×

A class of numerical methods for the solution of fourth-order ordinary differential equations in polar coordinates. (English) Zbl 1250.65105

Summary: Using only three grid points, we propose two sets of numerical methods in a coupled manner for the solution of a fourth-order ordinary differential equation \(u^{i,v}(x) = f(x, u(x), u'(x), u''(x), u'''(x)), ~a < x < b\), subject to boundary conditions \(u(a) = A_0, ~u'(a) = A_1, ~u(b) = B_0\), and \(u'(b) = B_1\), where \(A_0, A_1, B_0\), and \(B_1\) 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 a 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.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

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. · Zbl 0634.34009 · doi:10.1016/S0022-247X(86)80006-3
[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. · Zbl 0413.34020
[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. · Zbl 0467.34013
[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. · Zbl 0503.73061 · doi:10.1016/0771-050X(82)90035-3
[6] Z. Bai, B. Huang, and W. Ge, “The iterative solutions for some fourth-order p-Laplace equation boundary value problems,” Applied Mathematics Letters, vol. 19, no. 1, pp. 8-14, 2006. · Zbl 1092.34510 · doi:10.1016/j.aml.2004.10.010
[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. · Zbl 1119.34020 · doi:10.1016/j.na.2006.03.028
[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. · Zbl 1054.34038 · doi:10.1016/S0022-247X(03)00545-6
[9] M. D. Greenberg, Differential Equations and Linear Algebra, chapter 7, Prentice Hall, Engelwood Cliffs, NJ, USA, 2001. · Zbl 0998.11054
[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. · Zbl 0963.65083 · doi:10.1016/S0377-0427(99)00202-2
[11] R. K. Nagle and E. B. Saff, Fundamentals of Differential Equations, chapter 6, The Benjamin/Cummingspp, 1986. · Zbl 0604.34001
[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. · Zbl 1141.65375 · doi:10.1016/j.camwa.2006.12.057
[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. · Zbl 0795.34018 · doi:10.1016/0022-247X(91)90363-5
[14] J. Schröder, “Numerical error bounds for fourth order boundary value problems, simultaneous estimation of u(x) and u\(^{\prime\prime}\)(x),” Numerische Mathematik, vol. 44, no. 2, pp. 233-245, 1984. · Zbl 0536.34004 · doi:10.1007/BF01410108
[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. · Zbl 1025.65044 · doi:10.1016/S0096-3003(01)00040-6
[16] D. G. Zill and M. R. Cullen, Differential Equations with Boundary-Value Problems, chapter 5, Brooks Cole, NewYork, NY, USA, 1997. · Zbl 0802.34003
[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. · Zbl 0599.65057 · doi:10.1155/S0161171286000170
[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. · Zbl 0637.65081 · doi:10.1155/S0161171287000620
[19] R. A. Usmani and P. J. Taylor, “Finite difference methods for solving [p(x)y\(^{\prime\prime}\)]\(^{\prime}\)\(^{\prime}\)+q(x)y=r(x),” International Journal of Computer Mathematics, vol. 14, no. 3-4, pp. 277-293, 1983. · Zbl 0534.65040 · doi:10.1080/00207168308803391
[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. · Zbl 0619.65072
[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. · Zbl 0541.65055 · doi:10.1016/0377-0427(84)90058-X
[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. · Zbl 0807.34023 · doi:10.1006/jmaa.1994.1250
[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. · Zbl 1068.34013 · doi:10.1016/j.cam.2004.04.013
[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. · Zbl 1126.34013 · doi:10.1016/j.na.2006.03.042
[25] L. A. Hageman and D. M. Young, Applied Iterative Methods, Dover Publications, Mineola, NY, USA, 2004. · Zbl 1059.65028
[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. · Zbl 0922.68004 · doi:10.1080/10637199808947369
[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. · Zbl 0385.65038 · doi:10.1093/imamat/21.1.83
[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. · Zbl 0942.65083 · doi:10.1080/00207169908804883
[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. · Zbl 1003.65088 · doi:10.1080/00207160212704
[30] J. Prescott, Applied Elasticity, Dover Publications Inc, New York, NY, USA, 1961. · JFM 50.0554.12
[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. · Zbl 0347.76022 · doi:10.1007/BF00251865
[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. · Zbl 1031.65046
[34] R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2000. · Zbl 0998.65505
[35] D. M. Young, Iterative Solution of Large Linear Systems, Dover Publications, Mineola, NY, USA, 2003. · Zbl 1049.65022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.