An inverse problem of recovering a time dependent diffusion coefficient for a space-fractional diffusion equation has been considered. The space fractional derivative of order \(1<\alpha < 2\) is defined in the sense of Caputo. Due to an over-determination condition of integral type, we construct a mapping. Under certain conditions on the given data and application of Banach fixed point theorem ensured the unique local existence of the solution, moreover local solution is proved to be classical. The global existence of the solution of the inverse problem is shown by using Schauder fixed point theorem. Examples are also provided to support our analysis.

已经考虑了恢复针对空间分数扩散方程的时间相关扩散系数的反问题。在Caputo的意义上定义了阶数\（1 <\ alpha <2 \）的空间分数导数。由于整数类型的超确定条件，我们构造了一个映射。在给定数据的一定条件下，Banach不动点定理的应用确保了解的唯一局部存在，而且局部解被证明是经典的。利用Schauder不动点定理证明了反问题解的整体存在性。还提供了一些示例来支持我们的分析。