We study the inverse problem for a mixed type equation with the Riemann–Liouville and Caputo operator in a rectangular domain. A criterion for the uniqueness and existence of a solution to the inverse problem is established. The solution to the problem is constructed in the form of the sum of a series of eigenfunctions of the corresponding one-dimensional spectral problem. It is proved that the unique solvability of the inverse problem substantially depends on the choice of the boundary of a rectangular region. An example is constructed, in which the inverse problem with homogeneous conditions has a nontrivial solution. Estimates are obtained that allow substantiating the convergence of series in the class of regular solutions of this equation and the stability of the solution of the inverse problem on boundary data.

我们用矩形区域中的Riemann-Liouville和Caputo算子研究混合型方程的逆问题。建立了反问题解的唯一性和存在性的判据。该问题的解决方案以对应一维光谱问题的一系列本征函数之和的形式构造。事实证明，反问题的独特可解性基本上取决于矩形区域边界的选择。构造了一个例子，其中具有齐次条件的反问题具有非平凡的解。获得的估计值可以证实该方程组的正则解类别中级数的收敛性，以及边界数据上反问题的解的稳定性。