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Abstract <p> In this paper, a boundary value problem of the Dirichlet type is investigated for a degenerate inhomogeneous equation of even order with Gerasimov–Caputo derivative. The solution is constructed as a series in eigenfunctions of a one-dimensional spectral problem for a degenerate equation of even order. When constructing a solution to the problem, a boundary value problem for a one-dimensional equation of fractional order is also investigated depending on the sign of the constant coefficient <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3079_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\)</EquationSource> </InlineEquation> of the equation, and necessary estimates of the solution are obtained. Sufficient conditions are found for the convergence of the series that is a solution of the Dirichlet problem and the series obtained by differentiation. The uniqueness of the solution is shown by the spectral method. </p>

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Dirichlet-Type Problem for an Even-Order Degenerate Equation with Gerasimov–Caputo Fractional Derivative

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • B. I. Jamalov,
  • B. Yu. Irgashev

摘要

Abstract

In this paper, a boundary value problem of the Dirichlet type is investigated for a degenerate inhomogeneous equation of even order with Gerasimov–Caputo derivative. The solution is constructed as a series in eigenfunctions of a one-dimensional spectral problem for a degenerate equation of even order. When constructing a solution to the problem, a boundary value problem for a one-dimensional equation of fractional order is also investigated depending on the sign of the constant coefficient \(q\) of the equation, and necessary estimates of the solution are obtained. Sufficient conditions are found for the convergence of the series that is a solution of the Dirichlet problem and the series obtained by differentiation. The uniqueness of the solution is shown by the spectral method.