APPLICATION OF THE METHOD OF AN ADDITIONAL ARGUMENT TO PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, NONLINEAR WITH RESPECT TO AN UNKNOWN FUNCTION

Abstract

The paper considers the application of the additional argument method to second-order partial differential equations that are nonlinear with respect to an unknown function. In these studies, the advantages of the method of additional argument over other methods of studying similar equations appeared, consisting in the fact that the system of integral equations, to which the original problem is reduced, does not contain a superposition of unknown functions. In addition, the solution of the original problem is derived from the solution of integral equations by reducing the set dimension of the arguments, and not by inverting a nonlinear algebraic operator.  Using the basic ideas of the additional argument method, differential and integro-differential equations in partial derivatives of the Korteweg-de-Vries type, as well as nonlinear wave equations, were studied. The existence of a unique solution is proved.