Inverse Problem for a Fredholm Third Order Partial Integro-differential Equation

The solvability of various problems for partial differential equations of the third order is researched in many papers. But, partial Fredholm integro-differential equations of the third order are studied comparatively less. Integro-differential equations have traits in their one-valued solvability. The questions of solvability of linear inverse problems for partial differential equations are studied by many authors. We consider a nonlinear inverse problem, where the restore function appears in the equation nonlinearly and with delay. This equation with respect to the restore function is Fredholm implicit functional integral equation. The one- valued solvability of the nonlinear inverse problem for a partial Fredholm integro-differential equation of the third order is studied. First, the method of degenerate kernel designed for Fredholm integral equations is modified to the case of partial Fredholm integro-differential equations of the third order. The nonlinear Volterra integral equation of the first kind is obtained while solving the nonlinear inverse problem with respect to the restore function. This equation by the special non-classical integral transformation is reduced to a nonlinear Volterra integral equation of the second kind. Since the restore function, which entered into the integro-differential equation, is nonlinear and has delay time, we need an additional initial value condition with respect to restore function. This initial value condition ensures the uniqueness of solution of a nonlinear Volterra integral equation of the first kind and determines the value of the unknown restore function at the initial set. Further the method of successive approximations is used, combined with the method of contracting mapping.