Inverse problem for a nonlinear partial differential equation of the eighth order

Abstract

We study the questions of solvability of the inverse problem for a nonlinear partial differential equation of the eighth order, left-hand side of which is the superposition of pseudoparabolic and pseudohyperbolic operators of the fourth order. The applicability of the Fourier method of separation of variables is proved in study of mixed and inverse problems for a nonlinear partial differential equation of the eighth order. Using the method of separation of variables, the mixed problem is reduced to the study of the countable system of nonlinear Volterra integral equations of the second kind. Use the given additional conditions led us to study of nonlinear Volterra integral equation of the first kind with respect to the second unknown function (with respect to restore function). With the help of nonclassical integral transform the one-value restore of the second unknown function is reduced to study of the unique solvability of nonlinear Volterra integral equation of the second kind. As a result is obtained a system of two nonlinear Volterra integral equations of the second kind with respect to two unknown functions. This system is one-value solved by the method of successive approximations. Further the stability of solutions of the mixed and inverse problems is studied with respect to initial value and additional given functions.

Keywords

• inverse problem
• nonlinear partial differential equation
• equation of the eighth order
• superposition of two operators
• correctness of solution