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The Method of Potentials for the Airy Equation of Fractional Order

Rakhimov Kamoladdin

Abstract

In this work, initial-boundary value problems for the time-fractional Airy equation are considered on different intervals. We study the properties of potentials for this equation and, using these properties, construct solutions to the considered problems. The uniqueness of the solution is proved using an analogue of the Gronwall-Bellman inequality and an a priori estimate.

The Method of Potentials for the Airy Equation of Fractional Order

Abstract

In this work, initial-boundary value problems for the time-fractional Airy equation are considered on different intervals. We study the properties of potentials for this equation and, using these properties, construct solutions to the considered problems. The uniqueness of the solution is proved using an analogue of the Gronwall-Bellman inequality and an a priori estimate.

Paper Structure

This paper contains 6 sections, 11 theorems, 143 equations.

Table of Contents

  1. Formulation of problems and some properties of potentials
  2. Formulation of the problems
  3. Potential theory
  4. Well-posedness of problems
  5. Uniqueness of solutions
  6. Existence of solutions

Key Result

Lemma 1

Let $\tau_k \in C[0,T]$, $k=1,2$. Define Then: (i) The functions $w_k(x,t)$, $k=1,2$, satisfy the homogeneous equation (ii) For all $x\in\mathbb R$,

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 6 more