Approximate Solution of the Nonlinear Buckmaster Partial Differential Equation using Exponential Fourth-order Differentiable Functions

Authors

  • Usman Yusuf
    Department of Mathematics, Federal University of Lafia, Nasarawa State
  • Mbah Moses Anayo
    Department of Mathematics, Federal University of Lafia, Nasarawa State
  • Akpan Collins E.
    Department of Mathematics, Federal University of Lafia, Nasarawa State
  • Abubakar Abdulkarim
    Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State
  • Muhammad Sani Otto
    Department of Physics, Federal University of Lafia, Nasarawa State
  • Aminu Buhari
    Department of Mathematics, Federal University of Lafia, Nasarawa State
  • Godwin Ma'aji
    Department of Mathematics, Federal University of Lafia, Nasarawa State
  • Philip Nwanosike
    Department of Mathematics, Federal University of Lafia, Nasarawa State

Keywords:

Array, Array, Array, Array, Array

Abstract

In this paper, the nonlinear partial differential equation, Buckmaster equation is solved using the exponential cubic B-spline collocation method (ECBSM) and the approximate solutions from this method are compared with those of the hybrid cubic B-spline collocation method (HCBSM). In order to solve the equation, linearization technique is needed to linearize the nonlinear terms of the equation. This is done by the Taylor’s expansion approach. Further, the linearized equation is discretized into the fully implicit scheme and the Crank-Nicolson scheme. Three examples are used to test the proposed schemes by the fully implicit and Crank-Nicolson methods. The absolute errors of the methods are calculated and the comparison between the results of the ECBSM and the HCBSM is carried out. This is to analyze the accuracy of the methods of approximation. Both the ECBSM and HCBSM possess a free parameter which aids in determining accurate results. In general, the methods proved reliable with accuracy in approximating solutions of the equation.

Dimensions

Hafiz A. M., Faiq I. & Hassaballah M. (2024). Solving Partial Differential Equations using Large Data Models: A Literature Review. Springer Nature Open Journal. 57 (152), 1-24. https://doi.org/10.1007/s10462-024-10784-5.

Sanchez, V. (2016). Vessel Extraction using the Buckmaster-Airy Filter, Proc. SPIE 9863, Smart Biomedical and Physiological Sensor Technology XIII, 98630W. https://doi.org/10.1117/12.2222536.

Nirmala A. N., Kumbinarasaiah S., Shah N. A. and Alshehri M. (2023). A New Graph-theoretic Approach for the Study of the Surface of a Thin Sheet of a Viscous Liquid Model. Phys. Scr. 98 (2023) 095248. https://doi.org/10.1088/1402-4896/acecb1.

Bickley, W. G. (1968). Piecewise Cubic Interpolation and Two-Point Boundary Problems. The Computer Journal, 11(2), 206-208.

Mat Zin, S. (2016). Hybrid Cubic B-spline Collocation Method for Solving One-Dimensional Wave Equation, AIP Conference Proceedings, AIP Publishing. 1775(1), 030084.

Fyfe, D. J. (1969). The Use of Cubic Splines in the Solution of Two Point Boundary Value Problems. The Computer Journal, 12(2), 188-192.

Albasiny, E. L. & Hoskins, W. D. (1969). Cubic Spline Solutions to Two-point Boundary Value Problems. The Computer Journal, 12(2), 151-153.

Caglar, H., Caglar, N., & Elfaituri, K. (2006). B-Spline Interpolation Compared with Finite Difference, Finite Element and Finite Volume Methods which Applied to Two-Point Boundary Value Problems. Applied Mathematics and Computation, 175(1), 72-79.

Dehghan, M. & Lakestani, M. (2007). The Use of Cubic B-Spline Scaling Functions for Solving the One-dimensional Hyperbolic Equation with a Nonlocal Conservation Condition. Wiley InterScience, 23(6), 1277-1289.

Abd Hamid, N. N., Majid, A. A., & Ismail, A. I. M. (2010). Cubic Trigonometric B-spline Applied to Linear Two-point Boundary Value Problems of Order Two. World Academy of Science, Engineering and Technology, 47, 478-803.

