Table of Contents
Corrigendum

A corrigendum for this article has been published. To view the corrigendum, please click here.

ISRN Applied Mathematics
Volume 2011, Article ID 625908, 33 pages
http://dx.doi.org/10.5402/2011/625908
Research Article

On a Nonlinear Wave Equation Associated with Dirichlet Conditions: Solvability and Asymptotic Expansion of Solutions in Many Small Parameters

1Nhatrang Educational College, 01 Nguyen Chanh Street, Nhatrang City, Vietnam
2Department of Mathematics, University of Economics of Ho Chi Minh City, 59C Nguyen Dinh Chieu Street, District 3, Ho Chi Minh City, Vietnam
3Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Vietnam

Received 9 March 2011; Accepted 12 April 2011

Academic Editor: F. Jauberteau

Copyright © 2011 Le Thi Phuong Ngoc 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. E. L. Ortiz and A. P. N. Dinh, “Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the tau method,” SIAM Journal on Mathematical Analysis, vol. 18, no. 2, pp. 452–464, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. A. P. N. Dinh, “Sur un problème hyperbolique faiblement non-linéaire à une dimension,” Demonstratio Mathematica, vol. 16, no. 2, pp. 269–289, 1983. View at Google Scholar · View at Zentralblatt MATH
  3. A. P. N. Dinh and N. T. Long, “Linear approximation and asymptotic expansion associated to the nonlinear wave equation in one dimension,” Demonstratio Mathematica, vol. 19, no. 1, pp. 45–63, 1986. View at Google Scholar · View at Zentralblatt MATH
  4. N. T. Long and T. N. Diem, “On the nonlinear wave equation uttuxx=f(x,t,u,ux,ut) associated with the mixed homogeneous conditions,” Nonlinear Analysis, vol. 29, no. 11, pp. 1217–1230, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. N. T. Long, A. P. N. Dinh, and T. N. Diem, “On a shock problem involving a nonlinear viscoelastic bar,” Boundary Value Problems, no. 3, pp. 337–358, 2005. View at Google Scholar
  6. N. T. Long and L. X. Truong, “Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition,” Nonlinear Analysis, Theory, Methods & Applications. Series A: Theory and Methods, vol. 67, no. 3, pp. 842–864, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. N. T. Long and L. X. Truong, “Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary,” Electronic Journal of Differential Equations, vol. 2007, no. 48, pp. 1–19, 2007. View at Google Scholar · View at Zentralblatt MATH
  8. N. T. Long, N. C. Tam, and N. T. T. Truc, “On the nonlinear wave equation with the mixed nonhomogeneous conditions: linear approximation and asymptotic expansion of solutions,” Demonstratio Mathematica, vol. 38, no. 2, pp. 365–386, 2005. View at Google Scholar · View at Zentralblatt MATH
  9. L. T. P. Ngoc, L. N. K. Hang, and N. T. Long, “On a nonlinear wave equation associated with the boundary conditions involving convolution,” Nonlinear Analysis, Theory, Methods & Applications. Series A: Theory and Methods, vol. 70, no. 11, pp. 3943–3965, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. F. A. Ficken and B. A. Fleishman, “Initial value problems and time-periodic solutions for a nonlinear wave equation,” Communications on Pure and Applied Mathematics, vol. 10, pp. 331–356, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. P. H. Rabinowitz, “Periodic solutions of nonlinear hyperbolic partial differential equations,” Communications on Pure and Applied Mathematics, vol. 20, pp. 145–205, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. T. Kiguradze, “On bounded and time-periodic solutions of nonlinear wave equations,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 253–276, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. T. Caughey and J. Ellison, “Existence, uniqueness and stability of solutions of a class of nonlinear partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 51, pp. 1–32, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. L. T. P. Ngoc, L. K. Luan, T. M. Thuyet, and N. T. Long, “On the nonlinear wave equation with the mixed nonhomogeneous conditions: linear approximation and asymptotic expansion of solutions,” Nonlinear Analysis, Theory, Methods & Applications. Series A: Theory and Methods, vol. 71, no. 11, pp. 5799–5819, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires, Dunod-Gauthier Villars, Paris, France, 1969.
  16. K. Deimling, Nonlinear Functional Analysis, Springer, NewYork, NY, USA, 1985.
  17. N. T. Long, “On the nonlinear wave equation uttB(t,u2,ux2)uxx=f(x,t,u,ux,ut,u2,ux2) associated with the mixed homogeneous conditions,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 243–268, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. Boujot, A. P. N. Dinh, and J. P. Veyrier, “Oscillateurs harmoniques faiblement perturbés: L'algorithme numérique des “par de géants”,” RAIRO Analyse Numérique, vol. 14, no. 1, pp. 3–23, 1980. View at Google Scholar · View at Zentralblatt MATH
  19. N. T. Long, A. P. N. Dinh, and T. N. Diem, “Linear recursive schemes and asymptotic expansion associated with the Kirchoff—Carrier operator,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 116–134, 2002. View at Publisher · View at Google Scholar