In the present work, on the base of a numerical finite element method, the solution of the Dirichlet and

Neumann problems with respect to the oscillation equation for gravity-gyroscopic waves is discussed. The

approximation with respect to spatial variables is achieved by using linear splines, and the approximation with

respect to time is achieved by using cubic Hermitean splines. It is demonstrated that the use of such

approximation with respect to time allows the quality of the solution to be essentially improved as compared

with the traditional approximation ensuring the second order accuracy. The stability and accuracy of the method

are estimated. Using the method of regularization with spectrum shift, a new method is developed for solving

the spatial operator degeneration problem associated with the Neumann problem. The results of the numerical

calculations performed provide the possibility to make conclusions on the mode of behavior of the solution of

the Neumann problem depending on the problem variables.

**Author(s) Details**

**Mikhail Nikolayevich Moskalkov
**Berdakh Kara-Kalpak State University, Nukus, Uzbekistan.

**Dauletbay Utebaev
**Berdakh Kara-Kalpak State University, Nukus, Uzbekistan.

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