# High-Order Schemes for Nonlinear Fractional Differential Equations

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Overview

## 3. Approximation of the History Term

## 4. Approximation of the History Term

#### 4.1. Quadrature Rules on Non-Uniform Meshes

#### 4.2. The Spectral Deferred Correction Framework

#### 4.3. Order of Convergence

**Theorem**

**1.**

## 5. Adaptive Implementation

#### 5.1. Accuracy Control

- We verify that $\left(\right)open="\parallel "\; close="\parallel ">{E}_{n}^{[m+1]}$, where the vector ${E}_{n}^{[m+1]}$ (see (17)),$${E}_{n}^{[m+1]}=\left(\right)open="("\; close=")">{E}_{n,1}^{[m+1]},{E}_{n,2}^{[m+1]},\dots ,{E}_{n,p}^{[m+1]}$$
- Once ${Y}_{n}^{[m+1]}$ has been computed; the value of the approximate solution at an arbitrary time $t\in $$[{t}_{n},{t}_{n+1}]$ can be obtained by the Lagrange interpolant. We apply the interpolation process to both ${Y}_{n}^{[m+1]}$ and ${Y}_{n}^{\left[m\right]}$ and demand that the difference be less than $\u03f5$. This indicates that both the correction process and the discretization have converged to precision.

