**March 20, 2019**. Read More

# Numerical Investigation of Separated Solitary Waves Solution for KDV Equation through Finite Element Technique

10.5120/5049-7463 |

S Kapoor, S Rawat and S Dhawan. Article: Numerical Investigation of Separated Solitary Waves Solution for KDV Equation through Finite Element Technique. *International Journal of Computer Applications* 40(14):27-33, February 2012. Full text available. BibTeX

@article{key:article, author = {S. Kapoor and S. Rawat and S. Dhawan}, title = {Article: Numerical Investigation of Separated Solitary Waves Solution for KDV Equation through Finite Element Technique}, journal = {International Journal of Computer Applications}, year = {2012}, volume = {40}, number = {14}, pages = {27-33}, month = {February}, note = {Full text available} }

### Abstract

The Present manuscript reports the solution of well known non linear wave mechanics problem called KDV equation, here main emphasis is given on the Mathematical modeling of traveling waves and their solutions in the form of Korteweg-de Vries equation (KdV) It is a non-linear Partial Differential Equation (PDE) of third order which arises in a number of physical applications such as water waves, elastic rods, plasma physics etc. We present numerical solution of the above equation using B-spline FEM (Finite Element Method) approach. The ultimate goal of the paper is to solve the above problem using numerical simulation in which the accuracy of computed solutions is examined by making comparison with analytical solutions, which are found to be in good agreement with each other along with that we discussed the physical interpolation of the soliton study in which we found that the travel waves reaches to the maximum magnitude of the velocity in the short time of the interval and there is an uncertainty in the motion of the moving waves. Another important observation we found that the maximum magnitude of the velocity in the most of the time domain is around 1 but in some of the condition waves having a unnatural phenomena which is called the existence of the doubly soliton is seemed frequently. All above observation which is clearly indication of the generic outcome of a weakly nonlinear long-wave asymptotic analysis of many physical systems. The another achievement of the work is to implementation of the cubic B-spline FEM in the above non linear propagating waves phenomena.

### References

- 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. Phil. Mag. J. Science, 39 (1895) 422-443.
- Boussinesq, J. (1877), Essai sur la theorie des eaux courantes, Memoires presentes par divers savants ` l’Acad. des Sci. Inst. Nat. France, XXIII, pp. 1–680
- N.J. Zabusky and M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 6 (1965) 240-243.
- Miura, Robert M.; Gardner, Clifford S.; Kruskal, Martin D. (1968), "Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion", J. Mathematical Phys. 9: 1204–1209
- Lax, P. (1968), "Integrals of nonlinear equations of evolution and solitary waves", Comm. Pure Applied Math. 21: 467–490,
- A. C. Vliegenthart, On finite difference method for Korteweg-de Vries equation,J. Eng. Math., 5 (1971) 137-155.
- Miles., John W. (1981). "The Korteweg–De Vries equation: A historical essay". Journal of Fluid Mechanics 106: 131–147.
- Dingemans, M.W. (1997), Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering, 13, World Scientific, Singapore,, 2 Parts, 967 pages
- de Jager, E.M. (2006). "On the origin of the Korteweg–de Vries equation". arXiv;math/0602661vl [math.HO].
- Darvishi, M. T.; Kheybari, S.; Khani, F. (2007), A Numerical Solution of the Lax’s 7th-order KdV Equation by Pseudospectral Method and Darvishi’s Preconditioning, 2, pp. 1097–1106
- A. C. Scott, Y. F. Chu and D. W. McLaughlin, The Soliton: A New Concept In Applied Science, IEEE Proc., 61 (1973) 1443.
- Peregrine, D.H., Calculations of Development of an undular Bore, J.Fluid Mech, 25 (1966)321-330.
- Malik Zawwar Hussain., Muhammad Sarfraz., Ayesha Shakeel , Shape Preserving Surfaces for the Visualization of Positive and Convex Data using Rational Bi-quadratic Splines., International Journal of Computer Applications ., 27(10), 2011
- A. C. Vliegenthart.,On Finite difference method for Korteweg-de Vries equation, J. Eng. Math., 5 (1971)137-155.
- Zabusky, N. J.; Kruskal, M. D. (1965), “Interaction of “Soliton” in a collisionless Plasma and the Recurrence of Initial States”, Phys. Rev. Lett. 15: 240–243,
- K. Abe and 0. Inoue, Fourier expansion solution of the Korteweg-de Vries equation, J. Comp. Phys. 34 (1980) 202-210.
- T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations I. Analytic, J. Comp. Phys. 55 (1984) 192-202.
- T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations III. Numerical, J. Comp. Phys. 55 (1984) 231-253.
- B. Fornberg and G.B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Tra. Roy. Sot. London 289 (1978) 373-404.
- L. Iskander, New numerical solution of the Korteweg-de Vries equation, Appl. Num. Math. 5 (1989) 215-221.
- O.A. Karakashian and W. McKinney, On optimal high-order in time approximations for the Korteweg- de Vries equation, American Math. Sot. 55 (1990) 473-496.
- V.A. Dougalis and O.A. Karakashian, On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation, Math. Comp. 45 (1985) 329-345.
- O.A. Karakashian and W. Rust, On the parallel implementation of implicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput. 9 (1988) 1085-1090.
- S.Kapoor & S.Dhawan; “ A computational technique for the solution of Burgers’ equation” Int. J. of Appl. Math. and Mech. 6(3): 84-95, 2010.
- V.Dabral., S.Kapoor., S.Dhawan. S.Rawat “ Finite Element Based solution of Modified Equal Width equation (MEW) with Homogenous Boundary condition using B-spline Basis Function , Chiangmai University International Conference 2011 , Vol. 1, No. 1 (2010) 124 – 131
- V.Dabral., S.Kapoor., S.Dhawan., “Mathematical study of seprated solitry Wave solution for KDV equation: B-spline FEM Approach” . in 4th International conference on “Modeling ,Simulation and Applied Optimization “ (ICMSAO-2011) ” Kulalumpur (Malaysia), an IEEE conference held at Kula-Lumpur (Malaysia) Paper Published in conference Proceeding Paper code: 96508 in Applied Mathematics Track. P.P 771-775.,
- Yogesh Gupta., Manoj Kumar., “A Computer based Numerical Method for Singular Boundary Value Problems., International Journal of Computer Applications, 30(1)., 2011, pp. 21-25