  This paper concerns the use of asymptotic expansions for the efficient
solving of forward and inverse problems involving a nonlinear singularly
perturbed time-dependent reaction--diffusion--advection equation. By using an
asymptotic expansion with the local coordinates in the transition-layer region,
we prove the existence and uniqueness of a smooth solution with a sharp
transition layer for a three-dimensional partial differential equation.
Moreover, with the help of asymptotic expansion, a simplified model is derived
for the corresponding inverse source problem, which is close to the original
inverse problem over the entire region except for a narrow transition layer. We
show that such simplification does not reduce the accuracy of the inversion
results when the measurement data contain noise. Based on this simpler
inversion model, an asymptotic-expansion regularization algorithm is proposed
for efficiently solving the inverse source problem in the three-dimensional
case. A model problem shows the feasibility of the proposed numerical approach.
