First, we prove minimum principle to parabolic degenerate initial boundary value problems in order to provide positivity of the continuous solutions. Then we construct a fitted finite volume scheme and study the positivity and convergence of the discrete solution. This solution to the direct problem is incorporated into an inverse algorithm for estimating of source intensities. Numerical experiments for direct and inverse problems are performed to verify the effectiveness and usefulness of this new method.

In this paper we study the problem of a load, traveling along elastic structures, such as strings, beams or plates. The case of semi‐active viscous dampers is also considered. The numerical method, based on the Immersed Interface Finite Element Method is developed. The presence of Dirac‐delta function terms in the mathematical model leads to using of a special basis functions near the interfaces. Many numerical experiments are presented for different practical cases.

In this paper, we study the inverse problem for the reconstruction of the right-hand side of parabolic systems of equations with overdeterminations, given by point measurements. The numerical algorithm is based on a special decomposition of the numerical solution first proposed for the heat equation in [9, 11]. Numerical experiments show the capacity of the numerical algorithm for parabolic systems.