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An Efficient Analytical Approach for the Solution of Certain Fractional-Order Dynamical Systems

Author

Listed:
  • Ya Qin

    (Data Recovery Lab of Sichuan Province, Neijiang Normal University, Neijiang 641112, China
    School of Mathematics and Information Science, Neijiang Normal University, Neijiang 641112, China)

  • Adnan Khan

    (Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan)

  • Izaz Ali

    (Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan)

  • Maysaa Al Qurashi

    (Department of Mathematics, King Saud University, Riyadh 11495, Saudi Arabia)

  • Hassan Khan

    (Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan)

  • Rasool Shah

    (Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan)

  • Dumitru Baleanu

    (Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey
    Institute of Space Sciences, 077125 Magurele-Bucharest, Romania
    Department of Medical Research, China Medical University, Taichung 40402, Taiwan)

Abstract

Mostly, it is very difficult to obtained the exact solution of fractional-order partial differential equations. However, semi-analytical or numerical methods are considered to be an alternative to handle the solutions of such complicated problems. To extend this idea, we used semi-analytical procedures which are mixtures of Laplace transform, Shehu transform and Homotopy perturbation techniques to solve certain systems with Caputo derivative differential equations. The effectiveness of the present technique is justified by taking some examples. The graphical representation of the obtained results have confirmed the significant association between the actual and derived solutions. It is also shown that the suggested method provides a higher rate of convergence with a very small number of calculations. The problems with derivatives of fractional-order are also solved by using the present method. The convergence behavior of the fractional-order solutions to an integer-order solution is observed. The convergence phenomena described a very broad concept of the physical problems. Due to simple and useful implementation, the current methods can be used to solve problems containing the derivative of a fractional-order.

Suggested Citation

  • Ya Qin & Adnan Khan & Izaz Ali & Maysaa Al Qurashi & Hassan Khan & Rasool Shah & Dumitru Baleanu, 2020. "An Efficient Analytical Approach for the Solution of Certain Fractional-Order Dynamical Systems," Energies, MDPI, vol. 13(11), pages 1-14, May.
    • Handle: RePEc:gam:jeners:v:13:y:2020:i:11:p:2725-:d:364336
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    References listed on IDEAS

    as
    1. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    2. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    3. Hassan Khan & Adnan Khan & Maysaa Al-Qurashi & Rasool Shah & Dumitru Baleanu, 2020. "Modified Modelling for Heat Like Equations within Caputo Operator," Energies, MDPI, vol. 13(8), pages 1-14, April.
    4. Shehu Maitama, 2016. "A Hybrid Natural Transform Homotopy Perturbation Method for Solving Fractional Partial Differential Equations," International Journal of Differential Equations, Hindawi, vol. 2016, pages 1-7, September.
    5. He, Ji-Huan, 2005. "Application of homotopy perturbation method to nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 695-700.
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    Cited by:

    1. Constantin Bota & Bogdan Căruntu & Dumitru Ţucu & Marioara Lăpădat & Mădălina Sofia Paşca, 2020. "A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order," Mathematics, MDPI, vol. 8(8), pages 1-12, August.
    2. Mohamed M. Mousa & Fahad Alsharari, 2021. "Convergence and Error Estimation of a New Formulation of Homotopy Perturbation Method for Classes of Nonlinear Integral/Integro-Differential Equations," Mathematics, MDPI, vol. 9(18), pages 1-14, September.
    3. Faridul Islam & Aviral Kumar Tiwari & Wing-Keung Wong, 2021. "Editorial and Ideas for Research Using Mathematical and Statistical Models for Energy with Applications," Energies, MDPI, vol. 14(22), pages 1-4, November.
    4. Mohammed Kbiri Alaoui & Kamsing Nonlaopon & Ahmed M. Zidan & Adnan Khan & Rasool Shah, 2022. "Analytical Investigation of Fractional-Order Cahn–Hilliard and Gardner Equations Using Two Novel Techniques," Mathematics, MDPI, vol. 10(10), pages 1-19, May.

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