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Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering

Author

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  • Chaudry Masood Khalique

    (Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, Mafikeng Campus, North-West University, Private Bag X 2046, Mmabatho 2735, South Africa
    Department of Mathematics and Informatics, Azerbaijan University, Jeyhun Hajibeyli Str., 71, Baku AZ1007, Azerbaijan
    The authors contributed equally to this work.)

  • Karabo Plaatjie

    (Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, Mafikeng Campus, North-West University, Private Bag X 2046, Mmabatho 2735, South Africa
    The authors contributed equally to this work.)

Abstract

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

Suggested Citation

  • Chaudry Masood Khalique & Karabo Plaatjie, 2021. "Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering," Mathematics, MDPI, vol. 10(1), pages 1-17, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2021:i:1:p:24-:d:708213
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    References listed on IDEAS

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    1. Xu, Guoan & Zhang, Yi & Li, Jibin, 2022. "Exact solitary wave and periodic-peakon solutions of the complex Ginzburg–Landau equation: Dynamical system approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 191(C), pages 157-167.
    2. Çelik, Nisa & Seadawy, Aly R. & Sağlam Özkan, Yeşim & Yaşar, Emrullah, 2021. "A model of solitary waves in a nonlinear elastic circular rod: Abundant different type exact solutions and conservation laws," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    3. Kudryashov, Nikolai A., 2005. "Simplest equation method to look for exact solutions of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1217-1231.
    4. Oleksii Patsiuk & Sergii Kovalenko, 2015. "Symmetry reduction and exact solutions of the non-linear Black--Scholes equation," Papers 1512.06151, arXiv.org, revised Mar 2018.
    5. Wen, Xiaoyong & Lü, Dazhao, 2009. "Extended Jacobi elliptic function expansion method and its application to nonlinear evolution equation," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1454-1458.
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