Boundary Determination of the Inverse Heat Conduction Problem in One and Two Dimensions via the Collocation Method Based on the Satisfier Functions

Abstract

In this paper, we are concerned with the numerical solutions of the inverse heat conduction problems (IHCP) in one and two dimensions with free boundary conditions. For the one-dimensional problem, we first apply the Landau’s transformation to replace the physical domain with a rectangular one. Reciprocally, some nonlinear terms appear thus an iterative scheme based on the application of the satisfier function is proposed for solving the problem. Second, we treat with the nonlinear two-dimensional problem by providing a collocation technique which takes advantage of the satisfier functions. Throughout this work, the presented schemes make the reader free of solving any nonlinear system of algebraic equations. Moreover, an admissible regularization strategy, namely, the Landweber’s iterations method is used to overcome the numerical instability and achieve the acceptable approximations. Illustrative examples are included to show the efficiency of the presented algorithms.