Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes
- 1 Department of Math, King Abdulaziz University, Jeddah, Saudi Arabia
- 2 Department of Math, King Abdulaziz University, Jeddah, Saudi Arabia
Abstract
In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.
- Karakoc, S.B. (2018) Numerical Solutions of the Modified KdV Equation with Collocation Method. Malaya Journal of Mathematik, 6, 835-842.
- Kaya, D. (2005) An Application for the Higher Order Modified KdV Equation by Decomposition Method. Communications in Nonlinear Science and Numerical Simulation, 10, 693-702. https://doi.org/10.1016/j.cnsns.2003.12.009
- Biswas, A. and Raslan, K.R. (2011) Numerical Simulation of the Modified Korteweg-de Vries Equation. Physics of Wave Phenomena, 19, 142-147. https://doi.org/10.3103/S1541308X11020105
- Raslan, K.R. and Baghdady, H.A. (2015) A Finite Difference Scheme for the Modified Korteweg-de Vries Equation. General Mathematics Notes, 27, 101-113.
- Raslan, K.R. and Baghdady, H.A. (2014) New Algorithm for Solving the Modified Korteweg-de Vries (mKdV) Equation. International Journal of Research and Reviews in Applies Sciences, 18, 59-64.
- Wazwaz, A.-M. (2014) A Variety of (3+1)-Dimensional mKdV Equations Derived by Using the mKdV Recursion Operator. Computers and Fluids, 93, 41-45. https://doi.org/10.1016/j.compfluid.2014.01.010
- Wazwaz, A.-M. (2015) New (3+1)-Dimensional Nonlinear Evolution Equations with mKdV Equation Constituting Its Main Part: Multiple Soliton Solutions. Chaos, Solitons and Fractals, 76, 93-97. https://doi.org/10.1016/j.chaos.2015.03.018
- Ak, T., Karakoc, S.B.G. and Biswas, A. (2017) A New Approach for Numerical Solution of Modified Korteweg-de Vries Equation. Iranian Journal of Science and Technology, Transactions A: Science, 41, 1109-1121. https://doi.org/10.1007/s40995-017-0238-5
- Ak, T., Karakoc, S.B.G. and Biswas, A. (2017) Application of Petrov-Galerkin Finite Element Method to Shallow Water Waves Model: Modified Korteweg-de Vries equation, Scientia Iranica B, 24, 1148-1159. https://doi.org/10.24200/sci.2017.4096
- Mihaila, B., Cardenas, A., Cooper, F. and Saxena, A. (2010) Stability and Dynamical Properties of Rosenau-Hyman Compactons Using Padé Approximants. Physical Review E, 81, 056708. https://doi.org/10.1103/PhysRevE.81.056708
- Cooper, F., Hyman, J.M. and Khare, A. (2001) Compacton Solutions in a Class of Generalized Fifth-Order Korteweg-de Vries Equations. Physical Review E, 64, 026608. https://doi.org/10.1103/PhysRevE.64.026608
- Wazwaz, A.M. (2009) Partial Differential Equations and Solitary Waves Theory. Nonlinear Physical Science, Springer. https://doi.org/10.1007/978-3-642-00251-9