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Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions

Author

Listed:
  • Nabendra Parumasur

    (School of Mathematics Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
    These authors contributed equally to this work.)

  • Rasheed A. Adetona

    (School of Mathematics Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
    These authors contributed equally to this work.)

  • Pravin Singh

    (School of Mathematics Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
    These authors contributed equally to this work.)

Abstract

This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space.

Suggested Citation

  • Nabendra Parumasur & Rasheed A. Adetona & Pravin Singh, 2023. "Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions," Mathematics, MDPI, vol. 11(8), pages 1-21, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1847-:d:1122519
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    References listed on IDEAS

    as
    1. Ramadan, Mohamed A. & El-Danaf, Talaat S., 2005. "Numerical treatment for the modified burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(2), pages 90-98.
    2. Moghimi, Mahdi & Hejazi, Fatemeh S.A., 2007. "Variational iteration method for solving generalized Burger–Fisher and Burger equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1756-1761.
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