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New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi-Gauss collocation method. (English) Zbl 1217.65155
Summary: A new algorithm for solving the general nonlinear third-order differential equation is developed by means of a shifted Jacobi-Gauss collocation spectral method. The shifted Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithm, and some comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

MSC:
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
65L10Boundary value problems for ODE (numerical methods)
34B15Nonlinear boundary value problems for ODE
WorldCat.org
Full Text: DOI EuDML
References:
[1] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer, New York, NY, USA, 1988. · Zbl 0658.76001
[2] R. Peyret, Spectral Methods for Incompressible Viscous Flow, vol. 148, Springer, New York, NY, USA, 2002. · Zbl 1005.76001
[3] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, Pa, USA, 2000. · Zbl 0953.68643
[4] M. Dehghan and F. Shakeri, “Approximate solution of a differential equation arising in astrophysics using the variational iteration method,” New Astronomy, vol. 13, no. 1, pp. 53-59, 2008. · doi:10.1016/j.newast.2007.06.012
[5] K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi, “An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method,” Computer Physics Communications, vol. 181, no. 6, pp. 1096-1108, 2010. · Zbl 1216.65098 · doi:10.1016/j.cpc.2010.02.018
[6] A. H. Bhrawy and S. I. El-Soubhy, “Jacobi spectral Galerkin method for the integrated forms of second-order differential equations,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2684-2697, 2010. · Zbl 1204.65132 · doi:10.1016/j.amc.2010.08.006
[7] E. H. Doha, W. M. Abd-Elhameed, and A. H. Bhrawy, “Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations,” Applied Mathematical Modelling, vol. 33, no. 4, pp. 1982-1996, 2009. · Zbl 1205.65224 · doi:10.1016/j.apm.2008.05.005
[8] E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials,” Numerical Algorithms, vol. 42, no. 2, pp. 137-164, 2006. · Zbl 1103.65119 · doi:10.1007/s11075-006-9034-6
[9] E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1224-1244, 2008. · Zbl 1152.65112 · doi:10.1016/j.apnum.2007.07.001
[10] P. W. Livermore and G. R. Ierley, “Quasi-Lp norm orthogonal Galerkin expansions in sums of Jacobi polynomials. Orthogonal expansions,” Numerical Algorithms, vol. 54, no. 4, pp. 533-569, 2010. · Zbl 1197.65027 · doi:10.1007/s11075-009-9353-5
[11] E. H. Doha, A. H. Bhrawy, and W. M. Abd-Elhameed, “Jacobi spectral Galerkin method for elliptic Neumann problems,” Numerical Algorithms, vol. 50, no. 1, pp. 67-91, 2009. · Zbl 1169.65111 · doi:10.1007/s11075-008-9216-5
[12] S. Boatto, L. P. Kadanoff, and P. Olla, “Traveling-wave solutions to thin-film equations,” Physical Review E, vol. 48, no. 6, pp. 4423-4431, 1993. · doi:10.1103/PhysRevE.48.4423
[13] M. Gregu\vs, Third Order Linear Differential Equations, vol. 22 of Mathematics and Its Applications, D. Reidel, Dordrecht, The Netherlands, 1987.
[14] E. Momoniat, “Numerical investigation of a third-order ODE from thin film flow,” Meccanica. In press. · Zbl 1271.76215 · doi:10.1007/s11012-010-9310-3
[15] M. Bartu\vsek, M. Cecchi, and M. Marini, “On Kneser solutions of nonlinear third order differential equations,” Journal of Mathematical Analysis and Applications, vol. 261, no. 1, pp. 72-84, 2001. · Zbl 0995.34025 · doi:10.1006/jmaa.2000.7473
[16] Y. Feng, “Solution and positive solution of a semilinear third-order equation,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 153-161, 2009. · Zbl 1179.34021 · doi:10.1007/s12190-008-0121-9
[17] D. J. O’Regan, “Topological transversality: applications to third order boundary value problems,” SIAM Journal on Mathematical Analysis, vol. 18, no. 3, pp. 630-641, 1987. · Zbl 0628.34017 · doi:10.1137/0518048
[18] Q. Yao and Y. Feng, “The existence of solution for a third-order two-point boundary value problem,” Applied Mathematics Letters, vol. 15, no. 2, pp. 227-232, 2002. · Zbl 1008.34010 · doi:10.1016/S0893-9659(01)00122-7
[19] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, UK, 1991. · Zbl 0745.65049
[20] D. Sarafyan, “New algorithms for the continuous approximate solution of ordinary differential equations and the upgrading of the order of the processes,” Computers & Mathematics with Applications, vol. 20, no. 1, pp. 77-100, 1990. · Zbl 0702.65069 · doi:10.1016/0898-1221(90)90072-R
[21] S. O. Fatunla, Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1988. · Zbl 0651.65054
[22] S. Mehrkanoon, “A direct variable step block multistep method for solving general third-order ODEs,” Numerical Algorithms. In press. · Zbl 1215.65128 · doi:10.1007/s11075-010-9413-x
[23] D. O. Awoyemi, “A new sixth-order algorithm for general second order ordinary differential equations,” International Journal of Computer Mathematics, vol. 77, no. 1, pp. 117-124, 2001. · Zbl 0995.65076 · doi:10.1080/00207160108805054
[24] S. N. Jator and J. Li, “A self-starting linear multistep method for a direct solution of the general second-order initial value problem,” International Journal of Computer Mathematics, vol. 86, no. 5, pp. 827-836, 2009. · Zbl 1165.65038 · doi:10.1080/00207160701708250 · http://www.informaworld.com/smpp/./content~db=all~content=a794902936
[25] D. O. Awoyemi, “A P-stable linear multistep method for solving general third order ordinary differential equations,” International Journal of Computer Mathematics, vol. 80, no. 8, pp. 985-991, 2003. · Zbl 1040.65061 · doi:10.1080/0020716031000079572
[26] D. O. Awoyemi and O. M. Idowu, “A class of hybrid collocation methods for third-order ordinary differential equations,” International Journal of Computer Mathematics, vol. 82, no. 10, pp. 1287-1293, 2005. · Zbl 1117.65350 · doi:10.1080/00207160500112902
[27] B. Y. Guo and Z. R. Wang, “Numerical integration based on Laguerre-Gauss interpolation,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 37-40, pp. 3726-3741, 2007. · Zbl 1173.65312 · doi:10.1016/j.cma.2006.10.035
[28] B.-Y. Guo, Z.-Q. Wang, H.-J. Tian, and L.-L. Wang, “Integration processes of ordinary differential equations based on Laguerre-Radau interpolations,” Mathematics of Computation, vol. 77, no. 261, pp. 181-199, 2008. · Zbl 1127.65047 · doi:10.1090/S0025-5718-07-02035-2
[29] Y. Luke, The Special Functions and Their Approximations, vol. 2, Academic Press, New York, NY, USA, 1969. · Zbl 0193.01701
[30] H. N. Caglar, S. H. Caglar, and E. H. Twizell, “The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions,” International Journal of Computer Mathematics, vol. 71, no. 3, pp. 373-381, 1999. · Zbl 0929.65048 · doi:10.1080/00207169908804816
[31] L. Fox, “Determination and properties of Chebyshev expansions,” in Method of Numerical Approximation, D. C. Handscomb, Ed., Pergamon Press, Oxford, UK, 1966. · Zbl 0147.11703
[32] W. A. Light, “Are Chebyshev expansions really better?” Bulletin of the Institute of Mathematics and Its Applications, vol. 22, pp. 180-181, 1986. · Zbl 0635.41011