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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 614508, 10 pages
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.
- 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.
- M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Reviews of Modern Physics, vol. 65, no. 3, pp. 851–1112, 1993.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- R. P. Agarwal and P. R. Krishnamoorthy, “Boundary value problems for th order ordinary differential equations,” Bulletin of the Institute of Mathematics, vol. 7, no. 2, pp. 211–230, 1979.
- 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.
- J. Schröder, “Numerical error bounds for fourth order boundary value problems, simultaneous estimation of and ,” Numerische Mathematik, vol. 44, no. 2, pp. 233–245, 1984.
- 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.
- 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.
- 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.
- 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.
- R. A. Usmani and P. J. Taylor, “Finite difference methods for solving ,” International Journal of Computer Mathematics, vol. 14, no. 3-4, pp. 277–293, 1983.
- D. Britz, Digital Simulation in Electrochemistry, vol. 66 of Lecture Notes in Physics, Springer, Berlin, Germany, 2005.
- 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.
- 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.
- R. K. Mohanty, “A class of non-uniform mesh three point arithmetic average discretization for and the estimates of y′,” Applied Mathematics and Computation, vol. 183, no. 1, pp. 477–485, 2006.
- 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.
- 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.
- 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.
- W. Gautschi, Numerical Analysis, Birkhause, 2011.
- R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer, Berlin, 2000.
- P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1962.
- D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971.
- 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.
- 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.