Iterative solution of the retrospective inverse heat conduction problem by the Poisson integral

Abstract

The article studies an inverse problem of identifying the finite initial condition of the Cauchy problem for homogeneous heat equation in dimensionless variables, defined in a bounded domain by the Poisson integral. Discretization of the linear Fredholm integral equation of the first kind is carried out by the quadrature formula of rectangles. The authors suggest a conjugate gradient method for numerical implementation of the obtained system of linear algebraic equations with a complete, symmetric, positively defined, ill-conditioned matrix. The article gives examples of recovering smooth and discontinuous initial conditions in one-dimensional and two-dimensional cases, including “noise” injection as a typical additional condition for inverse problems.