# An Operator-Based Scheme for the Numerical Integration of FDEs

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## Abstract

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## 1. Introduction

## 2. Preliminaries and Motivation

#### 2.1. The Generalized Differential Operator Scheme for ODEs

#### 2.1.1. The Construction of Analytic Solutions to ODEs in the Series Form

#### 2.1.2. The Construction of Closed-Form Solutions to ODEs

#### 2.1.3. Truncated Series and Shifted Centers of the Expansion

#### 2.2. The Ordinary Riccati Equation and Its Solution

#### 2.3. The Fractional Power Series and Caputo Differentiation

#### 2.4. Motivation: The Fractional Riccati Equation

#### 2.5. Transformation of the FDE into the Characteristic ODE

## 3. The Development of the Numerical FDE Integration Scheme

#### 3.1. Adaptive Step-Size Selection Strategy for the FDE Integration Scheme

- Step 1. Let ${c}_{0}={s}_{0}=1$. The absolute differences ${\Delta}_{N}(x,{c}_{0},{s}_{0})=|\widehat{y}\left(x\right)-{\widehat{y}}_{N}(x,{c}_{0},{s}_{0})|$ are computed for $N=0,\cdots ,10$ and $x\in [1,L]$, where L is the upper bound of x. The contour plot depicting various levels of ${\Delta}_{N}(x,{c}_{0},{s}_{0})$ is presented in Figure 1a. It can be observed that for a fixed value of N the value of ${\Delta}_{N}(x,{c}_{0},{s}_{0})$ increases as x increases. New initial values ${c}_{1}$, ${s}_{1}$ for the next approximation are computed as follows:$${c}_{1}=\underset{x}{arg\; max}{\Delta}_{N}(x,{c}_{0},{s}_{0})\le \delta ;\phantom{\rule{1.em}{0ex}}{s}_{1}={\widehat{y}}_{N}({c}_{1},{c}_{0},{s}_{0}),$$
- Step $k=2,3,\cdots ,K$. Analogous computations are performed for steps $k=2,3,\cdots ,K$. Firstly, differences ${\Delta}_{N}(x,{c}_{k-1},{s}_{k-1})=|\widehat{y}\left(x\right)-{\widehat{y}}_{N}(x,{c}_{k-1},{s}_{k-1})|$ are evaluated for $N=0,\cdots ,10$ and $x\in [{c}_{k-1},L]$ and then new initial values ${c}_{k}$, ${s}_{k}$ are computed:$${c}_{k}=\underset{x}{arg\; max}{\Delta}_{N}(x,{c}_{k-1},{s}_{k-1})\le \delta ;\phantom{\rule{1.em}{0ex}}{s}_{1}={\widehat{y}}_{N}({c}_{k},{c}_{k-1},{s}_{k-1}).$$

