Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method. 1. Introduction Consider the fourth-order boundary value problem: subject to the necessary boundary conditions: where and , , , and are real constants and . Or equivalently subject to the natural boundary conditions: Fourth-order differential equations occur in various areas of mathematics such as viscoelastic and inelastic flows, beam theory, Lifshitz point in phase transition physics (e.g., nematic liquid crystal, crystals, and ferroelectric crystals) [1], the rolls in a Rayleigh-Benard convection cell (two parallel plates of different temperature with a liquid in between) [2], spontaneous pattern formation in second-order materials (e.g., polymeric fibres) [3], the waves on a suspension bridge [4, 5], geological folding of rock layers [6], buckling of a strut on a nonlinear elastic foundation [7], traveling water waves in a shallow channel [8], pulse propagation in optical fibers [9], system of two reaction diffusion equation [10], and so forth. The existence and uniqueness of the solution for the fourth and higher-order boundary value problems have been discussed in [11–14]. In the recent past, the numerical solution of fourth-order differential equations has been developed using multiderivative, finite element method, Ritz method, spline collocation, and finite difference method [15–18]. The determination of eigen values of self adjoint fourth-order differential equations was developed in [19] using finite difference scheme. The motivation of variable mesh technique for differential equations arises from the theory of electrochemical reaction-convection-diffusion problems in one-dimensional space geometry [20]. The geometric mesh method for self-adjoint singular perturbation problems using finite difference approximations was discussed in [21]. The use of geometric mesh in the context of boundary value problems was studied extensively in [22–24]. In this paper, we derive a geometric mesh finite difference method for the solution of fourth- and sixth-order differential boundary value problems with order
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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]
A. Doelman and V. Rottsch？fer, “Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms,” Journal of Nonlinear Science, vol. 7, no. 4, pp. 371–409, 1997.
[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.
[12]
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.
[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.
[14]
J. Schr？der, “Numerical error bounds for fourth order boundary value problems, simultaneous estimation of and ,” Numerische Mathematik, vol. 44, no. 2, pp. 233–245, 1984.
[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.
[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.
[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.
[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.
[19]
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.
[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.
[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.
[23]
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.
[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.
[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.
[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.
[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.
[29]
P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1962.
[30]
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971.
[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.
[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.
