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Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

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  • Kai Fan
  • Cunlong Zhou

Abstract

With the help of Maple, the precise traveling wave solutions of three fractal‐order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)‐expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional‐order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

Suggested Citation

  • Kai Fan & Cunlong Zhou, 2020. "Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order," Advances in Mathematical Physics, John Wiley & Sons, vol. 2020(1).
    • Handle: RePEc:wly:jnlamp:v:2020:y:2020:i:1:n:3758353
    DOI: 10.1155/2020/3758353
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    References listed on IDEAS

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    1. Hasan Bulut & Haci Mehmet Baskonus & Yusuf Pandir, 2013. "The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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    3. Hasan Bulut & Haci Mehmet Baskonus & Yusuf Pandir, 2013. "The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, September.
    4. Ahmet Bekir & Özkan Güner & Adem C. Cevikel, 2013. "Fractional Complex Transform and exp‐Function Methods for Fractional Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    5. Khater, Mostafa M.A. & Attia, Raghda A.M. & Abdel-Aty, Abdel-Haleem & Alharbi, W. & Lu, Dianchen, 2020. "Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
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    Cited by:

    1. Ricardo Almeida, 2025. "A Unified Approach to Implicit Fractional Differential Equations with Anti-Periodic Boundary Conditions," Mathematics, MDPI, vol. 13(17), pages 1-20, September.

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