The diffusion equation with a general convolutional derivative in time is considered on a bounded domain, as one of the boundary conditions is nonlocal. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. To find the source term and the solution, we resort to generalized eigenfunction expansion, using a bi-orthogonal pair of bases. Estimates for the time-dependent components in the spectral expansions are established and applied to prove uniqueness and existence in the classical sense. Analytical and numerical examples are provided.

中文翻译：

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通用时间分数扩散方程的非局部问题中与空间相关的源项的识别

由于边界条件之一是非局部的，因此在有界域上考虑了具有一般时间卷积导数的扩散方程。我们关注从给定的最终时间数据中恢复与空间有关的源项的逆源问题。为了找到源项和解，我们求助于使用双正交基对的广义本征函数展开。建立并估计光谱扩展中随时间变化的分量，并证明其在经典意义上的独特性和存在性。提供了分析和数值示例。