Conclusions. An efficient method for solution of the problem of recovering boundary value coefficients in the third boundary value problem for linear one-dimensional parabolic equation is described in the paper. The key advantages of the method are its simplicity, flexibility and also stability to perturbations of initial data of the problem. It is of significant theoretical and practical interest to extend the proposed method to a wider class of boundary problems and also to multidimensional and nonlinear parabolic equations.

Keywords

• inverse problems
• parabolic equations
• third boundary value problem
• stability theory
• continuous operator method