# On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives

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## Abstract

**:**

## 1. Introduction

## 2. Exact Solution of the Three-Term Fractional Equation

## 3. A Solution of the Three-Term Fractional Equation Using the Laplace Transform Technique

## 4. Numerical Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Atanackovic, T.; Pilipovic, S.; Stankovic, B.; Zorica, D. Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes; Wiley-ISTE: London, UK, 2014. [Google Scholar]
- Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models; Imperial College Press: London, UK, 2010. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: San Diego, CA, USA, 2006. [Google Scholar]
- Povstenko, Y. Fractional Thermoelasticity; Springer: New York, NY, USA, 2015. [Google Scholar]
- Sierociuk, D.; Dzieliński, A.; Sarwas, G.; Petras, I.; Podlubny, I.; Skovranek, T. Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2013**, 371, 20120146. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Machado, J.A.T.; Silva, M.F.; Barbosa, R.S.; Jesus, I.S.; Reis, C.M.; Marcos, M.G.; Galhano, A.F. Some applications of fractional calculus in engineering. Math. Probl. Eng.
**2010**, 2010, 639801. [Google Scholar] - Sun, H.G.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y.Q. A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 64, 213–231. [Google Scholar] [CrossRef] - Jiang, H.; Liu, F.; Turner, I.; Burrage, K. Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput. Math. Appl.
**2012**, 64, 3377–3388. [Google Scholar] - Kukla, S.; Siedlecka, U. A fractional single-phase-lag model of heat conduction for describing propagation of the maximum temperature in a finite medium. Entropy
**2018**, 20, 876. [Google Scholar] [CrossRef] [Green Version] - Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications; Academic Press: New York, NY, USA, 1999. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type; Springer: Berlin, Germany, 2010. [Google Scholar]
- Kiryakova, V. The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Comput. Math. Appl.
**2010**, 59, 1885–1895. [Google Scholar] - Garra, R.; Garrappa, R. The Prabhakar or three parameter Mittag-Leffler function: Theory and application. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 56, 314–329. [Google Scholar] [CrossRef] [Green Version] - Gorenflo, R.; Mainardi, F.; Rogosin, S. Mittag-Leffler Function: Properties and Applications. In Handbook of Fractional Calculus with Applications, Volume 1: Basic Theory; Kochubei, A., Luchko, Y., Eds.; De Gruyter: Berlin, Germany, 2019; pp. 269–296. [Google Scholar]
- Kilbas, A.A.; Saigo, M.; Trujillo, J.J. On the generalized Wright function. Fract. Calc. Appl. Anal.
**2002**, 5, 437–460. [Google Scholar] - Luchko, Y. The Wright function and its applications. In Handbook of Fractional Calculus with Applications, Volume 1: Basic Theory; Kochubei, A., Luchko, Y., Eds.; De Gruyter: Berlin, Germany, 2019; pp. 241–268. [Google Scholar]
- Kiryakova, V. The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput. Math. Appl.
**2010**, 59, 1128–1141. [Google Scholar] - Johansson, F. Computing hypergeometric functions rigorously. ACM Trans. Math. Softw.
**2019**, 45, 1–26. [Google Scholar] [CrossRef] [Green Version] - Kuhlman, K.L. Review of inverse Laplace transform algorithms for Laplace-space numerical approaches. Numer. Algorithms
**2013**, 63, 339–355. [Google Scholar] [CrossRef] [Green Version] - Rani, D.; Mishra, V.; Cattani, C. Numerical inverse Laplace transform for solving a class of fractional differential equations. Symmetry
**2019**, 11, 530. [Google Scholar] [CrossRef] [Green Version] - Cohen, A.M. Numerical Methods for Laplace Transform Inversion; Springer: New York, NY, USA, 2007. [Google Scholar]
- Gzyl, H.; Tagliani, A.; Milev, M. Laplace transform inversion on the real line is truly ill-conditioned. Appl. Math. Comput.
**2013**, 219, 9805–9809. [Google Scholar] - Kukla, S.; Siedlecka, U. A numerical-analytical solution of multi-term fractional-order differential equations. Math. Methods Appl. Sci.
**2020**, 43, 4883–4894. [Google Scholar] [CrossRef] - Atanackovic, T.M.; Zorica, D. On the Bagley-Torvik equation. Appl. Mech.
**2013**, 80, 041013. [Google Scholar] [CrossRef] - Kexue, L.; Jigen, P. Laplace transform and fractional differential equations. Appl. Math. Lett.
**2011**, 24, 2019–2023. [Google Scholar] [CrossRef] [Green Version]

**Table 1.**Values of the function ${G}_{2;3/2}^{0}\left(t\right)$ computed using the infinite series and the analytical–numerical method: (a) the series of generalized hypergeometric functions (Equations (5), (15) and (17)); (b) the double series (Equation (33)); (c) the asymptotic expansion (Equation (34)); (d) the analytical–numerical method (Equation (30)).

