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Green's Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative

Erkinjon Karimov, Doniyor Usmonov, Maftuna Mirzaeva

TL;DR

This work develops a Green-function framework for a time-fractional boundary-value problem driven by the Prabhakar derivative ${}^{PRL}D_{0t}^{\alpha,\beta,\gamma,\delta}$. By reducing the PDE to a first-order system and then to a Volterra integral equation, the authors construct a Green's function built from the Prabhakar kernel $\omega$ and a periodized kernel $W$, yielding a closed-form integral representation for the solution $u(t,x)$ in terms of boundary data and the source term $f$. They prove the existence and uniqueness of a regular solution on the rectangle $D=\{(t,x):0<t<T,0<x<a\}$ and provide explicit representations for the Green's functions $G$ and $G_s$, enabling analysis of boundary and inverse problems for Prabhakar-type fractional PDEs. The approach extends classical Green-function techniques to a broad class of fractional operators and furnishes analytical tools for future study of Prabhakar-type diffusion and diffusion–wave processes. Overall, this work bridges fractional calculus with boundary-value problem methods, offering rigorous solutions and kernels that can support numerical methods and inverse problems in systems with memory effects.

Abstract

We investigate a first boundary value problem for a second-order partial differential equation involving the Prabhakar fractional derivative in time. Using structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions, we reduce the problem to a Volterra type integral equation. This reduction enables the explicit construction of the corresponding Green's function. Based on the obtained Green's function, we derive a closed-form integral representation of the solution and prove its existence and uniqueness. The results extend classical Green-function techniques to a wider class of fractional operators and provide analytical tools for further study of boundary and inverse problems associated with Prabhakar-type fractional differential equations.

Green's Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative

TL;DR

This work develops a Green-function framework for a time-fractional boundary-value problem driven by the Prabhakar derivative . By reducing the PDE to a first-order system and then to a Volterra integral equation, the authors construct a Green's function built from the Prabhakar kernel and a periodized kernel , yielding a closed-form integral representation for the solution in terms of boundary data and the source term . They prove the existence and uniqueness of a regular solution on the rectangle and provide explicit representations for the Green's functions and , enabling analysis of boundary and inverse problems for Prabhakar-type fractional PDEs. The approach extends classical Green-function techniques to a broad class of fractional operators and furnishes analytical tools for future study of Prabhakar-type diffusion and diffusion–wave processes. Overall, this work bridges fractional calculus with boundary-value problem methods, offering rigorous solutions and kernels that can support numerical methods and inverse problems in systems with memory effects.

Abstract

We investigate a first boundary value problem for a second-order partial differential equation involving the Prabhakar fractional derivative in time. Using structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions, we reduce the problem to a Volterra type integral equation. This reduction enables the explicit construction of the corresponding Green's function. Based on the obtained Green's function, we derive a closed-form integral representation of the solution and prove its existence and uniqueness. The results extend classical Green-function techniques to a wider class of fractional operators and provide analytical tools for further study of boundary and inverse problems associated with Prabhakar-type fractional differential equations.
Paper Structure (14 sections, 2 theorems, 349 equations)

This paper contains 14 sections, 2 theorems, 349 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Formulation of a problem and main statement.
  4. Proof of Theorem 1.
  5. Reduction to the system of first-order differential equations and corresponding initial-boundary problems.
  6. Reduction to the integral equation.
  7. Appendices
  8. First appendix
  9. Second appendix
  10. Third appendix
  11. Fourth appendix
  12. Fifth appendix
  13. Sixth appendix
  14. Seventh appendix

Key Result

Theorem 2.1

Let ${{t}^{1-\beta }}{{\varphi }_{0}}\left( t \right),$${{t}^{1-\beta }}{{\varphi }_{1}}\left( t \right)\in C\left[ 0;T \right],$$\tau \left( x \right)\in C\left[ 0;a \right],$${{t}^{1-\beta }}f\left( t,x \right)\in C\left( \overline{D} \right)$ and $f\left( t,x \right)$ also satisfies the Hölder c Then there exists a unique regular solution of the equation eq2.1 in the domain $D,$ satisfying the

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1