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On the explicit formula linking a function to the order of its fractional derivative

Vasyl Semenov, Nataliya Vasylyeva

Abstract

In this paper, given a certain regularity of a function $v$, we derive an explicit formula relating the order $ν_0\in(0,1)$ of the leading fractional derivative in a fractional differential operator $\mathbf{D_t}$ with the variable coefficients $r_i=r_i(x,t)$ and the function $v$ on which this operator acts. Moreover, we discuss application of this result in the reconstruction of the memory order of semilinear subdiffusion with memory terms. To achieve this aim, we analyze some inverse problems to multi-term fractional in time ordinary and partial differential equations with smooth local or nonlocal additional measurements for small time. In conclusion, we discuss how this formula may be exploited to numerical computation of $ν_0$ in the case of discrete noisy observation in the corresponding inverse problems. Our theoretical results along with the computational algorithm are supplemented by numerical tests.

On the explicit formula linking a function to the order of its fractional derivative

Abstract

In this paper, given a certain regularity of a function , we derive an explicit formula relating the order of the leading fractional derivative in a fractional differential operator with the variable coefficients and the function on which this operator acts. Moreover, we discuss application of this result in the reconstruction of the memory order of semilinear subdiffusion with memory terms. To achieve this aim, we analyze some inverse problems to multi-term fractional in time ordinary and partial differential equations with smooth local or nonlocal additional measurements for small time. In conclusion, we discuss how this formula may be exploited to numerical computation of in the case of discrete noisy observation in the corresponding inverse problems. Our theoretical results along with the computational algorithm are supplemented by numerical tests.
Paper Structure (13 sections, 13 theorems, 143 equations, 3 tables)

This paper contains 13 sections, 13 theorems, 143 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Functional Spaces and Notation
  3. Explicit Formula for the Order $\nu_0$
  4. Reconstruction of the Memory Order in Multi-term Semilinear Subdiffusion
  5. Proof of Theorems \ref{['t3.1']}-\ref{['t3.2']}
  6. Key Lemmas
  7. Completion of the proof of Theorems \ref{['t3.1']}-\ref{['t3.2']}
  8. Proof of Theorems \ref{['t4.1']}-\ref{['t4.5']}
  9. Proof of Theorem \ref{['t4.1']}
  10. Verification of Theorems \ref{['t4.2']}-\ref{['t4.5']}
  11. Numerical simulation
  12. Algorithm of numerical reconstruction
  13. Numerical tests

Key Result

Theorem 3.1

Let $\mathbf{D}_t$ be the I type FDO and let assumptions h1 and h2 hold. If $\mathbf{D}_tv|_{t=0}\neq 0$, then and, besides, for all $t\in[0,T^*]$ the representation holds with

Theorems & Definitions (29)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.1
  • Remark 3.4
  • Corollary 3.1
  • Theorem 4.1
  • ...and 19 more