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Articles
DOI: 10.29303/emj.v7i1.190
Submitted: 2023-11-21
Published: 2024-02-15

Numerical Solution of the Korteweg-De Vries Equation Using Finite Difference Method

Universitas Mataram
Universitas Mataram
Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram
Universitas Mataram
Universitas Mataram
Universitas Mataram
Equation KdV Soliton Finite difference method Crank-nicolson scheme

Abstract

The Korteweg-de Vries (KdV) equation is a nonlinear partial differential equation that has a key role in wave physics and many other disciplines. In this article, we develop numerical solutions of the KdV equation using the finite difference method with the Crank-Nicolson scheme. We explain the basic theory behind the KdV equation and the finite difference method, and outline the implementation of the Crank-Nicolson scheme in this context. We also give an overview of the space and time discretization and initial conditions used in the simulation. The results of these simulations are presented through graphical visualizations, which allow us to understand how the KdV solution evolves over time. Through analysis of the results, we explore the behavior of the solutions and perform comparisons with exact solutions in certain cases. Our conclusion summarizes our findings and discusses the advantages and limitations of the method used. We also provide suggestions for future research in this area.

References

  1. Robbaniyyah, N. A. I. (2022). Pengembangan Metode Iterasi Petviashvili dalam Penentuan Solusi Gelombang Stasioner pada Persamaan Bertipe Schrödinger Nonlinear dengan Fungsi Potensial V (x). EIGEN MATHEMATICS JOURNAL, 47-53.
  2. Butcher, C., 2008. Numerical Methods for Ordinary Differential Equation 2nd Edition. England: John Wiley and Sons.
  3. Causon, D. & Mingham, C. 2010. Introductory Finite Difference Methods for PDEs. Manchester: Ventus Publishing.
  4. Daxois, T dan Michel, P. 2010. Physics of Solitons : An Introduction. Cambridge: Cambridge University Press.
  5. D.D Causon, C.G Mingham, 2010, introductory Finite Difference Methods for PDEs, Ventus Publishing Aps,ISBN 978-87-7681-642-1.
  6. Djojodihardjo, H. 2000. Metode Numerik. Jakarta: PT Gramedia Pustaka Utama.
  7. Hasan, dkk. (2016). Penerapan Metode Beda Hingga pada Model Matematika Aliran Banjir dari Persamaan saint Venant. Math Journal, 6-12.
  8. Lestari, Dwi. 2013. Persamaan Diferensial. UNY: Yogyakarta.
  9. Munir, R. 2010. Metode Numerik. Bandung: Informatika Bandung.
  10. Rumlawang, F. Y. (2013). Solusi numerik persamaan gelombang Kotewieg De Vries (KDV). Jurnal Barekeng, 1-7.
  11. Suryanto, A. (2017). Metode Numerik Untuk Persamaan Diferensial Biasa dan Aplikasinya dengan MATLAB. Malang: Universitas Malang.
  12. Strauss, W.A. 2007. Partial Differential Equations an Introduction Second Edition. New York: John Wiley and Sons Ltd.
  13. Yin, X., Cao, W. 2023. A Class of Efficient Hamiltonian Conservative Spectral Methods for Korteweg-de Vries Equations. J Sci Comput, 94, 10.
  14. Zhang, Da-Jun, Song-Lin Zhao, Ying-Ying Sun, and Jing Zhou. Solutions to the Modified Korteweg-de Vries Equation. Reviews in Mathematical Physics, 26, 07.

How to Cite

Haizar, M. R., Rizki, M., Robbaniyyah, N. A., Syechah, B. N., Salwa, S., & Awalushaumi, L. (2024). Numerical Solution of the Korteweg-De Vries Equation Using Finite Difference Method. EIGEN MATHEMATICS JOURNAL, 7(1), 1–7. https://doi.org/10.29303/emj.v7i1.190