On a Boundary-value Problem for Boussinesq type Nonlinear Integro-Differential Equation with Reflecting Argument

Abstract

In the three-dimensional domain a Boussinesq type nonlinear partial integro-differential equation of the fourth order with a degenerate kernel, integral form conditions, spectral parameters and reflecting argument is considered. The solution of this partial integro-differential equation is studied in the class of generality functions. The method of separation of variables and the method of a degenerate kernels are used. Using these methods, the nonlocal boundary value problem is integrated as a countable system of ordinary differential equations. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving countable system of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed and we obtained the countable system of nonlinear integral equations for each of five cases. To establish the unique solvability of this countable system of nonlinear integral equations we use the method of successive approximations and the method of compressing mappings. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution of the boundary value problem with respect to given functions in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed as Fourier series. For other cases, the absence of nontrivial solutions of the problem is proved. The corresponding theorems are formulated.