$\mathit{t}$ | Equation (15) | Equation (33) | Equation (34) | Equation (30) |
---|---|---|---|---|

6 | 0.282747 | 0.282747 | – | 0.282747 |

8 | 0.048191 | 0.048191 | −60.446402 | 0.048191 |

10 | −0.112090 | −0.112090 | −0.112087 | −0.112090 |

12 | −0.167477 | −0.167477 | −0.167474 | −0.167477 |

14 | −0.139679 | −0.139679 | −0.139677 | −0.139679 |

16 | −0.072722 | −0.072722 | −0.072720 | −0.072722 |

18 | −0.007246 | −0.007243 | −0.007244 | −0.007246 |

20 | 0.033270 | 2775.324 | 0.033271 | 0.033270 |

22 | 0.044207 | – | 0.044208 | 0.044207 |

**Table 2.**The absolute values of the relative errors $\delta \left(t\right)$ for different values of exponent $w$.

$\mathit{t}$ | $\mathit{w}=1.5$ | $\mathit{w}=2.0$ | $\mathit{w}=\hspace{0.17em}2.5$ |
---|---|---|---|

10 | $2.68070\hspace{0.17em}\times \hspace{0.17em}{10}^{-7}$ | $9.24022\hspace{0.17em}\times \hspace{0.17em}{10}^{-9}$ | $3.13867\hspace{0.17em}\times \hspace{0.17em}{10}^{-10}$ |

20 | $1.51514\hspace{0.17em}\times \hspace{0.17em}{10}^{-7}$ | $1.12149\hspace{0.17em}\times \hspace{0.17em}{10}^{-9}$ | $5.52770\hspace{0.17em}\times \hspace{0.17em}{10}^{-11}$ |

30 | $1.10786\hspace{0.17em}\times \hspace{0.17em}{10}^{-7}$ | $7.39357\hspace{0.17em}\times \hspace{0.17em}{10}^{-10}$ | $4.58730\hspace{0.17em}\times \hspace{0.17em}{10}^{-11}$ |

40 | $5.11753\hspace{0.17em}\times \hspace{0.17em}{10}^{-8}$ | $4.06154\hspace{0.17em}\times \hspace{0.17em}{10}^{-10}$ | $3.11811\hspace{0.17em}\times \hspace{0.17em}{10}^{-11}$ |

50 | $2.79989\hspace{0.17em}\times \hspace{0.17em}{10}^{-8}$ | $1.92111\hspace{0.17em}\times \hspace{0.17em}{10}^{-10}$ | $4.45135\hspace{0.17em}\times \hspace{0.17em}{10}^{-11}$ |

**Table 3.**The absolute values of the relative errors $\delta \left(t\right)$ for different values of order $\beta $.

$\mathit{t}$ | $\mathit{\beta}=0.75$ | $\mathit{\beta}=0.8$ | $\mathit{\beta}=0.9$ | $\mathit{\beta}=0.95$ |
---|---|---|---|---|

10 | $6.04445\hspace{0.17em}\times \hspace{0.17em}{10}^{-17}$ | $9.48103\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ | $7.72466\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ | $2.30385\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ |

20 | $1.37729\hspace{0.17em}\times \hspace{0.17em}{10}^{-16}$ | $1.67329\hspace{0.17em}\times \hspace{0.17em}{10}^{-13}$ | $8.58474\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ | $2.29894\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ |

30 | $7.69217\hspace{0.17em}\times \hspace{0.17em}{10}^{-17}$ | $6.44413\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ | $4.8862\hspace{0.17em}0\times \hspace{0.17em}{10}^{-14}$ | $2.32710\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ |

40 | $8.09329\hspace{0.17em}\times \hspace{0.17em}{10}^{-17}$ | $1.12672\hspace{0.17em}\times \hspace{0.17em}{10}^{-13}$ | $8.29622\times \hspace{0.17em}{10}^{-14}$ | $2.33347\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ |

50 | $1.72121\hspace{0.17em}\times \hspace{0.17em}{10}^{-16}$ | $2.04122\hspace{0.17em}\times \hspace{0.17em}{10}^{-13}$ | $8.99029\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ | $2.32054\hspace{0.17em}\times \hspace{0.17em}{10}^{-14}$ |

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**MDPI and ACS Style**

Kukla, S.; Siedlecka, U.
On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives. *Symmetry* **2020**, *12*, 1355.
https://doi.org/10.3390/sym12081355

**AMA Style**

Kukla S, Siedlecka U.
On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives. *Symmetry*. 2020; 12(8):1355.
https://doi.org/10.3390/sym12081355

**Chicago/Turabian Style**

Kukla, Stanisław, and Urszula Siedlecka.
2020. "On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives" *Symmetry* 12, no. 8: 1355.
https://doi.org/10.3390/sym12081355