SOLVABILITY AND STRUCTURE OF SOLUTIONS TO THE INITIAL PROBLEM OF HIGH-ORDER INTEGRO-DIFFERENTIAL PARTIAL DIFFERENTIAL EQUATIONS

Abstract

The relevance of the derivation of the initial problem of a nonlinear Integro-differential partial differential equation has not lost its relevance. Transformation is one of the research methods. Using this method, it is transformed into the nonlinear Volterra integral equation, the original problem, which is equivalent to the original problem. The topological method is applied to the resulting equation. For integro-differential equations with partial derivatives of the fourth order, the solution of the Cauchy problem was investigated, and the solution was found in the integral representation.

A special normal space was chosen to study the primary issue. Differentiation by parameter under the sign of the integral was also applied in the case when the limit of the integral depended on the parameter. The condition was determined under which the Volterra integral equation of the second kind has a unique nonlinear solution and is obtained in integral form.