New exact solutions and numerical approximations of the generalized KdV equation

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science and Art, Nevsehir Haci Bektas Veli University, Nevsehir, 50300, Turkey.

2 Department of Mathematics, Faculty of Science, AL-Azhar University, Nasr City, P.N.Box: 11884- Cairo, Egypt.


This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B splines over finite elements. Propagation of a single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann's theory is surveyed and indicated that our method is unconditionally stable.


[1] T. Ak, S. B. G. Karakoc, and A. Biswas, A New Approach for Numerical Solution of Modified Korteweg-de Vries Equation, Iran J. Sci. Technol. Trans. Sci., 41 (2017), 1109–1121.
[2] T. Ak, S. B. G. Karakoc, and A. Biswas, Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation, Scientia Iranica B, 24(3) (2017), 1148-1159.
[3] T. Ak , H. Triki, S. Dhawan, S. K. Bhowmik, S. P. Moshokoae, M. Z. Ullah, and A. Biswas, Computational analysis of shallow water waves with Korteweg-de Vries equation, Scientia Iranica B, 25(5) (2017), 2582-2597.
[4] E. N. Aksan and A. Ozdes, Numerical solution of Korteweg–de Vries equation by Galerkin Bspline finite element method, Applied Mathematics and Computation, 175 (2006), 1256–1265.
[5] A. Biswas, 1-soliton solution of the K(m,n) equation with generalized evolution, Physics Letters A, 372 (2008), 4601-4602.
[6] A. Biswas and D. Milovic, Bright and dark solitons of the generalized nonlinear Schr¨odinger’s equation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1473–1484.
[7] A. Biswas, 1-soliton solution of the K(m, n) equation with generalized evolution and timedependent damping and dispersion, Computers and Mathematics with Applications, 59 (2010), 2536–2540.
[8] A. Biswas and K. R .Raslan, Numerical simulation of the modified Korteweg-de Vries Equation, Physics of Wave Phenomena, 19(2) (2011), 142-147.
[9] P. Bracken, Specific Solutions of the Generalized Korteweg-de Vries Equation With Possible Physical Applications, Central European Journal of Physics, 3(1) (2005), 127-138.
[10] M. S. Bruzon, A. P. Marquez, T. M. Garrido, E. Recio, and R. de la Rosa, Conservation laws for a generalized seventh order KdV equation, Journal of Computational and Applied Mathematics, 354 (2019), 682-688.
[11] A. Canıvar, M. Sarı and I. Da˘g, A Taylor-Galerkin finite element method for the KdV equation using cubic B-splines, Physica B, 405 (2010), 3376-3383.
[12] I. Da˘g and Y. Dereli, Numerical solutions of KdV equation using radial basis function, Applied Mathematical Modelling, 32 (2008), 535–546.
[13] D. B. Dambaru and I.B. Muhammad, Numerical solution of KdV equation using modified Bernstein polynomials, Appl. Math. Comput.,174(2) (2006), 1255–1268.
[14] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H.C. Morris, Solitons and nonlinear wave equations, New York, Academic Press, 1982.
[15] O. Ersoy and I. Da˘g, The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation, Advances in Numerical Analysis, 2015 (2015), 1-8.
[16] B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc.,289 (1978), 373-404.
[17] C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput, 93 (1998), 73–82.
[18] M. G. Garcia Alvarado and G. A. Omel’yanov, Interaction of solitary waves for the generalized KdV equation, Commun Nonlinear Sci Numer Simul., 17 (2012), 3204–3218.
[19] C. S. Gardner, J. M. Green, M. D. Kruskal, and R.M. Miura, Method for solving Korteweg–de Vries equation, Physical Review Letters, 19 (1967), 1095–1097.
[20] G. A. Gardner, L. R. T. Gardner, and A. H. A. Ali, A finite element solution for the Korteweg– de Vries equation with cubic B-splines, UCNW Math Preprint, 1989.
[21] G. A. Gardner, L. R. T. Gardner, and A. H. A. Ali, Modelling solutions of the Korteweg–de Vries equation with quintic splines, UCNW Math Preprint, 1990.
[22] A. Ghiloufi, A. Rouatbi, and K. Omrani, A new conservative fourth-order accurate difference scheme for solving a model of nonlinear dispersive equations, Mathematical Methods in the Applied Sciences, 41 (2018), 5230- 5253.
[23] K. Goda, On instability of some finite difference schemes for Korteweg-de Vries equation, J. Phys. Soc. Japan, 39 (1975), 229-236.
[24] I. S. Greig and J. L. Morris, A hopscotch method for the Korteweg-de-Vries equation, J. Computational Phys., 20(1) (1976), 64–80.
[25] O. Guner, Shock waves solution of nonlinear partial differential equation system by using the ansatz method, Optik, 130 (2017), 448-454.
[26] I. E. Inan, Exact solutions for coupled KdV equation and KdV equations, Physics Letters A, 371 (2007), 90–95.
[27] M. Inc, Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method, Chaos Soliton Fractals, 34(4) (2007), 1075–1081.
[28] D. Irk, Quintic B-spline Galerkin method for the KdV equation, Anadolu University Journal of Science and Technology B- Theoritical Sciences, 5(2) (2017), 111-119.
[29] D. Irk, I. Da˘g, and B. Saka, A small time solutions for the Korteweg–de Vries equation using spline approximation, Appl. Math. Comput, 173(2) (2006), 834-846.
[30] M. S. Ismail and A. Biswas, 1-Soliton solution of the generalized KdV equation with generalized evolution, Applied Mathematics and Computation, 216 (2010), 1673–1679.
