IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i19p4079-d1247997.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/19/4079/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/19/4079/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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).
    4. 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.
    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. Meng An & Haixiang Zhang, 2023. "High-Dimensional Mediation Analysis for Time-to-Event Outcomes with Additive Hazards Model," Mathematics, MDPI, vol. 11(24), pages 1-11, December.
    2. Abdumauvlen Berdyshev & Dossan Baigereyev & Kulzhamila Boranbek, 2023. "Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model," Mathematics, MDPI, vol. 11(17), pages 1-27, September.
    3. Han-Sol Lee & Changgyun Jin & Chanwoo Shin & Seong-Eun Kim, 2023. "Sparse Diffusion Least Mean-Square Algorithm with Hard Thresholding over Networks," Mathematics, MDPI, vol. 11(22), pages 1-16, November.
    4. Elif Tan & Diana Savin & Semih Yılmaz, 2023. "A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields," Mathematics, MDPI, vol. 11(22), pages 1-14, November.
    5. Jun Zhang & Jingjing Zhang & Shangyou Zhang, 2023. "Explicit Symplectic Runge–Kutta–Nyström Methods Based on Roots of Shifted Legendre Polynomial," Mathematics, MDPI, vol. 11(20), pages 1-13, October.
    6. Yongou Zhang & Zhongjian Ling & Hao Du & Qifan Zhang, 2023. "Finite-Difference Frequency-Domain Scheme for Sound Scattering by a Vortex with Perfectly Matched Layers," Mathematics, MDPI, vol. 11(18), pages 1-11, September.
    7. Kumar, Ajay & Kumar, Sunil, 2022. "A study on eco-epidemiological model with fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    8. Jiaqi Wang & Jianbing Su, 2023. "Boundedness and Compactness of Weighted Composition Operators from α -Bloch Spaces to Bers-Type Spaces on Generalized Hua Domains of the First Kind," Mathematics, MDPI, vol. 11(20), pages 1-27, October.
    9. Ismagil T. Habibullin & Aigul R. Khakimova & Alfya U. Sakieva, 2023. "Miura-Type Transformations for Integrable Lattices in 3D," Mathematics, MDPI, vol. 11(16), pages 1-15, August.
    10. Badriah Alamri, 2023. "Solving Integral Equation and Homotopy Result via Fixed Point Method," Mathematics, MDPI, vol. 11(21), pages 1-19, October.
    11. Salman Khalid & Jinwoo Song & Muhammad Muzammil Azad & Muhammad Umar Elahi & Jaehun Lee & Soo-Ho Jo & Heung Soo Kim, 2023. "A Comprehensive Review of Emerging Trends in Aircraft Structural Prognostics and Health Management," Mathematics, MDPI, vol. 11(18), pages 1-42, September.
    12. Jiří Holman, 2023. "Numerical Solution of Transition to Turbulence over Compressible Ramp at Hypersonic Velocity," Mathematics, MDPI, vol. 11(17), pages 1-10, August.
    13. Jianyu Wang & Chunhua Fang & Guifeng Zhang, 2023. "Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels," Mathematics, MDPI, vol. 11(20), pages 1-19, October.
    14. Qiu Lin & Ruisheng Qi, 2023. "Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise," Mathematics, MDPI, vol. 12(1), pages 1-29, December.
    15. Jeffrey A. Hogan & Joseph D. Lakey, 2023. "Spatio–Spectral Limiting on Replacements of Tori by Cubes," Mathematics, MDPI, vol. 11(23), pages 1-14, November.
    16. Nazim & Nadeem Ur Rehman & Ahmad Alghamdi, 2023. "On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ n," Mathematics, MDPI, vol. 11(20), pages 1-14, October.
    17. Li Cheng & Wen-Xiu Ma, 2023. "Similarity Transformations and Nonlocal Reduced Integrable Nonlinear Schrödinger Type Equations," Mathematics, MDPI, vol. 11(19), pages 1-8, September.
    18. Jagdish S. Thakur & Archana Thakur & Lawrence G. Lum, 2023. "Mathematical Model to Predict Polyclonal T-Cell-Dependent Antibody Synthesis Responses," Mathematics, MDPI, vol. 11(18), pages 1-19, September.
    19. Amira F. Daghistani & Ahmed M. T. Abd El-Bar & Ahmed M. Gemeay & Mahmoud A. E. Abdelrahman & Samia Z. Hassan, 2023. "A Hyperbolic Secant-Squared Distribution via the Nonlinear Evolution Equation and Its Application," Mathematics, MDPI, vol. 11(20), pages 1-17, October.
    20. Sarfraz Nawaz Malik & Nazar Khan & Ferdous M. O. Tawfiq & Mohammad Faisal Khan & Qazi Zahoor Ahmad & Qin Xin, 2023. "Fuzzy Differential Subordination Associated with a General Linear Transformation," Mathematics, MDPI, vol. 11(22), pages 1-17, November.

    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:11:y:2023:i:19:p:4079-:d:1247997. 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.