Comparative Study of Some Numerical and Semi-analytical Methods for Some 1D and 2D Dispersive KdV-type Equations

Year 2022, Volume: 3 Issue: 1, 1 - 25, 30.01.2022

Abey Sherif Kelil

https://doi.org/10.54974/fcmathsci.1002281

Cited By: 2

Abstract

This paper aims to investigate an approximate-analytical and numerical solutions for some 1D
and 2D dispersive homogeneous and non-homogeneous KdV equations by employing two reliable methods namely reduced differential transform method (RDTM) and a classical finite-difference method. RDTM provides an analytical approximate solution in the form of a convergent series. The classical finite-difference method (FDM) to solve dispersive KdV equations is employed by primarily checking Von Neumann’s stability criterion. The performance of the mentioned methods for the considered experiments are compared by computing absolute and relative errors at some spatial nodes at a given time; and to the best of our knowledge, the comparison between these two methods for the considered experiments is novel. Knowledge acquired will enable us to build methods for other related PDEs such as KdV-Burgers, stochastic KdV and fractional KdV-type equations.

Keywords

Dispersive KdV equations, reduced dierential transform method, homogeneous, non-homogeneous, classical finite difference method

Supporting Institution

Nelson Mandela University

Thanks

Council Postdoc Fellowship at Nelson Mandela University and CoE-MaSS for Top-up funding for 2021.

References

• Abassy T.A., El-Tawil M.A., Saleh H.K., The solution of KdV and mKdV equations using Adomian Padé approximation, International Journal of Nonlinear Sciences and Numerical Simulation, 5(4), 327-339, 2004.
• Abassy T.A., El-Tawil M.A., El Zoheiry H., Toward a modified variational iteration method, Journal of Computational and Applied Mathematics, 207(1), 137-147, 2007.
• Aderogba A.A., Appadu A.R., Classical and multisymplectic schemes for linearized KdV equation: Numerical results and dispersion analysis, Fluids, 6(6), 214, 2021.
• Adomian G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, 1994.
• Adomian G., A review of decomposition method and some recent results for nonlinear equation, Mathematical and Computer Modelling, 13(7), 17-43, 1992.
• Adomian G., Rach R., Noise terms in decomposition solution series, Applied Mathematics and Computation, 24(11), 61-64, 1992.
• Al-Amr M.O., New applications of reduced differential transform method, Alexandria Engineering Journal, 53, 243-247, 2014.
• Appadu A.R., Kelil A.S., On semi-analytical solutions for linearized dispersive KdV equation, Mathematics, 8(10), 1769, 2020.
• Appadu A.R., Chapwanya M., Jejeniwa O.A., Some optimised schemes for 1D Korteweg-de-Vries equation, Progress in Computational Fluid Dynamics, 17(4), 250-266, 2017.
• Appadu, A.R., Kelil, A.S., Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations, Demonstratio Mathematica, 54(1), 377-409, 2021.
• Ascher U.M., Numerical Methods for Evolutionary Differential Equations, Society for Industrial and Applied Mathematics, 2008.
• Chen S.S., Chen C.K., Application to differential transformation method for solving systems of differential equations, Nonlinear Analysis Real World Applications, 10, 881-888, 2009.
• He J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262, 1999.
• He J.H., Homotopy perturbation method: A new non-linear analytical technique, Applied Mathematics and Computation, 135, 73-79, 2003.
• He J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26, 695-700, 2005.
• He J.H., Variational iteration method–a kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics, 34(4), 699-708, 1999.
• He J.H., A new approach to nonlinear partial differential equations, Communications in Nonlinear Science and Numerical Simulation, 2(4), 230-235, 1997.
• Keskin Y., Oturanc G., Reduced differential transform method for partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10(6), 741-750, 2009.
• Keskin Y., Oturanc G., Reduced differential transform method for solving linear and nonlinear wave equations, Iranian Journal of Science and Technology, Transaction A, 34(A2), 113-122, 2010.
• Korteweg D.J., de Vries G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine, 39, 422-443, 1895.
• Taha T.R., Ablowitz M.I., Analytical and numerical aspects of certain nonlinear evolution equations III: Numerical, Korteweg-de Vries equation, Journal of Computational Physics, 55(2), 231-253, 1984.
• Wazwaz A.M., An analytic study on the third-order dispersive partial differential equations, Applied Mathematics and Computation, 142(2-3), 511-520, 2003.
• Yildirim A., On the solution of the nonlinear Korteweg-de Vries equation by the homotopy perturbation method, Communications in Numerical Methods in Engineering, 25(12), 1127-1136, 2009.
• Zabusky N.J., Kruskal M.D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Physical Review Letters, 15(6), 240, 1965.
• Zhou J.K., Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, 1986.