Abstract Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

中文翻译：

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抛物线方程反源问题的一种数值方法及其在系数反问题中的应用

摘要 本文的两个主要目的是开发一种数值方法来求解抛物线方程的反源问题，并将其应用于求解非线性系数反问题。本文中的逆源问题是从外部观察重建源项的问题。我们解决这个逆源问题的方法包括两个阶段。我们首先建立抛物线方程解对时间变量的导数方程。然后，在第二阶段，我们通过拟可逆方法求解该方程。本文考虑的逆源问题是非线性系数逆问题的线性化。因此，迭代求解逆源问题提供了该系数逆问题的数值解。