Solvability of the inverse initial-boundary value problem with a known value on the line

The definitions of either the kernel or the right-hand sides of integro-differential equations, or the values of either the initial or boundary conditions for integro-differential equations or the definition of the right-hand side for an integro-differential equation with over determination at an interior point based on additional information about the solution of the original problem is called inverse problems. Mathematical models of modern problems of geophysics, oceanology, atmosphere, physics, technology and other sciences are described using integro-differential equations with partial derivatives of the fourth order. The present article is devoted to the solvability of the inverse problem, that is, the recovery of the kernel in the initial-boundary value problem for a fourth-order integro-differential equation with partial derivatives with a known value of the desired solution on the straight line $x = x_0$, $0 < x_0 < 1$, that is, with a new definition in the inner line. The authors have proved for the first time the existence and uniqueness of the solution of the inverse problem under consideration. Well-known methods are used to achieve this goal: the method of reducing the inverse problem to a linear integral Volterra equation of the second kind, the method of Green's functions for ordinary differential equations of the second order with homogeneous boundary conditions. When solving the formulated inverse problem, sufficient conditions for the existence and uniqueness of the solution of the inverse problem of recovering the kernel in a fourth order partial integro-differential equation are found. First, using transformations and the Green's function, the original problem is reduced to an equivalent problem, for which a theorem on the existence and uniqueness of a solution is proved. Further, using the methods of the theory of inverse problems, three Volterra integral equations of the second kind are compiled and the existence and uniqueness of the solution of systems of Volterra integral equations of the second kind are proved.