Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 548905, 11 pages
http://dx.doi.org/10.1155/2015/548905
Research Article

Numerical Approximation of Nonlinear Klein-Gordon Equation Using an Element-Free Approach

College of Information Technology, Shanghai Ocean University, Shanghai 201306, China

Received 1 July 2014; Accepted 22 August 2014

Academic Editor: Kim M. Liew

Copyright © 2015 Dong-mei Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A.-M. Wazwaz, “The modified decomposition method for analytic treatment of differential equations,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 165–176, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. A.-M. Wazwaz, “Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations,” Chaos, Solitons & Fractals, vol. 28, no. 4, pp. 1005–1013, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. S. M. El-Sayed, “The decomposition method for studying the Klein-Gordon equation,” Chaos, Solitons and Fractals, vol. 18, no. 5, pp. 1025–1030, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. D. Kaya and S. M. El-Sayed, “A numerical solution of the Klein-Gordon equation and convergence of the decomposition method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 341–353, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. A.-M. Wazwaz, “New travelling wave solutions to the Boussinesq and the Klein-Gordon equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 5, pp. 889–901, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. A.-M. Wazwaz, “The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1179–1195, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. Sirendaoreji, “A new auxiliary equation and exact travelling wave solutions of nonlinear equations,” Physics Letters A: General, Atomic and Solid State Physics, vol. 356, no. 2, pp. 124–130, 2006. View at Publisher · View at Google Scholar · View at Scopus
  8. A. Sirendaoreji, “Auxiliary equation method and new solutions of Klein-Gordon equations,” Chaos, Solitons & Fractals, vol. 31, no. 4, pp. 943–950, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Sirendaoreji, “Exact travelling wave solutions for four forms of nonlinear Klein-Gordon equations,” Physics Letters: A, vol. 363, no. 5-6, pp. 440–447, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. D. Kumar, J. Singh, and S. Kumar, “Numerical computation of Klein-Gordon equations arising in quantum field theory by using homotopy analysis transform method,” Alexandria Engineering Journal, vol. 53, no. 2, pp. 469–474, 2014. View at Publisher · View at Google Scholar · View at Scopus
  11. J. Rashidinia, M. Ghasemi, and R. Jalilian, “Numerical solution of the nonlinear Klein-Gordon equation,” Journal of Computational and Applied Mathematics, vol. 233, no. 8, pp. 1866–1878, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. M. Dehghan and A. Shokri, “Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 400–410, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. K. M. Liew, T. Y. Ng, X. Zhao, and J. N. Reddy, “Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 37-38, pp. 4141–4157, 2002. View at Publisher · View at Google Scholar · View at Scopus
  14. K. M. Liew, X. L. Chen, and J. N. Reddy, “Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrarily shaped shear deformable plates,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 3–5, pp. 205–224, 2004. View at Publisher · View at Google Scholar · View at Scopus
  15. K. M. Liew, Y. Cheng, and S. Kitipornchai, “Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems,” International Journal for Numerical Methods in Engineering, vol. 65, no. 8, pp. 1310–1332, 2006. View at Publisher · View at Google Scholar · View at Scopus
  16. F. X. Sun, C. Liu, and Y. M. Cheng, “An improved interpolating element-free Galerkin method based on nonsingular weight functions,” Mathematical Problems in Engineering, vol. 2014, Article ID 323945, 13 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. N. Zhao and H. Ren, “The interpolating element-free Galerkin method for 2D transient heat conduction problems,” Mathematical Problems in Engineering, vol. 2014, Article ID 712834, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. Q. Wei and R. Cheng, “The improved moving least-square Ritz method for the one-dimensional sine-Gordon equation,” Mathematical Problems in Engineering, vol. 2014, Article ID 383219, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. W. Shao and X. Wu, “The numerical solution of the nonlinear Klein-GORdon and sine-GORdon equations using the Chebyshev tau meshless method,” Computer Physics Communications, vol. 185, no. 5, pp. 1399–1409, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. A. Hussain, S. Haq, and M. Uddin, “Numerical solution of Klein-GORdon and sine-GORdon equations by meshless method of lines,” Engineering Analysis with Boundary Elements, vol. 37, no. 11, pp. 1351–1366, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229–256, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. K. M. Liew, Y. Q. Huang, and J. N. Reddy, “Moving least squares differential quadrature method and its application to the analysis of shear deformable plates,” International Journal for Numerical Methods in Engineering, vol. 56, no. 15, pp. 2331–2351, 2003. View at Publisher · View at Google Scholar · View at Scopus
  23. K. M. Liew and X. L. Chen, “Buckling of rectangular Mindlin plates subjected to partial in-plane edge loads using the radial point interpolation method,” International Journal of Solids and Structures, vol. 41, no. 5-6, pp. 1677–1695, 2004. View at Publisher · View at Google Scholar · View at Scopus
  24. J. J. Monaghan, “An introduction to SPH,” Computer Physics Communications, vol. 48, no. 1, pp. 89–96, 1988. View at Publisher · View at Google Scholar · View at Scopus
  25. W. Chen, “New RBF collocation methods and kernel RBF with applications,” in Meshfree Methods for Partial Differential Equations, vol. 1, pp. 75–86, Springer, 2000. View at Google Scholar
  26. K. M. Liew, X. Zhao, and T. Y. Ng, “The element-free kp-Ritz method for vibration of laminated rotating cylindrical panels,” International Journal of Structural Stability and Dynamics, vol. 2, pp. 523–558, 2002. View at Google Scholar
  27. X. Zhao and K. M. Liew, “Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 33–36, pp. 2796–2811, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. L. W. Zhang, Z. X. Lei, K. M. Liew, and J. L. Yu, “Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels,” Composite Structures, vol. 111, no. 1, pp. 205–212, 2014. View at Publisher · View at Google Scholar · View at Scopus
  29. L. W. Zhang, Y. J. Deng, and K. M. Liew, “An improved element-free Galerkin method for numerical modeling of the biological population problems,” Engineering Analysis with Boundary Elements, vol. 40, pp. 181–188, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. L. W. Zhang, Z. X. Lei, K. M. Liew, and J. L. Yu, “Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels,” Computer Methods in Applied Mechanics and Engineering, vol. 273, pp. 1–18, 2014. View at Publisher · View at Google Scholar · View at Scopus
  31. S. N. Atluri and T. Zhu, “A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics,” Computational Mechanics, vol. 22, no. 2, pp. 117–127, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  32. W. K. Liu, S. Jun, and Y. F. Zhang, “Reproducing kernel particle methods,” International Journal for Numerical Methods in Fluids, vol. 20, no. 8-9, pp. 1081–1106, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. L. W. Zhang, P. Zhu, and K. M. Liew, “Thermal buckling of functionally graded plates using a local Kriging meshless method,” Composite Structures, vol. 108, no. 1, pp. 472–492, 2014. View at Publisher · View at Google Scholar · View at Scopus