A double inverse problem for Fredholm integro-differential equation of elliptic type
In this paper the double inverse problem for partial differential equations is considered. The method of studying the one value solvability of the double inverse problem for a Fredholm integro-differential equation of elliptic type with degenerate kernel is offered. First the method of degenerate kernel designed for Fredholm integral equations is modiﬁed and developed to the case of Fredholm integro-differential equation of elliptic type. The system of differential-algebraic equations is obtained. The inverse problem is called double inverse problem if the problem consisted to restore the two unknown functions by the aid of given additional conditions. The ﬁrst restore function is nonlinear with respect to the second restore function. In solving the inverse problem with respect to the ﬁrst restore function the inhomogeneous differential equation of the second order is obtained, which is solved by the method of variation of arbitrary constants with initial value conditions. With respect to the second restore function the nonlinear integral equation of the ﬁrst kind is obtained, which is reduced by the aid of special nonclassical integral transform into nonlinear Volterra integral equation of the second kind. Further the method of successive approximations is used, combined with the method of compressing maps.