[31] H. N. A. Ismail, K. R. Raslan, and G. S. E. Salem, Solitary wave solutions for the general KDV equation by Adomian decomposition method, App. Mathematics and Comput., 154 (2004), 17–29.
[32] S. B. G. Karakoc, A quartic subdomain finite element method for the modified KdV equation, Stat., Optim. Inf. Comput., 6 (2018), 609–618.
[33] S. B. G. Karakoc, Numerical solutions of the mKdV equation via collocation finite element method, Anadolu University Journal of Science and Tech. B-Theoritical Sciences, 6(2) (2018), 1-13.
[34] D. Kaya, An application for the higher order modified KdV equation by decomposition method, Commun. in Nonlinear Science and Num. Simul., 10 (2005), 693-702.
[35] D. Kaya and M. Aassila, An application for a generalized KdV equation by the decomposition method, Physics Letters A, 299(2-3) (2002), 201-206.
[36] A. Korkmaz, Numerical Algorithms for Solutions of Korteweg–de Vries Equation, Numerical Methods for Partial Differential Equations, 26(6) (2010), 1504-1521.
[37] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. Mag., 39(5) (1895), 422–443.
[38] S. Kutluay, A. R. Bahadır, and A. Ozdes, A small time solutions for the Korteweg–de Vries equation, Appl. Math. Comput., 107 (2000), 203–210.
[39] X. Lai, Q. Cao, and E. H. Twizell, The global domain of attraction and the initial value problems of a kind of GKdV equations, Chaos Solitons and Fractals, 23 (2005), 1613–1628.
[40] M. S. A. Latif, Some exact solutions of KdV equation with variable coefficients, Commun Nonlinear Sci. Numer. Simulat., 16 (2011), 1783–1786.
[41] L. E. Lindgren, From Weighted Residual Methods to Finite Element Methods, 2009.
[42] H. Liu and N. Yi, A Hamiltonian preserving discontinuous Galerkin method forthe generalized Korteweg–de Vries equation, Journal of Computational Physics, 321 (2016), 776–796.
[43] P. M. Prenter, Splines and Variational Methods, John Wiley & Sons, New York, NY.USA, 1975.
[44] K. R. Raslan and H. A. Baghdady, A finite difference scheme for the modified Korteweg-de Vries equation, General Mathematics Notes, 27(1) (2015), 101-113.
[45] K. R. Raslan and H. A. Baghdady, New algorithm for solving the modified Korteweg-de Vries(mKdV) equation, International Journal of Research and Reviews in App. Sciences, 18(1) (2014), 59-64.
[46] A. Rouatbi, T. Achouri, and K. Omrani, High-order conservative difference scheme for a model of nonlinear dispersive equations, Computational and Applied Mathematics, 37 (2018), 4169-4195.
[47] A. Rouatbi and K. Omrani, Two conservative difference schemes for a model of nonlinear dispersive equations, Chaos, Solitons and Fractals, 104 (2017), 516-530.
[48] B. Saka, Cosine expansion-based differential quadrature method for numerical solution of the KdV equation, Chaos, Solitons and Fractals, 40 (2009), 2181–2190.
[49] A. Salih, Weighted Residual Methods, Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram – December 2016.
[50] M. Sarboland and A. Aminataei, On the numerical solution of the nonlinear Korteweg–de Vries equation, Systems Science & Control Engineering: An Open Access Journal, 3 (2015), 69–80.
[51] J. Sarma, Exact solutions for modified Korteweg–de Vries equation, Chaos, Solitons and Fractals, 42 (2009), 1599-1603.
[52] M. Sepulveda and O. V. Villagran, Numerical Methods for Generalized KdV equations, In Anais do XXXI Congresso Nacional de Matematica Aplicada e Computacional, 2008.
[53] J. M. Sanz Serna and I. Christie, Petrov Galerkin methods for non linear dispersive wave, J. Comput. Phys., 39 (1981), 94–102.
[54] A. A. Soliman, Collocation solution of the Korteweg-de Vries equation using septic splines, Int. J. Comput. Math., 81 (2004), 325-331.
[55] A. A. Soliman, A. H. A. Ali and K. R. Raslan, Numerical solution for the KdV equation based on similarity reductions, Applied Mathematical Modelling, 33 (2009), 1107–1115.
[56] H. Triki and A. M. Wazwaz, Bright and dark soliton solutions for a K(m,n) equation with t-dependent coefficients, Physics Letters A, 373 (2009), 2162–2165.
[57] O. O. Vaneeva, N. C. Papanicolaou, M. A. Christou, and C. Sophocleous, Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries, Communications in Nonlinear Science and Numerical Simulation, 19(9) (2014), 3074-3085.
[58] A. C. Vliengenthart, On finite difference methods for the Korteweg-de Vries equation, J. Eng. Math. 5 (1971), 137-155.
[59] A. M. Wazwaz, Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method, Chaos, Solitons and Fractals, 12 (2001), 2283–2293.
[60] A. M. Wazwaz, A variety of (3+1)-dimensional mKdV equations derived by using the mKdV recursion operator, Computers and Fluids, 93 (10) (2014), 41-45.
[61] A. M. Wazwaz, New (3+1)-dimensional nonlinear evolution equations with mKdV equation constituting its main part: multiple soliton solutions, Chaos, Solitons and Fractals, 76 (2015), 93-97.
[62] A. M. Wazwaz, A study on KdV and Gardner equations with time-dependent coefficients and forcing terms, Appl. Math. Comput., 217 (2010), 2277–2281.
[63] N. J. Zabusky, A synergetic approach to problem of nonlinear dispersive wave propagation and interaction, in:W. Ames(Ed.), Proceedings of the Symposium Nonlinear Partial Differential Equation Academic Press, 1967.
[64] N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.