On solution of solid state physics inverse problem by means of A. N. Tikhonov's regularization method and estimation of the error of this method

The paper considers one-dimensional Fredholm integral equation of the first kind with a closed core. It is known that the equation has a unique solution in the space $W^1_{2}[a, b]$. We use Tikhonov's regularization method of the first-order to solve the equation. The method allows us to reduce the equation to a variational problem. Solving the variational problem we get integro-differential equation of second order. We apply the finite-difference approximation method to reduce the original problem to a system of algebraic equations. regularization parameter.
We obtain an error estimate for the proposed algorithm taking into account the error of finite-difference approximation and state the relation between the approximation with the error and the regularization parameter and the error of the initial data.
This algorithm is used to solve the problem of determining the phonon spectrum of the crystal given its heat capacity.