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A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations. (English) Zbl 1216.65086
Summary: A Jacobi dual-Petrov-Galerkin (JDPG) method is introduced and used for solving fully integrated reformulations of third- and fifth-order ordinary differential equations (ODEs) with constant coefficients. The reformulated equation for the $$J$$-th order ODE involves $$n$$-fold indefinite integrals for $$n = 1, \dots, J$$. The extension of the JDPG method to ODEs with polynomial coefficients is treated using Jacobi-Gauss-Lobatto quadrature. Numerical results with comparisons are given to confirm the reliability of the proposed method for some constant and polynomial coefficients ODEs.

##### MSC:
 65L05 Numerical methods for initial value problems 34A30 Linear ordinary differential equations and systems, general 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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##### References:
 [1] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, Mineola, NY, USA, 2nd edition, 2001. · Zbl 0994.65128 [2] 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 [3] Y. Chen and T. Tang, “Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel,” Mathematics of Computation, vol. 79, no. 269, pp. 147-167, 2010. · Zbl 1207.65157 · doi:10.1090/S0025-5718-09-02269-8 [4] 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 [5] P. W. Livermore and G. R. Ierley, “Quasi- 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 [6] K. Aghigh, M. Masjed-Jamei, and M. Dehghan, “A survey on third and fourth kind of Chebyshev polynomials and their applications,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 2-12, 2008. · Zbl 1134.33300 · doi:10.1016/j.amc.2007.09.018 [7] E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of the integrated forms of second-order equations using ultraspherical polynomials,” The ANZIAM Journal, vol. 48, no. 3, pp. 361-386, 2007. · Zbl 1138.65104 · doi:10.1017/S1446181100003540 [8] M. Fernandino, C. A. Dorao, and H. A. Jakobsen, “Jacobi galerkin spectral method for cylindrical and spherical geometries,” Chemical Engineering Science, vol. 62, no. 23, pp. 6777-6783, 2007. · doi:10.1016/j.ces.2007.07.062 [9] W. Heinrichs, “Spectral approximation of third-order problems,” Journal of Scientific Computing, vol. 14, no. 3, pp. 275-289, 1999. · Zbl 0953.65072 · doi:10.1023/A:1023221619567 [10] P. W. Livermore, “Galerkin orthogonal polynomials,” Journal of Computational Physics, vol. 229, no. 6, pp. 2046-2060, 2010. · Zbl 1197.65027 · doi:10.1007/s11075-009-9353-5 [11] A. R. Aftabizadeh, C. P. Gupta, and J.-M. Xu, “Existence and uniqueness theorems for three-point boundary value problems,” SIAM Journal on Mathematical Analysis, vol. 20, no. 3, pp. 716-726, 1989. · Zbl 0704.34019 · doi:10.1137/0520049 [12] A. R. Aftabizadeh and K. Deimling, “A three-point boundary value problem,” Differential and Integral Equations, vol. 4, no. 1, pp. 189-194, 1991. · Zbl 0723.34016 [13] F. Bernis and L. A. Peletier, “Two problems from draining flows involving third-order ordinary differential equations,” SIAM Journal on Mathematical Analysis, vol. 27, no. 2, pp. 515-527, 1996. · Zbl 0845.34033 · doi:10.1137/S0036141093260847 [14] A. Boucherif, S. M. Bouguima, N. Al-Malki, and Z. Benbouziane, “Third order differential equations with integral boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1736-e1743, 2009. · Zbl 1238.34031 · doi:10.1016/j.na.2009.02.055 [15] A. R. Davies, A. Karageorghis, and T. N. Phillips, “Spectral Glarkien methods for the primary two-point boundary-value problems in modeling viscoelastic flows, Internat,” International Journal for Numerical Methods in Engineering, vol. 26, pp. 647-662, 1988. · Zbl 0635.73091 · doi:10.1002/nme.1620260309 [16] A. Karageorghis, T. N. Phillips, and A. R. Davies, “Spectral collocation methods for the primary two-point boundary-value problems in modeling viscoelastic flows,” International Journal for Numerical Methods in Engineering, vol. 26, no. 4, pp. 805-813, 1988. · Zbl 0637.76008 · doi:10.1002/nme.1620260404 [17] G. L. Liu, “New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique,” in Proceedings of the 7th Conference on Modern Mathematics and Mechanics, Shanghai, China, 1997. [18] R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Teaneck, NJ, USA, 1986. · Zbl 0681.76121 [19] E. H. Doha, “Explicit formulae for the coefficients of integrated expansions of Jacobi polynomials and their integrals,” Integral Transforms and Special Functions, vol. 14, no. 1, pp. 69-86, 2003. · Zbl 1051.33004 · doi:10.1080/10652460304541 [20] E. H. Doha and A. H. Bhrawy, “A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations,” Numerical Methods for Partial Differential Equations, vol. 25, no. 3, pp. 712-739, 2009. · Zbl 1170.65099 · doi:10.1002/num.20369 [21] B.-Y. Guo and L.-l. Wang, “Jacobi interpolation approximations and their applications to singular differential equations,” Advances in Computational Mathematics, vol. 14, no. 3, pp. 227-276, 2001. · Zbl 0984.41004 · doi:10.1023/A:1016681018268 [22] 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 [23] 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 [24] B.-Y. Guo and L.-l. Wang, “Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces,” Journal of Approximation Theory, vol. 128, no. 1, pp. 1-41, 2004. · Zbl 1057.41003 · doi:10.1016/j.jat.2004.03.008 [25] G. N. Watson, “A note on generalized hypergeometric series,” Proceedings London Mathematical Society, vol. 23, pp. 13-15, 1925. · JFM 51.0283.04 [26] Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, NY, USA, 1975. · Zbl 0318.33001 [27] E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1820-1832, 2011. · Zbl 1219.65077 · doi:10.1016/j.mcm.2011.01.002 [28] 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 [29] 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 [30] X. Lv and M. Cui, “An efficient computational method for linear fifth-order two-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1551-1558, 2010. · Zbl 1191.65103 · doi:10.1016/j.cam.2010.02.036 [31] H. N. \cCaglar, S. H. \cCaglar, and E. H. Twizell, “The numerical solution of fifth-order boundary value problems with sixth-degree -spline functions,” Applied Mathematics Letters, vol. 12, no. 5, pp. 25-30, 1999. · Zbl 0941.65073 · doi:10.1016/S0893-9659(99)00052-X [32] 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
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