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Boundary-value problem for a degenerate high-order equation with gluing conditions involving a fractional derivative

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Abstract

The article investigates a Dirichlet-type problem with conjugation conditions for a degenerate equation of high even order with variable coefficients, including the Riemann–Liouville fractional derivative, in a rectangular region consisting of two subdomains (\(y>0\) and \(y<0\)), in each of which the equation has various kind.The solution is constructed as a series in terms of eigenfunctions of the one-dimensional problem, the existence of eigenfunctions is proved by the method of the theory of integral equations with symmetric kernels. The theorem of expansion in terms of the system of obtained eigenfunctions is proved. Sufficent conditions are found for boundary functions under which the solution in the form of a series converges uniformly. When justifying the convergence of a series, the problem of “small denominators” arises.This problem has been successfully solved in the article. The uniqueness of the solution is proved by the spectral method.

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Authors and Affiliations

  1. New Uzbekistan University, 100174, Tashkent, Uzbekistan

    B. Yu. Irgashev

  2. Namangan Engineering-Construction Institute, 160100, Namangan, Uzbekistan

    B. Yu. Irgashev

  3. Institute of Mathematics, Academy of Science, 100174, Tashkent, Uzbekistan

    B. Yu. Irgashev

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Irgashev, B.Y. Boundary-value problem for a degenerate high-order equation with gluing conditions involving a fractional derivative. Rend. Circ. Mat. Palermo, II. Ser (2024). https://doi.org/10.1007/s12215-024-01039-x

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