A spring-damping regularization of the Fourier sine series solution to the inverse Cauchy problem for a 3D sideways heat equation

Abstract

For solving the inverse Cauchy problem of a three-dimensional (3D) non-homogeneous sideways heat equation in a cuboid, we first transform it into a partial homogeneous boundary value problem, using a time-dependent two-dimensional (2D) homogenization technique. After a time step-by-step approach of the inverse Cauchy problem, it becomes an inverse Cauchy problem of the 3D modified Helmholtz equation. A sequence of 2D sine functions and a set of second-order ordinary differential equations are derived to determine the expansion coefficients, which can be solved in closed-form. In order to stably solve the inverse Cauchy problem, we simply introduce a spring-damping regularization technique to modify the coefficients, which results in a regularized 2D Fourier sine series solution of the unknown Dirichlet data on the top of the cuboid. The novel method can find the solution in the whole domain, which is quite accurate with small errors, even a large noise up to is imposed on the given data. The present method together with the analyses of regularization parameter and the stability of solution is novel and shows promise in the 3D sideways problems of the heat equation.