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.

中文翻译：

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具有反射参数的Boussinesq型非线性积分微分方程的边值问题

摘要

在三维域中，考虑了具有退化核，积分形式条件，谱参数和反射自变量的四阶Boussinesq型非线性偏积分-微分方程。在泛化函数类中研究了该偏积分微分方程的解。使用了变量分离的方法和简并内核的方法。使用这些方法，非局部边值问题被整合为一个常微分方程的可数系统。当我们定义任意积分常数时，关于第一个光谱参数可能有五种情况。每种情况下光谱参数的计算值。此外，该问题被简化为求解线性代数方程的可数系统。确定第二光谱参数的不规则值。在第二光谱参数的不规则值下，Fredholm行列式退化。Fredholm行列式不会退化的第二光谱参数的其他值称为常规值。考虑到第一光谱参数的值和第二光谱参数的正则值，构造了相应的解，我们针对五种情况分别获得了可数的非线性积分方程组。为了建立这个可数的非线性积分方程组的唯一可解性，我们使用逐次逼近法和压缩映射法。利用柯西-施瓦兹不等式和贝塞尔不等式，我们证明了所获得傅里叶级数的绝对一致收敛。证明了积分条件下边值问题对于给定函数的解的稳定性。研究了边值问题解小的条件。对于第二频谱参数的不规则值，分别检查五种情况。使用正交性条件。确定问题具有无限数量解的情况，并将这些解构造为傅立叶级数。对于其他情况，证明没有问题的非平凡解决方案。制定了相应的定理。研究了边值问题解小的条件。对于第二频谱参数的不规则值，分别检查五种情况。使用正交性条件。确定问题具有无限数量解的情况，并将这些解构造为傅立叶级数。对于其他情况，证明没有问题的非平凡解决方案。制定了相应的定理。研究了边值问题解小的条件。对于第二频谱参数的不规则值，分别检查五种情况。使用正交性条件。确定问题具有无限数量解的情况，并将这些解构造为傅立叶级数。对于其他情况，证明没有问题的非平凡解决方案。制定了相应的定理。证明了没有非平凡的问题解决方案。制定了相应的定理。证明了没有非平凡的问题解决方案。制定了相应的定理。