Abstract and Applied Analysis

A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method

Mustafa Inc, Ali Akgül, and Adem Kiliçman

Full-text: Open access

Abstract

We propose a reproducing kernel method for solving the KdV equation with initial condition based on the reproducing kernel theory. The exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have also been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is effective.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2012), Article ID 578942, 11 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449863

Digital Object Identifier
doi:10.1155/2013/578942

Mathematical Reviews number (MathSciNet)
MR3034984

Zentralblatt MATH identifier
1266.65178

Citation

Inc, Mustafa; Akgül, Ali; Kiliçman, Adem. A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method. Abstr. Appl. Anal. 2013, Special Issue (2012), Article ID 578942, 11 pages. doi:10.1155/2013/578942. https://projecteuclid.org/euclid.aaa/1393449863


Export citation

References

  • P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, UK, 1989.
  • A. A. Soliman, “Collucation solution of the KdV equation using septic splines,” International Journal of Computer Mathematics, vol. 81, pp. 325–331, 2004.
  • A. A. Soliman, A. H. A. Ali, and K. R. Raslan, “Numerical solution for the KdV equation based on similarity reductions,” Applied Mathematical Modelling, vol. 33, no. 2, pp. 1107–1115, 2009.
  • N. J. Zabusky, “A synergetic approach to problem of nonlinear dispersive wave propagation and interaction,” in Proceeding of Symposium Nonlinear PDEs, W. Ames, Ed., pp. 223––258, Academic Press, New York, NY, USA, 1967.
  • D. J. Korteweg-de Vries and G. de Vries, “On the change in form of long waves advancing in rectangular canal and on a new type of long stationary waves,” Philosophical Magazine, vol. 39, pp. 422–443, 1895.
  • C. Gardner and G. K. Marikawa, “The effect of temperature of the width of a small amplitude solitary wave in a collision free plasma,” Communications on Pure and Applied Mathematics, vol. 18, pp. 35–49, 1965.
  • R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, NY, USA, 1982.
  • K. Goda, “On stability of some finite difference schemes for the KdV equation,” Journal of the Physical Society of Japan, vol. 39, pp. 229–236, 1975.
  • A. C. Vliengenthart, “On finite difference methods for KdV equation,” Journal of Engineering Mathematics, vol. 5, pp. 137–155, 1971.
  • M. Inc, “Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method,” Chaos, Solitons and Fractals, vol. 34, no. 4, pp. 1075–1081, 2007.
  • I. Dağ and Y. Dereli, “Numerical solution of KdV equation using radial basis functions,” Applied Mathematical Modelling, vol. 32, pp. 535–546, 2008.
  • A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Springer, London, UK, 2009.
  • M. I. Syam, “Adomian decomposition method for approximating the solution of the KdV equation,” Applied Mathematics and Computation, vol. 162, pp. 1465–1473, 2005.
  • N. Aronszajn, “Theory of reproducing kernels,” Transactions of the American Mathematical Society, vol. 68, pp. 337–404, 1950.
  • M. Inc, A. Akgül, and A. Kilicman, “Explicit solution of telegraph equation based on reproducing Kernel method,” Journal of Function Spaces and Applications, vol. 2012, Article ID 984682, 23 pages, 2012.
  • F. Geng and M. Cui, “Solving a nonlinear system of second order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 327, pp. 1167–1181, 2007.
  • H. Yao and M. Cui, “A new algorithm for a class of singular boundary value problems,” Applied Mathematics and Computation, vol. 186, pp. 1183–1191, 2007.
  • W. Wang, M. Cui, and B. Han, “A new method for solving a class of singular two-point boundary value prolems,” Applied Mathematics and Computation, vol. 206, pp. 721––727, 2008.
  • Y. Zhou, Y. Lin, and M. Cui, “An efficient computational method for second order boundary value problemsof nonlinear diffierential equations,” Applied Mathematics and Computation, vol. 194, pp. 357–365, 2007.
  • X. Lü and M. Cui, “Analytic solutions to a class of nonlinear infinite-delay-differential equations,” Journal of Mathematical Analysis and Applications, vol. 343, pp. 724–732, 2008.
  • Y. L. Wang and L. Chao, “Using reproducing kernel for solving a class of partial differential equation with variable-coefficients,” Applied Mathematics and Mechanics, vol. 29, pp. 129–137, 2008.
  • F. Li and M. Cui, “A best approximation for the solution of one-dimensional variable-coefficient Burger's equation,” Numerical Methods for Partial Differential Equations, vol. 25, pp. 1353–1365, 2009.
  • S. Zhou and M. Cui, “Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions,” International Journal of Computer Mathematics, vol. 86, pp. 2248–2258, 2009.
  • F. Geng and M. Cui, “New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 165–172, 2009.
  • J. Du and M. Cui, “Solving the forced Duffing equations with integral boundary conditions in the reproducing kernel space,” International Journal of Computer Mathematics, vol. 87, pp. 2088–2100, 2010.
  • X. Lv and M. Cui, “An efficient computational method for linear fifth-order two-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1551–1558, 2010.
  • W. Jiang and M. Cui, “Constructive proof for existence of nonlinear two-point boundary value problems,” Applied Mathematics and Computation, vol. 215, no. 5, pp. 1937–1948, 2009.
  • J. Du and M. Cui, “Constructive proof of existence for a class of fourth-order nonlinear BVPs,” Computers and Mathematics with Applications, vol. 59, no. 2, pp. 903–911, 2010.
  • M. Cui and H. Du, “Representation of exact solution for the nonlinear Volterra-Fredholm integral equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1795–1802, 2006.
  • B. Y. Wu and X. Y. Li, “Iterative reproducing kernel method for nonlinear oscillator with discontinuity,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1301–1304, 2010.
  • H. Jafari and M. A. Firoozjaee, “Homotopy analysis method for KdV equation,” Surveys in Mathematics and Its Applications, vol. 5, pp. 89–98, 2010.