Exploring Solutions of Geometry Problems for Inverse Cauchy Problems in Helmholtz and Modified Helmholtz Equations

Abstract In the present paper explores a reverse Cauchy problem for a heat transfer issue described by the Helmholtz and modified Helmholtz equation. Our goal is to identify an unknown defect  within a simply connected bounded domain , given the Dirichlet data (temperature)  on the boundary , and Neumann data (heat flux)  on the boundary . We assume that the temperature  satisfies the Helmholtz equation (or modified Helmholtz equation) that governs the heat condition in the fin. To solve this problem, we propose a method that involves two steps. First, we solve a Cauchy problem using the Helmholtz equation (or modified Helmholtz equation) to determine the temperature   Then, in the second phase, we solve a system of nonlinear scalar equations to determine the coordinates of the points defining the boundary . This can be achieved using an iterative method, such as Newton's method.