Computational identification of the time dependence of the right-hand side of a hyperbolic equation

Many applied problems lead to the necessity of solving inverse problems for partial differential equations. In particular, much attention is paid to the problem of identifying coefficients of equations using some additional information. The problem of determining the time dependence of the right-hand side of a multidimensional hyperbolic equation using information about the solution at an interior point of the computational domain is considered. An approximate solution is obtained using a standard finite element spatial approximation and implicit schemes for time approximations. The computational algorithm is based on a special decomposition of the solution of the inverse problem when the transition to a new time level is ensured by solving standard elliptic problems. Numerical results for a model two-dimensional problem are given, which demonstrate the potentialities of the computational algorithms proposed to approximately solve inverse problems.