#### 3.2. The Implementation of the Numerical FDE Integration Scheme

- Transform the FDE (40)–(41) into the characteristic ODE using the procedure described in Section 2.5:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \frac{\mathrm{d}\widehat{y}}{\mathrm{d}x}=P(\widehat{y},{v}_{0});\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \widehat{y}\left({c}_{0}\right)={s}_{0}.\hfill \end{array}$$
- Obtain analytic expressions of coefficients ${p}_{j}(c,s)$ in the series solution (35) to the ODE (42)–(43) (see Section 2.1.1).
- Fix the values of the following parameters: the order of the approximation N, the upper bound of the independent variable L, the upper bound for the step-size ${h}^{\left(U\right)}$, the upper bound for the change in the numerical solution $\Delta {\widehat{y}}_{N}^{\left(U\right)}$. Note that the recommended values for the parameters ${h}^{\left(U\right)}$ and $\Delta {\widehat{y}}_{N}^{\left(U\right)}$ are derived from the study presented in the previous section (Figure 5). The value ${h}^{\left(U\right)}$ corresponds to the highest value of h on the regression line, while the value $\Delta {\widehat{y}}_{N}^{\left(U\right)}$ corresponds to the highest value of $\Delta {\widehat{y}}_{N}$ on the regression line.
- Repeat the following steps until the upper bound L is reached ($k=0,1,2\cdots $):
- Evaluate coefficients ${p}_{j}({c}_{k},{s}_{k}),j=1,\cdots ,N$.
- Find the lowest value of x at which at least one of the following conditions is violated:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {h}_{k}\left(x\right)=x-{c}_{k}\le {h}^{\left(U\right)};\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \Delta {\widehat{y}}_{N}^{\left(k\right)}\left(x\right)=\underset{{c}_{k}\le \tilde{x}\le x}{max}{\widehat{y}}_{N}(\tilde{x},{c}_{k},{s}_{k})-\underset{{c}_{k}\le \tilde{x}\le x}{max}{\widehat{y}}_{N}(\tilde{x},{c}_{k},{s}_{k})\le \Delta {\widehat{y}}_{N}^{\left(U\right)};\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \Delta {\widehat{y}}_{N}^{\left(k\right)}\left(x\right)\le {\kappa}_{0}^{\left(N\right)}+{\kappa}_{1}^{\left(N\right)}{h}_{k}\left(x\right),\hfill \end{array}$$
- Assign new initial values:$${c}_{k+1}=x-\epsilon ;\phantom{\rule{1.em}{0ex}}{s}_{k+1}={\widehat{y}}_{N}({c}_{k+1},{c}_{k},{s}_{k}),$$

## 4. The Application of the Proposed Numerical FDE Integration Scheme

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analytical Expressions of the Coefficients p_{j} (c,s) for the ODE (30)–(31)

