Solving forward and inverse problems in a non-linear 3D PDE via an asymptotic expansion based approach
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This paper is concerned with the usage of the asymptotic expansion method for efficiently solving forward and inverse problems in a non-linear singularly perturbed time-dependent reaction-diffusion-advection equation. By using the 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 PDE. 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 result when the measurement data contains noise. Based on this simpler inversion model, an asymptotic expansion regularization algorithm is proposed for solving the inverse source problem in the three-dimensional case. Various numerical examples for both forward and inverse problems are given to show the efficiency of the proposed numerical approach.
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