In this piece of work using only three grid points, we propose two sets of numerical methods in a coupled manner for the solution of fourth-order ordinary differential equation , , subject to boundary conditions , , , and , where , , , and are real constants. We do not require to discretize the boundary conditions. The derivative of the solution is obtained as a byproduct of the discretization procedure. We use block iterative method and tridiagonal solver to obtain the solution in both cases. Convergence analysis is discussed and numerical results are provided to show the accuracy and usefulness of the proposed methods. 1. Introduction Consider the fourth-order boundary value problem subject to the prescribed natural boundary conditions or equivalently, for , subject to the natural boundary conditions where , , , and are real constants and . Fourth-order differential equations occur in a number of areas of applied mathematics, such as in beam theory, viscoelastic and inelastic flows, and electric circuits. Some of them describe certain phenomena related to the theory of elastic stability. A classical fourth-order equation arising in the beam-column theory is the following (see Timoshenko [1]): where is the lateral deflection, is the intensity of a distributed lateral load, is the axial compressive force applied to the beam, and represents the flexural rigidity in the plane of bending. Various generalizations of the equation describing the deformation of an elastic beam with different types of two-point boundary conditions have been extensively studied via a broad range of methods. The existence and uniqueness of solutions of boundary value problems are discussed in the papers and book of Agarwal and Krishnamoorthy, Agarwal and Akrivis (see [2–5]). Several authors have investigated solving fourth-order boundary value problem by some numerical techniques, which include the cubic spline method, Ritz method, finite difference method, multiderivative methods, and finite element methods (see [6–16]). In the 1980s, Usmani et al. (see [17–19]) worked on finite difference methods for solving and finite difference methods for computing eigenvalues of fourth-order linear boundary value problem. In 1984, Twizell and Tirmizi (see [20]) developed multi-derivative methods for linear fourth-order boundary value problems. In 1984, Agarwal and Chow (see [21]) developed iterative methods for a fourth-order boundary value problem. In 1991, O’Regan (see [13]) worked on the solvability of some fourth-(and higher) order singular boundary value problems. In 1994, Cabada (see
References
[1]
S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book, New York, NY, USA, 2nd edition, 1961.
[2]
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.
[3]
R. P. Agarwal and P. R. Krishnamoorthy, “Boundary value problems for nth order ordinary differential equations,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 7, no. 2, pp. 211–230, 1979.
[4]
R. P. Agarwal, “Boundary value problems for higher order differential equations,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 9, no. 1, pp. 47–61, 1981.
[5]
R. P. Agarwal and G. Akrivis, “Boundary value problems occurring in plate deflection theory,” Journal of Computational and Applied Mathematics, vol. 8, no. 3, pp. 145–154, 1982.
[6]
Z. Bai, B. Huang, and W. Ge, “The iterative solutions for some fourth-order -Laplace equation boundary value problems,” Applied Mathematics Letters, vol. 19, no. 1, pp. 8–14, 2006.
[7]
J. R. Graef and L. Kong, “A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems,” Nonlinear Analysis, vol. 66, no. 11, pp. 2389–2412, 2007.
[8]
J. R. Graef, C. Qian, and B. Yang, “A three point boundary value problem for nonlinear fourth order differential equations,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 217–233, 2003.
[9]
M. D. Greenberg, Differential Equations and Linear Algebra, chapter 7, Prentice Hall, Engelwood Cliffs, NJ, USA, 2001.
[10]
R. K. Mohanty, “A fourth-order finite difference method for the general one-dimensional nonlinear biharmonic problems of first kind,” Journal of Computational and Applied Mathematics, vol. 114, no. 2, pp. 275–290, 2000.
[11]
R. K. Nagle and E. B. Saff, Fundamentals of Differential Equations, chapter 6, The Benjamin/Cummingspp, 1986.
[12]
M. A. Noor and S. T. Mohyud-Din, “An efficient method for fourth-order boundary value problems,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1101–1111, 2007.
[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]
D. G. Zill and M. R. Cullen, Differential Equations with Boundary-Value Problems, chapter 5, Brooks Cole, NewYork, NY, USA, 1997.
[17]
R. A. Usmani, “Finite difference methods for computing eigenvalues of fourth order boundary value problems,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 1, pp. 137–143, 1986.
[18]
R. A. Usmani and M. Sakai, “Two new finite difference methods for computing eigenvalues of a fourth order linear boundary value problem,” International Journal of Mathematics and Mathematical Sciences, vol. 10, no. 3, pp. 525–530, 1987.
[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]
E. H. Twizell and S. I. A. Tirmizi, “Multiderivative methods for linear fourth order boundary value problems,” Tech. Rep. TR/06/84, Department of Mathematics And Statistics, Brunel University, 1984.
[21]
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.
[22]
A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302–320, 1994.
[23]
D. Franco, D. O'Regan, and J. Perán, “Fourth-order problems with nonlinear boundary conditions,” Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 315–327, 2005.
[24]
G. Han and F. Li, “Multiple solutions of some fourth-order boundary value problems,” Nonlinear Analysis, vol. 66, no. 11, pp. 2591–2603, 2007.
[25]
L. A. Hageman and D. M. Young, Applied Iterative Methods, Dover Publications, Mineola, NY, USA, 2004.
[26]
R. K. Mohanty and D. J. Evans, “Block iterative methods for one-dimensional nonlinear biharmonic problems on a parallel computer,” Parallel Algorithms and Applications, vol. 13, no. 3, pp. 239–263, 1999.
[27]
M. M. Chawla, “A fourth-order tridiagonal finite difference method for general non-linear two-point boundary value problems with mixed boundary conditions,” Journal of the Institute of Mathematics and its Applications, vol. 21, no. 1, pp. 83–93, 1978.
[28]
R. K. Mohanty and D. J. Evans, “New algorithms for the numerical solution of one dimensional singular biharmonic problems of second kind,” International Journal of Computer Mathematics, vol. 73, no. 1, pp. 105–124, 1999.
[29]
D. J. Evans and R. K. Mohanty, “Alternating group explicit method for the numerical solution of non-linear singular two-point boundary value problems using a fourth order finite difference method,” International Journal of Computer Mathematics, vol. 79, no. 10, pp. 1121–1133, 2002.
[30]
J. Prescott, Applied Elasticity, Dover Publications Inc, New York, NY, USA, 1961.
[31]
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.
[32]
C. T. Kelly, Iterative Methods for Linear and Non-Linear Equations, SIAM Publication, Philadelphia, Pa, USA, 1995.
[33]
Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2nd edition, 2003.
[34]
R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2000.
[35]
D. M. Young, Iterative Solution of Large Linear Systems, Dover Publications, Mineola, NY, USA, 2003.