Abd Hamid, N. N., Majid, A. A., & Ismail, A. I. M. (2012). Quartic B-spline Interpolation Method for Linear Two-point Boundary Value Problem. World Applied Sciences Journal, 17, 39-43.

Khan, A., Noor, M. A., & Aziz, T. (2004). Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary Value Problem. Journal of Optimization Theory and Applications, 122(2), 309-322.

McCartin, B. J. (1981), Theory, Computation and Application of Exponential Splines, PhD Thesis, New York University.

McCartin, B. J. (1990). Computation of Exponential Splines. SIAM J. Sci. Stat. Comput., 11(2), 242-262.

McCartin, B. J. & Jameson, A. (1990). Numerical Solution of Nonlinear Hyperbolic Conservation Laws Using Exponential Splines. Computational Mechanics, 6, 77-91.

Ersoy, O. & Dag, I. (2015a). The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation. Hindawi Publishing Corporation, Advances in Numerical Analysis, 2015.

Ersoy, O. & Dag, I. (2015b). Numerical Solutions of the Reaction-Diffusion System Using Exponential Cubic B-Spline Collocation Algorithms. Open Physics, 13(1), 414-427.

Eroy, O. & Dag, I. (2016). The Exponential Cubic B-Spline Collocation Method for the Kuramoto-Sivashinsky Equation. Faculty of Sciences and Mathematics, University of Niš, Serbia, 30(3), 853-861.

Zhu, X., Nie, Y., Yuan, Z., Wang, J., & Yang, Z. (2017). An Exponential B-Spline Collocation Method for the Fractional Sub-Diffusion Equation, Advances in Difference Equations, SpringerOpen Journal, 2017, 285.

Singh, S., Singh, S., & Arora, R. (2017). Numerical Solution of Second-Order One-Dimensional Hyperbolic Equation by Exponential B-Spline Collocation Method, Numerical Analysis and Applications. Pleiades Publishing Ltd, 10(2), 164-176.

Shukla, H. S. & Tamsir, M. (2018). An Exponential Cubic B-Spline Algorithm for the Multi-Dimensional Convection-Diffusion Equations. Alexandria Engineering Journal, 57(3), 1999-2007.

Hussain, E. A. & Alwan, Z. M. (2013). The Finite Volume Method for Solving Buckmaster’s Equation, Fisher’s Equation and Sine Gordon’s Equation for PDEs. International Mathematical Forum, 8(13), 599-617.

Iyiola, O. S. & Olayinka, O. G. (2014). Analytical Solutions of Time-Fractional Models for Homogeneous Gardner and Non-Homogeneous Differential Equations. Ain Shams Engineering Journal, 5(3), 999-1004.

Chanthrasuwan M., Mohd Asri N. A., Abd Hamid N. N., Abd. Majid A., & Azmi A. (2017). Solving Buckmaster Equation using Cubic B-spline and Cubic Trigonometric B-spline Collocation Methods. AIP Conference Proceedings. 1870, 040027. http://dx.doi.org/10.1063/1.4995859.

Farin, G. (2002). A History of Curves and Surfaces in CAGD. In Handbook of Computer Aided Geometric Design, (Farin, G., Hoschek, J., Kim, M. S., ed.) p. 1-14. Amsterdam: Elsevier Science B.V.

cover

Published

2025-05-09

How to Cite

Approximate Solution of the Nonlinear Buckmaster Partial Differential Equation using Exponential Fourth-order Differentiable Functions. (2025). Lafia Journal of Scientific and Industrial Research, 3(2), 14-20. https://doi.org/10.62050/ljsir2025.v3n2.564

Issue

Volume 3, Issue 2 (October, 2025), Lafia Journal of Scientific and Industrial Research (LJSIR)

Section

Physical Sciences
Copyright & Licensing

Copyright (c) 2025 Lafia Journal of Scientific and Industrial Research

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

How to Cite

Approximate Solution of the Nonlinear Buckmaster Partial Differential Equation using Exponential Fourth-order Differentiable Functions. (2025). Lafia Journal of Scientific and Industrial Research, 3(2), 14-20. https://doi.org/10.62050/ljsir2025.v3n2.564

Most read articles by the same author(s)

Similar Articles

11-20 of 27

You may also start an advanced similarity search for this article.