Asymptotic-numerical solvers for diffusion equation with time-like highly oscillatory forcing terms

The paper is concerned with an asymptotic expansion for the oscillatory term of the solution of diffusion equation with time-like highly oscillatory forcing terms. We derive an asymptotic expansion in inverse of powers of the oscillatory parameter, for the Dirichlet case and the Neumann case. Each term of the asymptotic expansion can be computed, at a lower cost. Numerical examples are given and show that with few terms of the asymptotic expansion, we can effectively approximate the oscillatory term of the solution.