#
A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. A Diffusive-Representation-Based Numerical Scheme for Computing Fractional Derivatives

- (1)
- The computational complexity is $O\left(N\right)$ where N is the number of time steps over which the solution to the fractional differential equation is sought, whereas traditional methods usually have a cost of $O\left({N}^{2}\right)$ when implemented in a straightforward manner [11,13] or of $O(NlogN)$ or $O\left(N{log}^{2}N\right)$ [15,16,17,18] if more sophisticated implementations are used.
- (2)
- Due to the completely different way in which the inherent memory of the fractional differential operators is dealt with, the active memory requirements of the method are only $O\left(1\right)$ and not $O\left(N\right)$ for the traditional methods or at best $O(logN)$ for their modified versions [15,19].
- (3)
- Whereas some (but not all) traditional schemes require the use of a uniform mesh, this approach gives the user complete freedom to use any discretization whatsoever of the interval on which the fractional differential equation is to be solved.

## 3. Convergence Properties of the Numerical Method

**Remark**

**1.**

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Remark**

**2.**

**Proof**

**of**

**Lemma**

**1.**

**Remark**

**3.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Remark**

**4.**

## 4. Comments and Further Remarks

#### 4.1. The Stiffness of the Differential Equation for the Integrand

#### 4.2. The Error Bounds for the ODE Solver

#### 4.3. Choice of the Quadrature Formula

#### 4.4. The Smoothness of the Function y

#### 4.5. Diffusive Representations for Fractional Integrals

#### 4.6. The Discretization of the Interval $[a,a+T]$

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Diethelm, K.; Kiryakova, V.; Luchko, Y.; Machado, J.A.T.; Tarasov, V.E. Trends, directions for further research, and some open problems of fractional calculus. Nonlinear Dyn.
**2022**, 107, 3245–3270. [Google Scholar] [CrossRef] - Diethelm, K. A new diffusive representation for fractional derivatives, Part I: Construction, implementation and numerical examples. In Fractional Differential Equations: Modeling, Discretization, and Numerical Solvers; Springer: Heidelberg, Germany, 2022. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations; Springer: Berlin, Germany, 2010. [Google Scholar]
- Baffet, D. A Gauss-Jacobi kernel compression scheme for fractional differential equations. J. Sci. Comput.
**2019**, 79, 227–248. [Google Scholar] [CrossRef][Green Version] - Hinze, M.; Schmidt, A.; Leine, R.I. Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation. Fract. Calc. Appl. Anal.
**2019**, 22, 1321–1350. [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] - McLean, W. Exponential sum approximations for t
^{−β}. In Contemporary Computational Mathematics; Dick, J., Kuo, F.Y., Woźniakowski, H., Eds.; Springer: Cham, Switzerland, 2018; pp. 911–930. [Google Scholar] - Singh, S.J.; Chatterjee, A. Galerkin projections and finite elements for fractional order derivatives. Nonlinear Dyn.
**2006**, 45, 183–206. [Google Scholar] [CrossRef] - Yuan, L.; Agrawal, O.P. A numerical scheme for dynamic systems containing fractional derivatives. J. Vib. Acoust.
**2002**, 124, 321–324. [Google Scholar] [CrossRef][Green Version] - Zhang, W.; Capilnasiu, A.; Sommer, G.; Holzapfel, G.A.; Nordsletten, D. An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials. Comput. Methods Appl. Mech. Eng.
**2020**, 362, 112834. [Google Scholar] [CrossRef] [PubMed][Green Version] - Diethelm, K.; Ford, N.J.; Freed, A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn.
**2002**, 29, 3–22. [Google Scholar] [CrossRef] - Diethelm, K.; Ford, N.J.; Freed, A.D. Detailed error analysis for a fractional Adams method. Numer. Algorithms
**2004**, 36, 31–52. [Google Scholar] [CrossRef][Green Version] - Lubich, C. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput.
**1985**, 45, 463–469. [Google Scholar] [CrossRef] - Lubich, C. Discretized fractional calculus. SIAM J. Math. Anal.
**1986**, 17, 704–719. [Google Scholar] [CrossRef] - Ford, N.J.; Simpson, A.C. The numerical solution of fractional differential equations: Speed versus accuracy. Numer. Algorithms
**2001**, 26, 333–346. [Google Scholar] [CrossRef] - Garrappa, R. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics
**2018**, 6, 16. [Google Scholar] [CrossRef][Green Version] - Hairer, E.; Lubich, C.; Schlichte, M. Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput.
**1985**, 6, 532–541. [Google Scholar] [CrossRef] - Hairer, E.; Lubich, C.; Schlichte, M. Fast numerical solution of weakly singular Volterra integral equations. J. Comput. Appl. Math.
**1988**, 23, 87–98. [Google Scholar] [CrossRef] - Diethelm, K.; Freed, A.D. An efficient algorithm for the evaluation of convolution integrals. Comput. Math. Appl.
**2006**, 51, 51–72. [Google Scholar] [CrossRef][Green Version] - Mastroianni, G.; Monegato, G. Error Estimates for Gauss-Laguerre and Gauss-Hermite Quadrature Formulas. In Approximation and Computation; Zahar, R.V.M., Ed.; Int. Ser. Numer. Math. 119; Birkhäuser: Boston, MA, USA, 1994; pp. 421–434. [Google Scholar]
- Plato, R. Concise Numerical Mathematics; American Mathematical Society: Providence, RI, USA, 2003. [Google Scholar]
- Szego, G. Orthogonal Polynomials, 4th ed.; American Mathematical Society: Providence, RI, USA, 1975. [Google Scholar]
- Iserles, A. A First Course in the Numerical Analysis of Differential Equations; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Mastroianni, G.; Monegato, G. Truncated quadrature rules over (0,∞) and Nyström-type methods. SIAM J. Numer. Anal.
**2003**, 41, 1870–1892. [Google Scholar] [CrossRef] - Diethelm, K. Fast solution methods for fractional differential equations in the modeling of viscoelastic materials. In Proceedings of the 9th International Conference on Systems and Control (ICSC 2021), Caen, France, 24–26 November 2021. [Google Scholar]

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

© 2022 by the author. 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**

Diethelm, K. A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme. *Mathematics* **2022**, *10*, 1245.
https://doi.org/10.3390/math10081245

**AMA Style**

Diethelm K. A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme. *Mathematics*. 2022; 10(8):1245.
https://doi.org/10.3390/math10081245

**Chicago/Turabian Style**

Diethelm, Kai. 2022. "A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme" *Mathematics* 10, no. 8: 1245.
https://doi.org/10.3390/math10081245