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Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations

BaoLi Yin, Yang Liu, Hong Li, Zhimin Zhang

TL;DR

Two families of novel fractional $\theta$-methods are introduced by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator $\mathit{I}^{\alpha}$ with a second order convergence rate.

Abstract

In this article, we introduce two families of novel fractional $θ$-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator $\mathit{I}^α$ with a second order convergence rate. A new fractional BT-$θ$ method connects the fractional BDF2 (when $θ=0$) with fractional trapezoidal rule (when $θ=1/2$), and another novel fractional BN-$θ$ method joins the fractional BDF2 (when $θ=0$) with the second order fractional Newton-Gregory formula (when $θ=1/2$). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different $θ$-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional $θ$-methods are A($\vartheta$)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis.

Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations

TL;DR

Two families of novel fractional -methods are introduced by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator with a second order convergence rate.

Abstract

In this article, we introduce two families of novel fractional -methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator with a second order convergence rate. A new fractional BT- method connects the fractional BDF2 (when ) with fractional trapezoidal rule (when ), and another novel fractional BN- method joins the fractional BDF2 (when ) with the second order fractional Newton-Gregory formula (when ). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different -methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional -methods are A()-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis.

Paper Structure

This paper contains 16 sections, 9 theorems, 55 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Two families of new fractional $\theta$ methods
  3. New fractional BT-$\theta$ method
  4. New fractional BN-$\theta$ method
  5. Convergence analysis
  6. Convergence for fractional BT-$\theta$ method
  7. Convergence for fractional BN-$\theta$ method
  8. Correction technique
  9. Stability region
  10. Stability regions for the fractional BT-$\theta$ method
  11. Stability regions for the fractional BN-$\theta$ method
  12. Numerical tests
  13. Example with a sufficiently smooth solution
  14. Example of solution with weak regularity
  15. Concluding remarks
  16. ...and 1 more sections

Key Result

Lemma 3.2

\newlabelConv.lemma.20 The LMM (conv.2) is zero-stable and consistent of order $2$. Hence by Lax-Richtmyer theorem, (conv.2) is convergent of order $2$ provided the error on the initial data tends to zeros as $O(h^2)$.

Figures (10)

  • Figure 1: $\alpha=\frac{1}{2}$, $\theta \in [0,\frac{1}{2}]$.
  • Figure 2: $\alpha=\frac{1}{2}$, $\theta \in (-\infty,0]$.
  • Figure 3: $\alpha=\frac{2}{3}$, $\theta \in [0,\frac{1}{2}]$.
  • Figure 4: $\alpha=\frac{2}{3}$, $\theta \in (-\infty,0]$.
  • Figure 5: $\alpha=\frac{1}{2}$, $\theta \in [0,1]$.
  • ...and 5 more figures

Theorems & Definitions (16)

  • Remark 2.1
  • Definition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Remark 3.6
  • Lemma 3.7
  • Remark 3.8
  • Definition 4.1
  • ...and 6 more