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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 614508, 10 pages
http://dx.doi.org/10.1155/2013/614508
Research Article

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

1Department of Mathematics, South Asian University, Akbar Bhawan, Chanakya Puri, New Delhi 110021, India
2Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India

Received 9 June 2013; Revised 8 August 2013; Accepted 23 August 2013

Academic Editor: Rüdiger Weiner

Copyright © 2013 Navnit Jha 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. R. M. Hornreich, M. Luban, and S. Shtrikman, “Critical behavior at the onset of k→-space instability on the λ line,” Physical Review Letters, vol. 35, no. 25, pp. 1678–1681, 1975. View at Publisher · View at Google Scholar · View at Scopus
  2. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Reviews of Modern Physics, vol. 65, no. 3, pp. 851–1112, 1993. View at Publisher · View at Google Scholar · View at Scopus
  3. A. Leizarowitz and V. J. Mizel, “One dimensional infinite-horizon variational problems arising in continuum mechanics,” Archive for Rational Mechanics and Analysis, vol. 106, no. 2, pp. 161–194, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. C. Lazer and P. J. Mckenna, “Large-amplitude periodic oscillations in suspension bridges. Some new connections with nonlinear analysis,” SIAM Review, vol. 32, no. 4, pp. 537–578, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Y. Chen and P. J. McKenna, “Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations,” Journal of Differential Equations, vol. 136, no. 2, pp. 325–355, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. J. Budd, G. W. Hunt, and M. A. Peletier, “Self-similar fold evolution under prescribed end shortening,” Mathematical Geology, vol. 31, no. 8, pp. 989–1004, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. C. J. Amick and J. F. Toland, “Homoclinic orbits in the dynamic phase-space analogy of an elastic strut,” European Journal of Applied Mathematics, vol. 3, no. 2, pp. 97–114, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. B. Buffoni, A. R. Champneys, and J. F. Toland, “Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system,” Journal of Dynamics and Differential Equations, vol. 8, no. 2, pp. 221–279, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. N. N. Akhmediev, A. V. Buryak, and M. Karlsson, “Radiationless optical solitons with oscillating tails,” Optics Communications, vol. 110, no. 5-6, pp. 540–544, 1994. View at Scopus
  10. A. Doelman and V. Rottschäfer, “Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms,” Journal of Nonlinear Science, vol. 7, no. 4, pp. 371–409, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. R. Aftabizadeh, “Existence and uniqueness theorems for fourth-order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. P. Agarwal and P. R. Krishnamoorthy, “Boundary value problems for nth order ordinary differential equations,” Bulletin of the Institute of Mathematics, vol. 7, no. 2, pp. 211–230, 1979. View at MathSciNet
  13. D. O'Regan, “Solvability of some fourth (and higher) order singular boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 161, no. 1, pp. 78–116, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. Schröder, “Numerical error bounds for fourth order boundary value problems, simultaneous estimation of u(x) and u(x),” Numerische Mathematik, vol. 44, no. 2, pp. 233–245, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. V. Shanthi and N. Ramanujam, “A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations,” Applied Mathematics and Computation, vol. 129, no. 2-3, pp. 269–294, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. P. Agarwal and Y. M. Chow, “Iterative methods for a fourth order boundary value problem,” Journal of Computational and Applied Mathematics, vol. 10, no. 2, pp. 203–217, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. W. K. Zahra, “A smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems,” Applied Mathematics and Computation, vol. 217, no. 21, pp. 8447–8457, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. Rashidinia and M. Ghasemi, “B-spline collocation for solution of two-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2325–2342, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. A. Usmani and P. J. Taylor, “Finite difference methods for solving pxy+qxy=rx,” International Journal of Computer Mathematics, vol. 14, no. 3-4, pp. 277–293, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. Britz, Digital Simulation in Electrochemistry, vol. 66 of Lecture Notes in Physics, Springer, Berlin, Germany, 2005.
  21. M. K. Kadalbajoo and D. Kumar, “Geometric mesh FDM for self-adjoint singular perturbation boundary value problems,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1646–1656, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. K. Jain, S. R. K. Iyengar, and G. S. Subramanyam, “Variable mesh methods for the numerical solution of two-point singular perturbation problems,” Computer Methods in Applied Mechanics and Engineering, vol. 42, no. 3, pp. 273–286, 1984. View at Scopus
  23. R. K. Mohanty, “A class of non-uniform mesh three point arithmetic average discretization for y=f(x,y,y) and the estimates of y′,” Applied Mathematics and Computation, vol. 183, no. 1, pp. 477–485, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. N. Jha, “A fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems,” Applied Mathematics and Computation, vol. 219, no. 16, pp. 8425–8434, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  25. S. R. K. Iyengar and P. Jain, “Spline finite difference methods for singular two point boundary value problems,” Numerische Mathematik, vol. 50, no. 3, pp. 363–376, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. D. Russell and L. F. Shampine, “Numerical methods for singular boundary value problems,” SIAM Journal on Numerical Analysis, vol. 12, pp. 13–36, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. W. Gautschi, Numerical Analysis, Birkhause, 2011.
  28. R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer, Berlin, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  29. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1962. View at MathSciNet
  30. D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971. View at MathSciNet
  31. J. Talwar and R. K. Mohanty, “A class of numerical methods for the solution of fourth-order ordinary differential equations in polar coordinates,” Advances in Numerical Analysis, vol. 2012, Article ID 626419, 20 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. A. R. Elcrat, “On the radial flow of a viscous fluid between porous disks,” Archive for Rational Mechanics and Analysis, vol. 61, no. 1, pp. 91–96, 1976. View at Zentralblatt MATH · View at MathSciNet