Rate of convergence and error estimates for finite-difference schemes of solving linear ill-posed Cauchy problems of the second order

Finite-difference schemes of solving ill-posed Cauchy problems for linear second-order differential operator equations in Banach spaces are considered. Several time-uniform rate of convergence and error estimates are obtained for the considered schemes under the assumption that the sought solution satisfies the sourcewise condition. Necessary and sufficient conditions are found in terms of sourcewise index for a class of schemes with the power convergence rate with respect to the discretization step. A number of full discretization schemes for second-order ill-posed Cauchy problems are proposed on the basis of combining the half-discretization in time with the discrete approximation of the spaces and the operators.