A DOUBLE INVERSE PROBLEM FOR A FREDHOLM PARTIAL INTEGRO-DIFFERENTIAL EQUATION OF FOURTH ORDER
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It is offered the method of studying the one-value solvability of the double inverse problem for a Fredholm partial integro-differential equation of fourth order with degenerate kernel and deviation. First, it is modified and development to the case of Fredholm integro-differential equation of fourth order the method of degenerate kernel designed for Fredholm integral equations. It is obtained the system of differential equations. The inverse problem is called as a 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 is obtained the inhomogeneous differential equation of the fourth order, which is solved by the method of variation of arbitrary constants with initial value conditions. With respect to the second restore function it is obtained the nonlinear integral equation of the first kind with deviation, which is reduced by the aid of special nonclassical integral transform into nonlinear Volterra integral equation of the second kind. Further it is used the method of successive approximations, combined it with the method of compressing maps.
Abstract
It is offered the method of studying the one-value solvability of the double inverse problem for a Fredholm partial integro-differential equation of fourth order with degenerate kernel and deviation. First, it is modified and development to the case of Fredholm integro-differential equation of fourth order the method of degenerate kernel designed for Fredholm integral equations. It is obtained the system of differential equations. The inverse problem is called as a 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 is obtained the inhomogeneous differential equation of the fourth order, which is solved by the method of variation of arbitrary constants with initial value conditions. With respect to the second restore function it is obtained the nonlinear integral equation of the first kind with deviation, which is reduced by the aid of special nonclassical integral transform into nonlinear Volterra integral equation of the second kind. Further it is used the method of successive approximations, combined it with the method of compressing maps.
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