## Appendix B. Transformation of FDE (49) into the Characteristic ODE (51)

## Appendix C. Analytical Expressions of the Coefficients p_{j}(c,s) for the ODE (51)–(52)

## References

- Heymans, N. Fractional calculus description of non-linear viscoelastic behaviour of polymers. Nonlinear Dyn.
**2004**, 38, 221–231. [Google Scholar] [CrossRef] - Li, X.; Tian, X. Fractional order thermo-viscoelastic theory of biological tissue with dual phase lag heat conduction model. Appl. Math. Model.
**2021**, 95, 612–622. [Google Scholar] [CrossRef] - Dubey, V.P.; Dubey, S.; Kumar, D.; Singh, J. A computational study of fractional model of atmospheric dynamics of carbon dioxide gas. Chaos Solitons Fractals
**2021**, 142, 110375. [Google Scholar] [CrossRef] - Acay, B.; Inc, M. Fractional modeling of temperature dynamics of a building with singular kernels. Chaos Solitons Fractals
**2021**, 142, 110482. [Google Scholar] [CrossRef] - Chen, Y.; Liu, F.; Yu, Q.; Li, T. Review of fractional epidemic models. Appl. Math. Model.
**2021**, 97, 281–307. [Google Scholar] [CrossRef] - Biala, T.A.; Khaliq, A. A fractional-order compartmental model for the spread of the COVID-19 pandemic. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 98, 105764. [Google Scholar] [CrossRef] [PubMed] - Rezaei, M.; Yazdanian, A.; Ashrafi, A.; Mahmoudi, S. Numerical pricing based on fractional Black–Scholes equation with time-dependent parameters under the CEV model: Double barrier options. Comput. Math. Appl.
**2021**, 90, 104–111. [Google Scholar] [CrossRef] - Tarasov, V.E. Fractional econophysics: Market price dynamics with memory effects. Phys. A Stat. Mech. Appl.
**2020**, 557, 124865. [Google Scholar] [CrossRef] - Betancur-Herrera, D.E.; Muñoz-Galeano, N. A numerical method for solving Caputo’s and Riemann-Liouville’s fractional differential equations which includes multi-order fractional derivatives and variable coefficients. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 84, 105180. [Google Scholar] [CrossRef] - Alchikh, R.; Khuri, S. Numerical solution of a fractional differential equation arising in optics. Optik
**2020**, 208, 163911. [Google Scholar] [CrossRef] - Mendes, E.M.; Salgado, G.H.; Aguirre, L.A. Numerical solution of Caputo fractional differential equations with infinity memory effect at initial condition. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 69, 237–247. [Google Scholar] [CrossRef] - Firoozjaee, M.; Jafari, H.; Lia, A.; Baleanu, D. Numerical approach of Fokker–Planck equation with Caputo–Fabrizio fractional derivative using Ritz approximation. J. Comput. Appl. Math.
**2018**, 339, 367–373. [Google Scholar] [CrossRef] - Kheybari, S. Numerical algorithm to Caputo type time–space fractional partial differential equations with variable coefficients. Math. Comput. Simul.
**2021**, 182, 66–85. [Google Scholar] [CrossRef] - Esmaeili, S.; Shamsi, M.; Luchko, Y. Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput. Math. Appl.
**2011**, 62, 918–929. [Google Scholar] [CrossRef][Green Version] - Han, W.; Chen, Y.M.; Liu, D.Y.; Li, X.L.; Boutat, D. Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis. Adv. Differ. Equ.
**2018**, 2018, 1–22. [Google Scholar] [CrossRef] - 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] - Maleknejad, K.; Rashidinia, J.; Eftekhari, T. Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach. Numer. Methods Partial Differ. Equ.
**2021**, 37, 707–731. [Google Scholar] [CrossRef] - Babaaghaie, A.; Maleknejad, K. Numerical solutions of nonlinear two-dimensional partial Volterra integro-differential equations by Haar wavelet. J. Comput. Appl. Math.
**2017**, 317, 643–651. [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] - Jafari, H.; Tajadodi, H. He’s variational iteration method for solving fractional Riccati differential equation. Int. J. Differ. Equ.
**2010**, 2010, 764738. [Google Scholar] [CrossRef][Green Version] - Khan, N.A.; Ara, A.; Jamil, M. An efficient approach for solving the Riccati equation with fractional orders. Comput. Math. Appl.
**2011**, 61, 2683–2689. [Google Scholar] [CrossRef][Green Version] - Gohar, M. Approximate Solution to Fractional Riccati Differential Equations. Fractals
**2019**, 27, 1950128. [Google Scholar] [CrossRef] - Timofejeva, I.; Navickas, Z.; Telksnys, T.; Marcinkevičius, R.; Yang, X.J.; Ragulskis, M.K. The extension of analytic solutions to FDEs to the negative half-line. AIMS Math.
**2021**, 6, 3257–3271. [Google Scholar] [CrossRef] - Navickas, Z.; Bikulciene, L.; Ragulskis, M. Generalization of Exp-function and other standard function methods. Appl. Math. Comput.
**2010**, 216, 2380–2393. [Google Scholar] [CrossRef] - Navickas, Z.; Marcinkevicius, R.; Telksnys, T.; Ragulskis, M. Existence of second order solitary solutions to Riccati differential equations coupled with a multiplicative term. IMA J. Appl. Math.
**2016**, 81, 1163–1190. [Google Scholar] [CrossRef] - Kurakin, V.; Kuzmin, A.; Mikhalev, A.; Nechaev, A. Linear recurring sequences over rings and modules. J. Math. Sci.
**1995**, 76, 2793–2915. [Google Scholar] [CrossRef] - Zaitsev, V.F.; Polyanin, A.D. Handbook of Exact Solutions for Ordinary Differential Equations; CRC Press: Boca Raton, FL, USA, 2002. [Google Scholar]
- Navickas, Z.; Ragulskis, M. How far one can go with the Exp-function method? Appl. Math. Comput.
**2009**, 211, 522–530. [Google Scholar] [CrossRef] - Andrews, G.E.; Askey, R.; Roy, R. Special Functions; Number 71; Cambridge University Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Navickas, Z.; Telksnys, T.; Timofejeva, I.; Marcinkevičius, R.; Ragulskis, M. An operator-based approach for the construction of closed-form solutions to fractional differential equations. Math. Model. Anal.
**2018**, 23, 665–685. [Google Scholar] [CrossRef] - Zhou, Y.; Wang, J.; Zhang, L. Basic Theory of Fractional Differential Equations; World Scientific: Singapore, 2016. [Google Scholar]
- Navickas, Z.; Telksnys, T.; Marcinkevicius, R.; Ragulskis, M. Operator-based approach for the construction of analytical soliton solutions to nonlinear fractional-order differential equations. Chaos Solitons Fractals
**2017**, 104, 625–634. [Google Scholar] [CrossRef] - Scott, A. Encyclopedia of Nonlinear Science; Routledge: Abingdon-on-Thames, UK, 2006. [Google Scholar]

