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Solvability of a Mixed Problem for a Time-Fractional PDE with Time-Space Degenerating Coefficients

Bakhodirjon Toshtemirov, Azizbek Mamanazarov

Abstract

In this paper, we investigate the unique solvability of a mixed boundary value problem for a fractional partial differential equation featuring a degenerate coefficient. By introducing a novel operator and applying the method of separation of variables, we establish the existence of eigenvalues and eigenfunctions for the associated spectral problem and prove that the operator possesses a discrete spectrum. Additionally, we establish the relationship between the given data and the unique solvability of the problem, offering new insights into how degeneracy influences fractional diffusion processes.

Solvability of a Mixed Problem for a Time-Fractional PDE with Time-Space Degenerating Coefficients

Abstract

In this paper, we investigate the unique solvability of a mixed boundary value problem for a fractional partial differential equation featuring a degenerate coefficient. By introducing a novel operator and applying the method of separation of variables, we establish the existence of eigenvalues and eigenfunctions for the associated spectral problem and prove that the operator possesses a discrete spectrum. Additionally, we establish the relationship between the given data and the unique solvability of the problem, offering new insights into how degeneracy influences fractional diffusion processes.

Paper Structure

This paper contains 11 sections, 12 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mittag-Leffler Function
  4. Caputo-like counterpart of the hyper-Bessel FDO with arbitrary starting point
  5. Relevant function spaces
  6. Dependence of the boundary conditions on the degree of degeneration
  7. Main Problem Outline and Exploration
  8. Spectral problem
  9. The order of Fourier coefficients
  10. The existence and uniqueness of the solution to Problem \ref{['problem4.1']}
  11. Uniqueness of solution

Key Result

Lemma 2.1

(see Podl) Let $\alpha<2, \, \beta\in\mathbb{R}$ and $\frac{\pi\alpha}{2}<\mu<\min\{\pi, \pi\alpha\}$. Then the following estimate holds true $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • ...and 14 more