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Karakalpak Scientific Journal

Abstract

In this paper, we study a boundary value problem for degenerate fourth-order mixed type partial differential equation in a rectangular domain. The unique regular solvability of the boundary value problem posed is investigated. The solution is constructed in the form of the sum of the biorthogonal series in an explicit form, and the rationale for the convergence of the series in the class of regular solutions is given. To prove the solution of this problem, the estimates of the coefficients of the series and the system of eigenfunctions are used, which are established by asymptotic formulas for the Bessel function of the first kind for large values of the argument and zero values of this function. Sufficient conditions are obtained with respect to the data of the problem, which guarantee the convergence of the constructed series in the class of regular solutions. The uniqueness of the solution to the problem is proved on the basis of the completeness of the system of eigenfunctions corresponding to the one-dimensional eigenvalue problem in the space of quadratically summable functions.

First Page

16

Last Page

24

References

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