#### 5.2. Computational Details and Numerical Tests

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Dastgerdi, M.; Bastani, A. Solving Parametric Fractional Differential Equations Arising from the Rough Heston Model Using Quasi-Linearization and Spectral Collocation. SIAM J. Financ. Math.
**2020**, 11, 1063–1097. [Google Scholar] [CrossRef] - Li, C.; Zeng, F. Numerical Methods for Fractional Calculus. In Chapman & Hall/CRC Numerical Analysis and Scientific Computing; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Metzler, R.; Klafter, J. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A
**2004**, 37, 161–208. [Google Scholar] [CrossRef] - Perdikaris, P.; Karniadakis, G.E. Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng.
**2014**, 42, 1012–1023. [Google Scholar] [CrossRef] - Wang, W.; Chen, X.; Ding, D.; Lei, S.L. Circulant preconditioning technique for barrier options pricing under fractional diffusion models. Int. J. Comput. Math.
**2015**, 92, 2596–2614. [Google Scholar] [CrossRef] - Gao, G.H.; Sun, Z.Z.; Zhang, H.W. A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys.
**2014**, 259, 33–50. [Google Scholar] [CrossRef] - Jiang, S.; Zhang, J.; Zhang, Q.; Zhang, Z. Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys.
**2017**, 21, 650–678. [Google Scholar] [CrossRef] [Green Version] - Li, C.; Yi, Q.; Chen, A. Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys.
**2016**, 316, 614–631. [Google Scholar] [CrossRef] - Lv, C.W.; Xu, C.J. Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput.
**2016**, 38, A2699–A2724. [Google Scholar] [CrossRef] - Zayernouri, M.; Karniadakis, G.E. Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys.
**2014**, 257, 460–480. [Google Scholar] [CrossRef] - Zeng, F.; Turner, I.; Burrage, K. A Stable Fast Time-Stepping Method for Fractional Integral and Derivative Operators. J. Sci. Comput.
**2018**, 77, 283–307. [Google Scholar] [CrossRef] - Yan, Y.G.; Sun, Z.Z.; Zhang, J.W. Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: A second-order scheme. Commun. Comput. Phys.
**2017**, 22, 1028–1048. [Google Scholar] [CrossRef] - Baffet, D.; Hesthaven, J.S. High-order accurate adaptive kernel compression time-stepping schemes for fractional differential equations. J. Sci. Comput.
**2019**, 72, 1169–1195. [Google Scholar] [CrossRef] - Baffet, D.; Hesthaven, J.S. A kernel compression scheme for fractional differential equations. SIAM J. Numer. Anal.
**2017**, 55, 496–520. [Google Scholar] [CrossRef] [Green Version] - Cao, W.; Zhang, Z.; Karniadakis, G.E. Time-splitting schemes for fractional differential equations I: Smooth solutions. SIAM J. Sci. Comput.
**2015**, 37, A1752–A1776. [Google Scholar] [CrossRef] - Li, J.R. A fast time stepping method for evaluating fractional integrals. SIAM J. Sci. Comput.
**2010**, 31, 4696–4714. [Google Scholar] [CrossRef] [Green Version] - Jin, B.; Lazarov, R.; Pasciak, J.; Zhou, Z. Error analysis of a finite element method for the space-fractional parabolic equation. SIAM J. Numer. Anal.
**2014**, 52, 2272–2294. [Google Scholar] [CrossRef] [Green Version] - Tian, W.Y.; Deng, W.; Wu, Y. Polynomial spectral collocation method for space fractional advection-diffusion equation. Numer. Methods Partial. Differ. Equ.
**2014**, 30, 514–535. [Google Scholar] [CrossRef] [Green Version] - Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Vinagre Jara, B.M. Matrix approach to discrete fractional calculus. II. Partial fractional differential equations. J. Comput. Phys.
**2009**, 228, 3137–3153. [Google Scholar] [CrossRef] [Green Version] - Stynes, M.; O’Riordan, E.; Gracia, J. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal.
**2017**, 55, 1057–1079. [Google Scholar] [CrossRef] [Green Version] - Rauh, A.; Jaulin, L. Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations. Fractal Fract.
**2021**, 5, 17. [Google Scholar] [CrossRef] - Jin, S.; Xie, J.; Qu, J.; Chen, Y. A Numerical Method for Simulating Viscoelastic Plates Based on Fractional Order Model. Fractal Fract.
**2022**, 6, 150. [Google Scholar] [CrossRef] - Bohmer, K.; Stetter, H.J. Defect Correction Methods Theory and Applications; Springer: New York, NY, USA, 1984. [Google Scholar]
- Buvoli, T. A Class of Exponential Integrators Based on Spectral Deferred Correction. SIAM J. Sci. Comput.
**2020**, 42, A1–A27. [Google Scholar] [CrossRef] [Green Version] - Dutt, A.; Greengard, L.; Rokhlin, V. Spectral deferred correction methods for ordinary differential equations. BIT
**2000**, 40, 241–266. [Google Scholar] [CrossRef] [Green Version] - Causley, M.F.; Seal, D.C. On the convergence of spectral deferred correction methods. Commun. Appl. Math. Comput. Sci.
**2019**, 14, 33–64. [Google Scholar] [CrossRef] [Green Version] - Ong, B.W.; Spiteri, R.J. Deferred Correction Methods for Ordinary Differential Equations. J. Sci. Comput.
**2020**, 38, 60–83. [Google Scholar] [CrossRef] - Cafagna, D.; Grassi, G. Fractional-Oder Chua’s Circuit: Time-Domain Analysis, Bifurcation, Chaotic Behavior and Test for Chaos. Int. J. Bifurcat Chaos
**2008**, 18, 615–639. [Google Scholar] [CrossRef]

**Figure 1.**(

**top left**)—solution plot; (

**top right**)—accuracy vs. time; (

**bottom left**)—step-size vs. time; (

**bottom right**)—accuracy vs. ${h}_{avg}$.

**Figure 2.**Chua’s circuit: (

**left**)—solution plot; (

**right**)—the curve $\left(\right)$ in the $\left(\right)$ plane.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alsayyed, O.; Awawdeh, F.; Al-Shara’, S.; Rawashdeh, E.
High-Order Schemes for Nonlinear Fractional Differential Equations. *Fractal Fract.* **2022**, *6*, 748.
https://doi.org/10.3390/fractalfract6120748

**AMA Style**

Alsayyed O, Awawdeh F, Al-Shara’ S, Rawashdeh E.
High-Order Schemes for Nonlinear Fractional Differential Equations. *Fractal and Fractional*. 2022; 6(12):748.
https://doi.org/10.3390/fractalfract6120748

**Chicago/Turabian Style**

Alsayyed, Omar, Fadi Awawdeh, Safwan Al-Shara’, and Edris Rawashdeh.
2022. "High-Order Schemes for Nonlinear Fractional Differential Equations" *Fractal and Fractional* 6, no. 12: 748.
https://doi.org/10.3390/fractalfract6120748