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Spectral shifted Jacobi tau and collocation methods for solving fifth-order boundary value problems. (English) Zbl 1221.65171
Summary: We present an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth-order two-point boundary value problems (BVPs). An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth-order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.

MSC:
65L10Boundary value problems for ODE (numerical methods)
34B05Linear boundary value problems for ODE
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
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References:
[1] A. R. Davies, A. Karageoghis, and T. N. Phillips, “Spectral Glarkien methods for the primary twopoint boundary-value problems in modeling viscelastic flows,” International Journal for Numerical Methods in Engineering, vol. 26, pp. 647-662, 1988. · Zbl 0635.73091 · doi:10.1002/nme.1620260309
[2] D. J. Fyfe, “Linear dependence relations connecting equal interval Nth degree splines and their derivatives,” Journal of the Institute of Mathematics and Its Applications, vol. 7, pp. 398-406, 1971. · Zbl 0219.65010 · doi:10.1093/imamat/7.3.398
[3] A. Karageoghis, T. N. Phillips, and A. R. Davies, “Spectral collocation methods for the primary two-point boundary-value problems in modeling viscelastic flows,” International Journal for Numerical Methods in Engineering, vol. 26, pp. 805-813, 1998. · Zbl 0637.76008 · doi:10.1002/nme.1620260404
[4] G. L. Liu, “New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique,” in Proceeding of the 7th Conference of the Modern Mathematics and Mechanics, Shanghai, China, 1997.
[5] R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986. · Zbl 0681.76121
[6] H. N. \cCaglar, S. H. \cCaglar, and E. H. Twizell, “The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions,” Applied Mathematics Letters, vol. 12, no. 5, pp. 25-30, 1999. · Zbl 0941.65073 · doi:10.1016/S0893-9659(99)00052-X
[7] A.-M. Wazwaz, “The numerical solution of fifth-order boundary value problems by the decomposition method,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 259-270, 2001. · Zbl 0986.65072 · doi:10.1016/S0377-0427(00)00618-X
[8] M. A. Khan, Siraj-ul-Islam, I. A. Tirmizi, E. H. Twizell, and S. Ashraf, “A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 651-660, 2006. · Zbl 1096.65070 · doi:10.1016/j.jmaa.2005.08.023
[9] S. S. Siddiqi and G. Akram, “Sextic spline solutions of fifth order boundary value problems,” Applied Mathematics Letters, vol. 20, no. 5, pp. 591-597, 2007. · Zbl 1125.65071 · doi:10.1016/j.aml.2006.06.012
[10] S. S. Siddiqi, G. Akram, and S. A. Malik, “Nonpolynomial sextic spline method for the solution along with convergence of linear special case fifth-order two-point value problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 532-541, 2007. · Zbl 1125.65072 · doi:10.1016/j.amc.2007.01.071
[11] 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
[12] Siraj-ul-Islam and M. Azam Khan, “A numerical method based on polynomial sextic spline functions for the solution of special fifth-order boundary-value problems,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 356-361, 2006. · Zbl 1148.65312 · doi:10.1016/j.amc.2006.01.042
[13] H. Caglar and N. Caglar, “Solution of fifth order boundary value problems by using local polynomial regression,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 952-956, 2007. · Zbl 1118.65347 · doi:10.1016/j.amc.2006.08.046
[14] M. El-Gamel, “Sinc and the numerical solution of fifth-order boundary value problems,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1417-1433, 2007. · Zbl 1121.65087 · doi:10.1016/j.amc.2006.09.049
[15] J. Rashidinia, R. Jalilian, and K. Farajeyan, “Spline approximate solution of fifth-order boundary-value problem,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 107-112, 2007. · Zbl 1193.65132 · doi:10.1016/j.amc.2007.02.124
[16] 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
[17] B. Fornberg, A Practical Guide to Pseudospectral Methods, vol. 1 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, Mass, USA, 1996. · Zbl 0844.65084 · doi:10.1017/CBO9780511626357
[18] L. N. Trefethen, Spectral Methods in MATLAB, vol. 10 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2000.
[19] E. H. Doha, A. H. Bhrawy, and M. A. Saker, “Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations,” Applied Mathematics Letters, vol. 24, no. 4, pp. 559-565, 2011. · Zbl 1236.65091 · doi:10.1016/j.aml.2010.11.013
[20] E. H. Doha, A. H. Bhrawy, and M. A. Saker, “On the derivatives of bernstein polynomials: an application for the solution of high even-order differential equations,” Boundary Value Problems, vol. 2011, Article ID 829543, 16 pages, 2011. · Zbl 1220.33006 · doi:10.1155/2011/829543 · eudml:224410
[21] A. H. Bhrawy and W. M. Abd-Elhameed, “New algorithm for the numerical solution of nonlinear third-order differential equation using Jacobi-Gauss collocation metho,” Mathematical Problems in Engineering, vol. 2011, Article ID 837218, 14 pages, 2011. · Zbl 1217.65155 · doi:10.1155/2011/837218 · eudml:229863
[22] A. H. Bhrawy and A. S. Alofi, “A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations,” Communications in Nonlinear Science and Numerical Simulation, 2011. · Zbl 1244.65099 · doi:10.1016/j.cnsns.2011.04.025
[23] 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
[24] 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
[25] E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 947230, 21 pages, 2011. · Zbl 1216.65086 · doi:10.1155/2011/947230 · eudml:223203
[26] 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
[27] E. L. Ortiz, “The tau method,” SIAM Journal on Numerical Analysis, vol. 6, pp. 480-492, 1969. · Zbl 0195.45701 · doi:10.1137/0706044
[28] N. Mai-Duy, “An effective spectral collocation method for the direct solution of high-order ODEs,” Communications in Numerical Methods in Engineering, vol. 22, no. 6, pp. 627-642, 2006. · Zbl 1105.65342 · doi:10.1002/cnm.841
[29] E. H. Doha, “On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials,” Journal of Physics A, vol. 37, no. 3, pp. 657-675, 2004. · Zbl 1055.33007 · doi:10.1088/0305-4470/37/3/010
[30] Y. Luke, The Special Functions and Their Approximations, vol. 2, Academic Press, New York, NY, USA, 1969. · Zbl 0193.01701
[31] M. A. Noor and S. T. Mohyud-Din, “An efficient algorithm for solving fifth-order boundary value problems,” Mathematical and Computer Modelling, vol. 45, no. 7-8, pp. 954-964, 2007. · Zbl 1133.65052 · doi:10.1016/j.mcm.2006.09.004
[32] M. A. Noor and S. T. Mohyud-Din, “Variational iteration technique for solving higher order boundary value problems,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1929-1942, 2007. · Zbl 1122.65374 · doi:10.1016/j.amc.2006.12.071
[33] M. A. Noor and S.T. Mohyud-Din, “Modified decomposition method for solving linear and nonlinear fifth-order boundary value problems,” International Journal of Applied Mathematics and Computer Science. In press. · Zbl 1133.65052
[34] M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for fifth-order boundary value problems using He’s polynomials,” Mathematical Problems in Engineering, vol. 2008, Article ID 954794, 12 pages, 2008. · Zbl 1151.65334 · doi:10.1155/2008/954794 · eudml:54822