Study of the Solution of one Nonlinear Integro-differential Equation of Fourth Order.

There is considered a nonlinear integro-differential equation in partial derivatives of the fourth order, the existence and uniqueness of a local solution are proved, and the conditions for the existence of a local solution are determined. The proof is carried out using the method of an additional argument. Recently, this method has been used to reduce nonlinear differential equations in partial derivatives of high order into integral equations and systems of integral equations. This method was developed by Kyrgyz scientists and used to solve nonlinear differential equations in partial derivatives of the first order. Currently, the method of the additional argument is being developed for various new classes of nonlinear partial differential equations of higher order and systems of nonlinear partial differential equations. The nonlinear partial differential equation of the fourth order discussed in the article is presented in operator form, with the method of the additional argument applied consecutively several times. As a result, an integral equation is derived, and the existence and uniqueness of the solution to this integral equation are determined using the contraction mapping principle. The unknown function in the integral equation contains an additional argument, and by equating this additional argument to time, a local solution to the original initial value problem can be obtained. The results presented in the article can be applied in the study of solutions to other nonlinear differential and integro-differential equations of higher order.