In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs). The key idea of this approach is the use of the asymptotic-expansion theory, which allows us to determine the conditions for the existence and uniqueness of a solution to a given PDE with a sharp transition layer. As a by-product, we derive a simpler link equation between the source function and first-order asymptotic approximation of the measurable quantities, and based on that equation we propose an efficient inversion algorithm, AER, for inverse source problems. We prove that this simplification will not decrease the accuracy of the inversion result, especially for inverse problems with noisy data. Various numerical examples are provided to demonstrate the efficiency of our new approach.

翻译：在本文中,我们为二维非线性和非静态奇扰的局部偏差方程(PDEs)的反源问题开发了一种无症状扩张-正规化(AER)方法。这种方法的关键理念是使用无症状扩展理论,这使我们能够确定某种具有尖锐过渡层的PDE解决方案的存在条件和独特性。作为一个副产品,我们从源函数和可测量数量的第一阶无症状近似之间得出一个更简单的等式,并在此基础上,我们建议一种高效的反源算法(AER),用于反源问题。我们证明,这种简化不会降低反位结果的准确性,特别是对于噪音数据的反面问题。我们提供了各种数字实例,以展示我们新方法的效率。