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Representations of solutions of time-fractional multi-order systems of differential-operator equations

Sabir Umarov

Abstract

This paper is devoted to the general theory of systems of time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single order systems, and hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations this question remains open even in the case of ordinary differential equations. In this paper we obtain representation formulas for solutions of arbitrary fractional multi-order systems of differential-operator equations along with proving the existence and uniqueness theorems in appropriate topological-vector spaces. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.

Representations of solutions of time-fractional multi-order systems of differential-operator equations

Abstract

This paper is devoted to the general theory of systems of time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single order systems, and hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations this question remains open even in the case of ordinary differential equations. In this paper we obtain representation formulas for solutions of arbitrary fractional multi-order systems of differential-operator equations along with proving the existence and uniqueness theorems in appropriate topological-vector spaces. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.
Paper Structure (18 sections, 12 theorems, 249 equations)

This paper contains 18 sections, 12 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional derivatives
  4. Matrix-valued functions
  5. Classical ML functions
  6. Matrix-valued ML functions
  7. Vector-indiced matrix-valued ML functions
  8. Matrix-valued operators with singular symbols
  9. Main results
  10. Fractional multi-order systems of differential-operator equations: $m=2$
  11. Fractional multi-order systems of differential-operator equations: $m \ge 2$
  12. Fractional multi-order systems of differential-operator equations with triangular matrix-valued operators
  13. Commensurate and non-commensurate rational order systems
  14. The Riemann-Liouville case
  15. The case $m=2$ and $\beta_1 \ne \beta_2$
  16. ...and 3 more sections

Key Result

Lemma 2.1

For $0<\beta_1<\beta_2, \ \nu>0,$ and $k = 0,1,\dots,$ the following relations hold: where $I$ is the identity operator and $J^{\beta}$ is the fractional integral of order $\beta.$

Theorems & Definitions (22)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • Example 2.5
  • Remark 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 12 more