IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2021i1p24-d708213.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/1/24/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/1/24/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Seadawy, Aly R. & Ali, Safdar & Rizvi, Syed T.R., 2022. "On modulation instability analysis and rogue waves in the presence of external potential: The (n + 1)-dimensional nonlinear Schrödinger equation," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    2. Kudryashov, Nikolay A. & Zakharchenko, Anastasia S., 2014. "Painlevé analysis and exact solutions for the Belousov–Zhabotinskii reaction–diffusion system," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 111-117.
    3. Eslami, Mostafa, 2016. "Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 141-148.
    4. Kudryashov, N.A., 2015. "On nonlinear differential equation with exact solutions having various pole orders," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 173-177.
    5. Yusuf Pandir & Halime Ulusoy, 2013. "New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations," Journal of Mathematics, Hindawi, vol. 2013, pages 1-5, January.
    6. Tian, Yang & Tian, Hui & Cui, Qimei & Zhu, Xuzhen, 2024. "Phase transition phenomena in social propagation with dynamic fashion tendency and individual contact," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    7. Oke Davies Adeyemo & Lijun Zhang & Chaudry Masood Khalique, 2022. "Bifurcation Theory, Lie Group-Invariant Solutions of Subalgebras and Conservation Laws of a Generalized (2+1)-Dimensional BK Equation Type II in Plasma Physics and Fluid Mechanics," Mathematics, MDPI, vol. 10(14), pages 1-46, July.
    8. Innocent Simbanefayi & Chaudry Masood Khalique, 2020. "Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation," Mathematics, MDPI, vol. 8(6), pages 1-20, June.
    9. Kudryashov, Nikolay A. & Ivanova, Yulia S., 2016. "Painleve analysis and exact solutions for the modified Korteweg–de Vries equation with polynomial source," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 377-382.
    10. Fahmy, E.S., 2008. "Travelling wave solutions for some time-delayed equations through factorizations," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1209-1216.
    11. Vitanov, Nikolay K. & Dimitrova, Zlatinka I. & Vitanov, Kaloyan N., 2015. "Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 363-378.
    12. Mustafa Inc & Rubayyi T. Alqahtani & Ravi P. Agarwal, 2023. "W-Shaped Bright Soliton of the (2 + 1)-Dimension Nonlinear Electrical Transmission Line," Mathematics, MDPI, vol. 11(7), pages 1-13, April.
    13. Yang, Lijuan & Du, Xianyun & Yang, Qiongfen, 2016. "New variable separation solutions to the (2 + 1)-dimensional Burgers equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1271-1275.
    14. Md Nur Hossain & Faisal Alsharif & M. Mamun Miah & Mohammad Kanan, 2024. "Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization," Mathematics, MDPI, vol. 12(10), pages 1-19, May.
    15. Ramírez, J. & Romero, J.L. & Muriel, C., 2016. "Reductions of PDEs to second order ODEs and symbolic computation," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 122-136.
    16. Kudryashov, N.A. & Lavrova, S.F., 2021. "Dynamical features of the generalized Kuramoto-Sivashinsky equation," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    17. Zayed, E.M.E. & Alurrfi, K.A.E., 2016. "Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 111-131.
    18. Kudryashov, Nikolay A., 2019. "Exact solutions of the equation for surface waves in a convecting fluid," Applied Mathematics and Computation, Elsevier, vol. 344, pages 97-106.
    19. Andrei D. Polyanin & Alexander V. Aksenov, 2024. "Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions," Mathematics, MDPI, vol. 12(13), pages 1-29, July.
    20. Yıldırım, Yakup & Yaşar, Emrullah, 2018. "A (2+1)-dimensional breaking soliton equation: Solutions and conservation laws," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 146-155.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2021:i:1:p:24-:d:708213. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.