A numerical algorithm for solving inverse filtration problems with the pointwise overdetermination

The inverse problems of recovering the right-hand side in a pseudoparabolic equations of filtration with the use of the pointwise overdetermination are studied. We expose some existence and uniqueness theorems which are the base of an numerical algorithm of recovering the right-hand side (the source function) and a solution. The problem is well-posed and the stability estimates hold. It can be reduced to a Volterra-type integral equation, where the operator has a small norm for small time segments. The finite element method is used to reduce the problem to a system of ordinary differential equations which is solved by the finite difference method. The idea of the predictor-corrector method is employed in the algorithm. The results of numerical experiments are presented. They show a good convergence of an approximate solutions to a solution.