Polynomial Approximation of a Nonlinear Inverse Cauchy Problem

Abstract

In this paper, a class of nonlinear inverse boundary problem in the context of heat transfer is considered. We consider a class of nonlinear inverse boundary problems in the context of heat transfer. The problem involves determining the temperature distribution within a domain subject to a Cauchy boundary condition on a part of its boundary. We introduce a transformed variable, which allows us to reformulate the problem as a linear Cauchy problem followed by a series of nonlinear equations. We propose a polynomial expansion method to solve the linear Cauchy problem for the Laplace equation, and we employ the Newton method to solve the resulting nonlinear equations. Importantly, our approach does not rely on mesh-based discretization, allowing for parallel computation and preserving the mesh-free nature of the problem. We present numerical results obtained using our methodology and discuss the effectiveness of the proposed approach. The results show that the method provides a robust and efficient framework for solving nonlinear inverse boundary problems in heat transfer, with potential applications in various engineering and scientific fields.