We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we decouple the full inverse problem into two Dirichlet problems at each time level. On this base, we develop decomposition techniques to obtain exact formulas for the unknown boundary conditions at point measurements. A discrete version of the analytical approach is realized on time adaptive mesh for different fractional order of the equations in each of the disjoint domains. A variety of numerical examples are discussed.

Based on transformation techniques used in analysis of initial boundary value problems, we propose and discuss two numerical approaches for nonlinear models in option pricing. The first one exploits difference schemes for a degenerate parabolic problem on a finite interval. The second one solves the problem on infinite interval by Rothe’s method. An appropriate substitution reduces the matter to a semilinear integral equation at each time step. Numerical experiments are discussed.