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On systems of fractional nonlinear partial differential equations

Ravshan Ashurov, Oqila Muhiddinova

Abstract

The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the equation does not depend on the solution of the problem and 2) it depends on the solution, but at the same time satisfies the classical Lipschitz condition with respect to this variable and an additional condition which guarantees a global existence of the solution. Sufficient conditions are found (in some cases they are necessary) on the initial function and on the right-hand side of the equation, which ensure the existence of a classical solution. In previously known works, linear but more general systems of fractional pseudodifferential equations were considered and the existence of a weak solution was proven in the special classes of distributions.

On systems of fractional nonlinear partial differential equations

Abstract

The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the equation does not depend on the solution of the problem and 2) it depends on the solution, but at the same time satisfies the classical Lipschitz condition with respect to this variable and an additional condition which guarantees a global existence of the solution. Sufficient conditions are found (in some cases they are necessary) on the initial function and on the right-hand side of the equation, which ensure the existence of a classical solution. In previously known works, linear but more general systems of fractional pseudodifferential equations were considered and the existence of a weak solution was proven in the special classes of distributions.
Paper Structure (8 sections, 14 theorems, 144 equations)

This paper contains 8 sections, 14 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliaries
  3. The Fourier and Laplace transforms. The Mittag-Leffler functions
  4. Fractional integrals and derivatives
  5. Matrix differential operators with constant symbols
  6. Linear system of differential equations
  7. Nonlinear system of differential equations
  8. Conclusions

Key Result

Lemma 2.1

(see KST, p. 43). For any $t\geq 0$ one has where constant $C$ does not depend on $t$ and $\mu$.

Theorems & Definitions (20)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 10 more