**Figure 1.**The determination of the second set of initial values for the numerical solution (35). The first set of initial values is ${c}_{0}=1$, ${s}_{0}=1$. Part (

**a**) depicts a contour plot of the error for different values of N. Part (

**b**) depicts the next initial points for different errors for $N=6$.

**Figure 2.**The determination of the third set of initial values for the numerical solution (35). The second set of initial values is ${c}_{1}=1.229$, ${s}_{1}=0.750$. Part (

**a**) depicts a contour plot of the error for different values of N. Part (

**b**) depicts the next initial points for different errors for $N=6$.

**Figure 3.**The determination of the fourth set of initial values for the numerical solution (35). The third set of initial values is ${c}_{2}=1.409$, ${s}_{2}=0.578$. Part (

**a**) depicts a contour plot of the error for different values of N. Part (

**b**) depicts the next initial points for different errors for $N=6$.

**Figure 4.**Gray and black solid lines correspond to the exact solution and the piecewise-polynomial approximation to (32)–(33), respectively ($N=6$, $\delta ={10}^{-5}$). Black dashed lines separate the parts of the numerical solution obtained at different steps. Circled digits denote the step number.

**Figure 5.**The relationship between $\Delta {\widehat{y}}_{N}$ (the change in the numerical solution ${\widehat{y}}_{N}\left(x\right)$) and the step-size h. Parts (

**a**,

**b**) correspond to $N=6$ and $N=7$, respectively.

Step k | ${\mathit{h}}_{\mathit{k}}$ | $\Delta {\widehat{\mathit{y}}}_{6}^{\left(\mathit{k}\right)}$ |
---|---|---|

1 | 0.080 | 0.1992 |

2 | 0.128 | 0.1990 |

3 | 0.217 | 0.2000 |

4 | 0.311 | 0.1743 |

5 | 0.393 | 0.1500 |

6 | 0.399 | 0.1185 |

7 | 0.399 | 0.1006 |

8 | 0.073 | 0.0170 |

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**MDPI and ACS Style**

Timofejeva, I.; Navickas, Z.; Telksnys, T.; Marcinkevicius, R.; Ragulskis, M.
An Operator-Based Scheme for the Numerical Integration of FDEs. *Mathematics* **2021**, *9*, 1372.
https://doi.org/10.3390/math9121372

**AMA Style**

Timofejeva I, Navickas Z, Telksnys T, Marcinkevicius R, Ragulskis M.
An Operator-Based Scheme for the Numerical Integration of FDEs. *Mathematics*. 2021; 9(12):1372.
https://doi.org/10.3390/math9121372

**Chicago/Turabian Style**

Timofejeva, Inga, Zenonas Navickas, Tadas Telksnys, Romas Marcinkevicius, and Minvydas Ragulskis.
2021. "An Operator-Based Scheme for the Numerical Integration of FDEs" *Mathematics* 9, no. 12: 1372.
https://doi.org/10.3390/math9121372