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 = x0, 0