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.

翻译：本文研究利用渐近展开高效求解涉及非线性奇异摄动时间依赖反应-扩散-对流方程的正问题与反问题。通过引入过渡层区域的局部坐标渐近展开，我们证明了一类具有陡峭过渡层的三维偏微分方程光滑解的存在唯一性。此外，借助渐近展开方法，建立了对应源项反问题的简化模型，该模型在全区域（除狭窄过渡层外）与原反问题高度近似。研究表明，当测量数据包含噪声时，此类简化不会降低反演结果的精度。基于该简化反演模型，提出了适用于三维情形下高效求解反源问题的渐近展开正则化算法。数值算例验证了所提数值方法的可行性。