An Inverse Problem for Integro-Differential Pseudoparabolic Equation with Nonlocal Initial and Non-Classical Boundary Conditions

We study an inverse problem for determining the kernel of an integro-differential pseudoparabolic equation. The direct problem is formulated with nonlocal initial and nonclassical boundary conditions. First, we analyze the corresponding spectral problem, where the completeness of the eigenfunctions, the construction of a biorthogonal system in the space $$L_{2}(0,1)$$ , and the properties of the eigenfunctions are established. By applying the Fourier method, the direct problem is reduced to a Volterra–Fredholm type integral equation of the second kind. The existence and uniqueness of its solution are proved using Gronwall’s inequality and the properties of functional series. Using the solution of the direct problem together with an additional condition, we derive an integral equation for the unknown kernel function. Finally, the existence and uniqueness of the solution to the inverse problem are proved using the fixed-point theorem.

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