In this work, we present a numerical method based on a splitting algorithm to find the solution of an inverse source problem with the integral condition. The source term is reconstructed by using the specified data and by employing the Lie splitting method, we decompose the equation into linear and nonlinear parts. Each subproblem is solved by the Fourier transform and then by combining the solutions of subproblems, the solution of the original problem is computed. Moreover, the framework of strongly continuous semigroup (or C₀‐semigroup) is employed in error analysis of operator splitting method for the inverse problem. The convergence of the proposed method is also investigated and proved. Finally, some numerical examples in one, two, and three‐dimensional spaces are provided to confirm the efficiency and capability of our work compared with some other well‐known methods.

中文翻译：

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抛物型方程反源问题的分解方法：误差分析与应用

在这项工作中，我们提出了一种基于分裂算法的数值方法，以找到具有积分条件的逆源问题的解。通过使用指定的数据并通过使用Lie拆分方法来重构源项，我们将方程分解为线性和非线性部分。每个子问题都通过傅立叶变换求解，然后通过组合子问题的解来计算原始问题的解。此外，强连续半群（或C ₀-semigroup）用于针对反问题的算子拆分方法的误差分析。研究并证明了该方法的收敛性。最后，在一维，二维和三维空间中提供了一些数值示例，以确认我们的工作与其他一些知名方法相比的效率和能力。