×

Cubic Hermite collocation method for solving boundary value problems with Dirichlet, Neumann, and Robin conditions. (English) Zbl 1337.65165

Summary: Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. H. Ahlberg and T. Ito, “A collocation method for two point boundary value problems,” Mathematics of Computation, vol. 29, no. 131, pp. 761-776, 1975. · Zbl 0312.65056 · doi:10.2307/2005287
[2] M. A. Soliman and A. A. Ibrahim, “Studies on the method of orthogonal collocation III: the use of Jacobi orthogonal polynomials for the solution of boundary value problems,” Journal of King Saud University, vol. 11, no. 2, pp. 191-202, 1999.
[3] H. Caglar, N. Caglar, and K. Elfaituri, “B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 72-79, 2006. · Zbl 1088.65069 · doi:10.1016/j.amc.2005.07.019
[4] W. R. Dyksen and R. E. Lynch, “A new decoupling technique for the Hermite cubic collocation equations arising from boundary value problems,” Mathematics and Computers in Simulation, vol. 54, no. 4-5, pp. 359-372, 2000. · Zbl 0985.65088 · doi:10.1016/S0378-4754(00)00194-4
[5] D. J. Higham, “Monotonic piecewise cubic interpolation, with applications to ODE plotting,” Journal of Computational and Applied Mathematics, vol. 39, no. 3, pp. 287-294, 1992. · Zbl 0768.65002 · doi:10.1016/0377-0427(92)90205-C
[6] P. M. Prenter, Splines and Variational Methods, Wiley interscience, New York, NY, USA, 1975. · Zbl 0344.65044
[7] Q. Fang, T. Tsuchiya, and T. Yamamoto, “Finite difference, finite element and finite volume methods applied to two-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 139, no. 1, pp. 9-19, 2002. · Zbl 0993.65082 · doi:10.1016/S0377-0427(01)00392-2
[8] M. Kumar, “A second order finite difference method and its convergence for a class of singular two-point boundary value problems,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 873-878, 2003. · Zbl 1032.65086 · doi:10.1016/S0096-3003(02)00645-8
[9] V. R. Subramanian and R. E. White, “Symbolic solutions for boundary value problems using Maple,” Computers and Chemical Engineering, vol. 24, no. 11, pp. 2405-2416, 2000.
[10] S. N. Ha, “A nonlinear shooting method for two-point boundary value problems,” Computers and Mathematics with Applications, vol. 42, no. 10-11, pp. 1411-1420, 2001. · Zbl 0999.65077 · doi:10.1016/S0898-1221(01)00250-4
[11] M. Krajnc, “Geometric Hermite interpolation by cubic G1 splines,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 7, pp. 2614-2626, 2009. · Zbl 1163.41303 · doi:10.1016/j.na.2008.03.048
[12] F.-G. Lang and X.-P. Xu, “Quintic B-spline collocation method for second order mixed boundary value problem,” Computer Physics Communications, vol. 183, no. 4, pp. 913-921, 2012. · Zbl 1264.65123 · doi:10.1016/j.cpc.2011.12.017
[13] L.-B. Liu, H.-W. Liu, and Y. Chen, “Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6872-6882, 2011. · Zbl 1213.65109 · doi:10.1016/j.amc.2011.01.047
[14] M. A. Ramadan, I. F. Lashien, and W. K. Zahra, “Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 476-484, 2007. · Zbl 1114.65092 · doi:10.1016/j.amc.2006.06.053
[15] S. S. Siddiqi and G. Akram, “Solution of fifth order boundary value problems using nonpolynomial spline technique,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1574-1581, 2006. · Zbl 1094.65072 · doi:10.1016/j.amc.2005.09.004
[16] S. S. Siddiqi, G. Akram, and A. Elahi, “Quartic spline solution of linear fifth order boundary value problems,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 214-220, 2008. · Zbl 1135.65352 · doi:10.1016/j.amc.2007.05.060
[17] W. Sun, “Hermite cubic spline collocation methods with upwind features,” ANZIAM Journal, vol. 42, pp. 1379-1397, 2000.
[18] Z. Xiong and Y. Chen, “Finite volume element method with interpolated coefficients for two-point boundary value problem of semilinear differential equations,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 37-40, pp. 3798-3804, 2007.
[19] C. de Boor, A Practical Guide To Splines, Springer, New York, NY, USA, 2001. · Zbl 0987.65015
[20] J. Douglas and T. Dupont, “A finite element collocation method for quasi-linear parabolic equations,” Mathematics of Computation, vol. 121, pp. 17-28, 1973. · Zbl 0256.65050 · doi:10.2307/2005243
[21] I. A. Ganaie, B. Gupta, N. Parumasur, P. Singh, and V. K. Kukreja, “Asymptotic convergence of cubic Hermite collocation method for axial dispersion model,” Applied Mathematics and Computation, vol. 220, no. 1, pp. 560-567, 2013. · Zbl 1329.65235 · doi:10.1016/j.amc.2013.05.073
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.