ON AN INITIAL BOUNDARY–VALUE PROBLEM FOR A DEGENERATE EQUATION OF HIGH EVEN ORDER

Akhmadjon K. Urinov     (Fergana State University, 19, Murabbiylar st., Fergana, 150100; V.I. Romanovskiy Institute of Mathematics of Uzbekistan Academy of Sciences, 9 University Str., 100174 Tashkent, Uzbekistan)
Dastonbek D. Oripov     (Fergana State University, 19, Murabbiylar st., Fergana, 150100, Uzbekistan)

Abstract


In this paper, we formulate and study an initial boundary-value problem of the type of the third boundary condition for a degenerate partial differential equation of high even order in a rectangle. Using the Fouriers method, based on separation of variables, a spectral problem for an ordinary differential equation is obtained. Using the Green's function method, the latter problem is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and a system of eigenfunctions of the spectral problem. Using the found integral equation and Mercer's theorem, the uniform convergence of certain bilinear series depending on the eigenfunctions is proved. The order of the Fourier coefficients has been established. The solution to the considered problem has been written as a sum of the Fourier series over the system of eigenfunctions of the spectral problem. The uniqueness of the solution to the problem was proved using the method of energy integrals. An estimate for solution of the problem was obtained, which implies its continuous dependence on the given functions.


Keywords


Degenerate equation, Initial boundary-value problem, Method of separation of variables, Spectral problem, Green’s function method, Integral equation, Fourier series

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References


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DOI: http://dx.doi.org/10.15826/umj.2025.1.010

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