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An Innovative Numerical Method Utilizing Novel Cubic B-Spline Approximations to Solve Burgers’ Equation

Author

Listed:
  • Ishtiaq Ali

    (Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia)

  • Muhammad Yaseen

    (Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan)

  • Muhammad Abdullah

    (Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan)

  • Sana Khan

    (Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan)

  • Fethi Bin Muhammad Belgacem

    (Department of Mathematics, Faculty of Basic Education, Public Authority for Applied Education and Training, Al-Ardhiya 92400, Kuwait)

Abstract

Burgers’ equation is a nonlinear partial differential equation that appears in various areas of physics and engineering. Finding accurate and efficient numerical methods to solve this equation is crucial for understanding complex fluid flow phenomena. In this study, we propose a spline-based numerical technique for the numerical solution of Burgers’ equation. The space derivative is discretized using cubic B-splines with new approximations for the second order. Typical finite differences are used to estimate the time derivative. Additionally, the scheme undergoes a stability study to ensure minimal error accumulation, and its convergence is investigated. The primary advantage of this scheme is that it generates an approximate solution as a smooth piecewise continuous function, enabling approximation at any point within the domain. The scheme is subjected to a numerical study, and the obtained results are compared to those previously reported in the literature to demonstrate the effectiveness of the proposed approach. Overall, this study aims to contribute to the development of efficient and accurate numerical methods for solving Burgers’ equation. The spline-based approach presented herein has the potential to advance our understanding of complex fluid flow phenomena and facilitate more reliable predictions in a range of practical applications.

Suggested Citation

  • Ishtiaq Ali & Muhammad Yaseen & Muhammad Abdullah & Sana Khan & Fethi Bin Muhammad Belgacem, 2023. "An Innovative Numerical Method Utilizing Novel Cubic B-Spline Approximations to Solve Burgers’ Equation," Mathematics, MDPI, vol. 11(19), pages 1-19, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4079-:d:1247997
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    References listed on IDEAS

    as
    1. Yang, Xuehua & Wu, Lijiao & Zhang, Haixiang, 2023. "A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity," Applied Mathematics and Computation, Elsevier, vol. 457(C).
    2. Nejib Smaoui & Fethi Belgacem, 2002. "Connections between the convective diffusion equation and the forced Burgers equation," International Journal of Stochastic Analysis, Hindawi, vol. 15, pages 1-17, January.
    3. Idris Dag & Dursun Irk & Ali Sahin, 2005. "B-spline collocation methods for numerical solutions of the Burgers' equation," Mathematical Problems in Engineering, Hindawi, vol. 2005, pages 1-18, January.
    4. Iqbal, Muhammad Kashif & Abbas, Muhammad & Wasim, Imtiaz, 2018. "New cubic B-spline approximation for solving third order Emden–Flower type equations," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 319-333.
    Full references (including those not matched with items on IDEAS)

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