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A novel expansion iterative method for solving linear partial differential equations of fractional order

Author

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  • El-Ajou, Ahmad
  • Abu Arqub, Omar
  • Momani, Shaher
  • Baleanu, Dumitru
  • Alsaedi, Ahmed

Abstract

In this manuscript, we implement a relatively new analytic iterative technique for solving time–space-fractional linear partial differential equations subject to given constraints conditions based on the generalized Taylor series formula. The solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations. This method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics, physics, and engineering. Some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality, generality, and accuracy for solving such equations with different constraints conditions. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm.

Suggested Citation

  • El-Ajou, Ahmad & Abu Arqub, Omar & Momani, Shaher & Baleanu, Dumitru & Alsaedi, Ahmed, 2015. "A novel expansion iterative method for solving linear partial differential equations of fractional order," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 119-133.
    • Handle: RePEc:eee:apmaco:v:257:y:2015:i:c:p:119-133
    DOI: 10.1016/j.amc.2014.12.121
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    Cited by:

    1. Alquran, Marwan & Al-Khaled, Kamel & Sardar, Tridip & Chattopadhyay, Joydev, 2015. "Revisited Fisher’s equation in a new outlook: A fractional derivative approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 81-93.
    2. Mine Aylin Bayrak & Ali Demir & Ebru Ozbilge, 2022. "An Improved Version of Residual Power Series Method for Space‐Time Fractional Problems," Advances in Mathematical Physics, John Wiley & Sons, vol. 2022(1).
    3. Mohammed Shqair & Ahmad El-Ajou & Mazen Nairat, 2019. "Analytical Solution for Multi-Energy Groups of Neutron Diffusion Equations by a Residual Power Series Method," Mathematics, MDPI, vol. 7(7), pages 1-20, July.
    4. Omar Abu Arqub & Mohamed S. Osman & Abdel-Haleem Abdel-Aty & Abdel-Baset A. Mohamed & Shaher Momani, 2020. "A Numerical Algorithm for the Solutions of ABC Singular Lane–Emden Type Models Arising in Astrophysics Using Reproducing Kernel Discretization Method," Mathematics, MDPI, vol. 8(6), pages 1-15, June.
    5. Aylin Bayrak, Mine & Demir, Ali, 2018. "A new approach for space-time fractional partial differential equations by residual power series method," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 215-230.
    6. Benedict Barnes & Martin Anokye & Mohammed Muniru Iddrisu & Bismark Gawu & Emmanuel Afrifa, 2023. "The Analytic Solutions of the Fractional‐Order Model for the Spatial Epidemiology of the COVID‐19 Infection," Advances in Mathematical Physics, John Wiley & Sons, vol. 2023(1).
    7. Abu Arqub, Omar & Al-Smadi, Mohammed, 2020. "An adaptive numerical approach for the solutions of fractional advection–diffusion and dispersion equations in singular case under Riesz’s derivative operator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    8. Samir A. El-Tantawy & Rasool Shah & Albandari W. Alrowaily & Nehad Ali Shah & Jae Dong Chung & Sherif. M. E. Ismaeel, 2023. "A Comparative Study of the Fractional-Order Belousov–Zhabotinsky System," Mathematics, MDPI, vol. 11(7), pages 1-15, April.
    9. Abu Arqub, Omar & Al-Smadi, Mohammed, 2018. "Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 161-167.
    10. Wenjin Li & Yanni Pang, 2018. "An Iterative Method for Time‐Fractional Swift‐Hohenberg Equation," Advances in Mathematical Physics, John Wiley & Sons, vol. 2018(1).
    11. Al-Smadi, Mohammed & Arqub, Omar Abu, 2019. "Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 280-294.
    12. Dubey, Ved Prakash & Singh, Jagdev & Alshehri, Ahmed M. & Dubey, Sarvesh & Kumar, Devendra, 2022. "Forecasting the behavior of fractional order Bloch equations appearing in NMR flow via a hybrid computational technique," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    13. Arqub, Omar Abu & Maayah, Banan, 2019. "Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 394-402.

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