On a Benney–Luke Type Differential Equation with Nonlinear Boundary Value Conditions
T. K. Yuldashev, Farhod Dustmurodovich Rakhmonov
Abstract
Abstract In three-dimensional domain a nonlocal boundary value problem for a Benney–Luke type partial differential equation of the even order with nonlinear integral conditions and positive spectral parameter is considered. The solution of this partial differential equation is studied in the class of regular functions. The Fourier series method is used. A countable system of ordinary nonlinear differential equations is obtained. In determination of the arbitrary integration constants the five cases with respect to the spectral parameter are considered. The method of successive approximations is used. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the Fourier series and its derivatives is proved.