The Inverse Problem of Recovering the Kernel and the Right-Hand Side of a Fifth Order Integro-Differential Equation

In this paper, sufficient conditions for the solvability of inverse problems are found for one class of fifth-order partial differential equations. The study of inverse problems, in which is reconstructed the kernel and the right-hand side of the equation. This research is important for several reasons. Completeness of information: restoring both the kernel and the right side allows you to obtain a more complete picture of the process or system under study. Modeling accuracy: such tasks improve the accuracy of mathematical models used to describe physical, biological, or engineered systems. Improved diagnosis and prognosis: in medical imaging, for example, this could lead to more accurate diagnoses and a better understanding of pathologies. Process optimization: in geophysics and non-destructive testing, this can help in optimizing mineral prospecting or defect detection processes. These problems are complex and require the development of new regularization methods and computational algorithms, making them an active area of research in applied mathematics and in computational physics.