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Analytical Investigation of Fractional-Order Cahn–Hilliard and Gardner Equations Using Two Novel Techniques

Author

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  • Mohammed Kbiri Alaoui

    (Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia)

  • Kamsing Nonlaopon

    (Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand)

  • Ahmed M. Zidan

    (Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia)

  • Adnan Khan

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

  • Rasool Shah

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

Abstract

In this paper, we used the natural decomposition approach with non-singular kernel derivatives to find the solution to nonlinear fractional Gardner and Cahn–Hilliard equations arising in fluid flow. The fractional derivative is considered an Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF) throughout this paper. We implement natural transform with the aid of the suggested derivatives to obtain the solution of nonlinear fractional Gardner and Cahn–Hilliard equations followed by inverse natural transform. To show the accuracy and validity of the proposed methods, we focused on two nonlinear problems and compared it with the exact and other method results. Additionally, the behavior of the results is demonstrated through tables and figures that are in strong agreement with the exact solutions.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:10:p:1643-:d:813386
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    References listed on IDEAS

    as
    1. Anas Arafa & Ghada Elmahdy, 2018. "Application of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flow," International Journal of Differential Equations, Hindawi, vol. 2018, pages 1-10, July.
    2. 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.
    3. Longfei Lin & Yansheng Liu & Daliang Zhao, 2021. "Study on Implicit-Type Fractional Coupled System with Integral Boundary Conditions," Mathematics, MDPI, vol. 9(4), pages 1-15, February.
    4. Longfei Lin & Yansheng Liu & Daliang Zhao, 2021. "Controllability of Impulsive ψ -Caputo Fractional Evolution Equations with Nonlocal Conditions," Mathematics, MDPI, vol. 9(12), pages 1-14, June.
    5. Meshari Alesemi & Naveed Iqbal & Thongchai Botmart, 2022. "Novel Analysis of the Fractional-Order System of Non-Linear Partial Differential Equations with the Exponential-Decay Kernel," Mathematics, MDPI, vol. 10(4), pages 1-17, February.
    6. 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.
    7. Noufe H. Aljahdaly & Ravi P. Agarwal & Rasool Shah & Thongchai Botmart, 2021. "Analysis of the Time Fractional-Order Coupled Burgers Equations with Non-Singular Kernel Operators," Mathematics, MDPI, vol. 9(18), pages 1-24, September.
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    Citations

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    Cited by:

    1. Wedad Albalawi & Rasool Shah & Nehad Ali Shah & Jae Dong Chung & Sherif M. E. Ismaeel & Samir A. El-Tantawy, 2023. "Analyzing Both Fractional Porous Media and Heat Transfer Equations via Some Novel Techniques," Mathematics, MDPI, vol. 11(6), pages 1-19, March.
    2. 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.

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