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Abstract <p>This paper investigates a boundary value problem with deviation from the characteristic (Problem <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8419_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--LobJMat2560964Islomov-m1--> </InlineEquation>) for a parabolic–hyperbolic equation with a time-fractional derivative in the sense of Gerasimov–Caputo. To construct the solution of Problem <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8419_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--LobJMat2560964Islomov-m2--> </InlineEquation>, the first boundary value problem is solved in the parabolic part of the mixed domain, based on which the existence and uniqueness of the classical and strong solutions of Problem <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8419_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--LobJMat2560964Islomov-m3--> </InlineEquation> are proven. The existence of eigenvalues for the boundary value Problem <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8419_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--LobJMat2560964Islomov-m4--> </InlineEquation> is established.</p>

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Unique Solvability and Existence of Eigenvalues for a Boundary Value Problem for a Parabolic-Hyperbolic Type Equation with a Fractional Derivative

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • B. I. Islomov,
  • I. A. Akhmadov

摘要

Abstract

This paper investigates a boundary value problem with deviation from the characteristic (Problem \(M\) ) for a parabolic–hyperbolic equation with a time-fractional derivative in the sense of Gerasimov–Caputo. To construct the solution of Problem \(M\) , the first boundary value problem is solved in the parabolic part of the mixed domain, based on which the existence and uniqueness of the classical and strong solutions of Problem \(M\) are proven. The existence of eigenvalues for the boundary value Problem \(M\) is established.