Exact and approximating boundary conditions for the parabolic problems on unbounded domains

In this paper, the numerical solutions of the problems of heat equation in two dimensions on unbounded domains are considered. For a given problem, we introduce an artificial boundary Γ to finite the computational domain. On the artificial boundary Γ, we propose an exact boundary condition to reduce the given problem to an initial-boundary problem of heat equation on the finite computational domain, which is equivalent to the original problem. Furthermore, a series of approximating artificial boundary conditions is given. Then the finite difference method and finite element method are used to solve the reduced problem on the finite computational domain. Finally, the numerical examples show the feasibility and effectiveness of the method given in this paper.