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

This paper considers the inverse problem of identification of the finite initial condition of the Cauchy problem for the homogeneous heat conduction equation using the first kind linear Fredholm integral equation. Its discretization is carried out with the help of the quadrature rectangular formula. For the numerical realization of the obtained system of linear algebraic equations with almost complete, symmetric, positively determined, ill-conditioned matrix it is proposed to use the method of conjugate gradients. Examples of reconstruction of smooth, nonsmooth and discontinuous initial conditions in one- and two-dimensional cases, including the introduction of «noise», characteristic of redefinition conditions of inverse problems, are given.