×

zbMATH — the first resource for mathematics

A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations. (English) Zbl 1302.65175
Summary: This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational-Gauss collocation points. The proposed Jacobi rational-Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line.

MSC:
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ajello, W. G.; Freedman, H. I.; Wu, J., A model of stage structured population growth with density depended time delay, SIAM J. Appl. Math., 52, 855-869, (1992) · Zbl 0760.92018
[2] Alomari, A. K.; Noorani, M. S.M.; Nazar, R., Solution of delay differential equation by means of homotopy analysis method, Acta Appl. Math., 108, 395-412, (2009) · Zbl 1187.34081
[3] Askey, R., Orthogonal polynomials and special functions, Reg. Conf. Appl. Math., 21, 1258-1288, (1975)
[4] Bergh, J.; Löfström, J., Interpolation spaces, an introduction, (1976), Springer-Verlag Berlin · Zbl 0344.46071
[5] Bhrawy, A. H.; Alofi, A. S., A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations, Commun. Nonlinear Sci. Numer. Simul., 17, 62-70, (2012) · Zbl 1244.65099
[6] Boyd, J. P., Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys., 70, 63-88, (1987) · Zbl 0614.42013
[7] Boyd, J. P., Chebyshev and Fourier spectral methods, (2000), Dover New York
[8] Boyd, J. P.; Rangan, C.; Bucksbaum, P. H., Pseudospectral methods on a semi-infinite interval with application to the hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions, J. Comput. Phys., 188, 56-74, (2003) · Zbl 1028.65086
[9] Buhmann, M. D.; Iserles, A., Stability of the discretized pantograph differential equation, Math. Comp., 60, 575-589, (1993) · Zbl 0774.34057
[10] Burnett, D. S., A three-dimensional acoustic infinite element based an a prolate spheroidal multipole expansion, J. Acoust. Soc. Amer., 96, 2798-2816, (1994)
[11] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods: fundamentals in single domains, (2006), Springer-Verlag New York · Zbl 1093.76002
[12] Demkowicz, L.; Gerdes, K., Convergence of the infinite element methods for the Helmholtz equation in separable domains, Numer. Math., 79, 11-42, (1998) · Zbl 0898.65067
[13] Derfel, G.; Iserles, A., The pantograph equation in the complex plane, J. Math. Anal. Appl., 213, 117-132, (1997) · Zbl 0891.34072
[14] Doha, E. H.; Bhrawy, A. H., Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math., 58, 1224-1244, (2008) · Zbl 1152.65112
[15] Doha, E. H.; Bhrawy, A. H., An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method, Comput. Math. Appl., 64, 558-571, (2012) · Zbl 1252.65194
[16] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., On shifted Jacobi spectral method for high-order multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 17, 3802-3810, (2012) · Zbl 1251.65112
[17] Evans, D. J.; Raslan, K. R., The adomain decomposition method for solving delay differential equation, Int. J. Comput. Math., 82, 49-54, (2005) · Zbl 1069.65074
[18] Feldstein, A.; Liu, Y., On neutral functional differential equations with variable time delays, Math. Proc. Cambridge Philos. Soc., 124, 371-384, (1998) · Zbl 0913.34067
[19] Funaro, D., Computational aspects of pseudospectral Laguerre approximations, Appl. Numer. Math., 6, 447-457, (1990) · Zbl 0708.65072
[20] Guo, B. Y., Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl., 243, 373-408, (2000) · Zbl 0951.41006
[21] Guo, B. Y.; Shen, J.; Wang, Z. Q., A rational approximation and its applications to differential equations on the half line, J. Sci. Comput., 15, 117-147, (2000) · Zbl 0984.65104
[22] Guo, B. Y.; Shen, J.; Wang, Z. Q., Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Int. J. Numer. Methods Eng., 53, 65-84, (2002) · Zbl 1001.65129
[23] Guo, B. Y.; Wang, Z. Q.; Wan, Z.; Chu, D., Second order Jacobi approximation with applications to fourth-order differential equations, Appl. Numer. Math., 55, 480-502, (2005) · Zbl 1081.65080
[24] Leis, R., Initial boundary value problems in mathematical physics, (1996), Teubner Stuttgurt · Zbl 0414.73082
[25] Li, D.; Liu, M. Z., Runge-Kutta methods for the multi-pantograph delay equation, Appl. Math. Comput., 163, 383-395, (2005) · Zbl 1070.65060
[26] Li, X. Y.; Wu, B. Y., A continuous method for nonlocal functional differential equations with delayed or advanced arguments, J. Math. Anal. Appl., 409, 485-493, (2014) · Zbl 1306.65225
[27] Liu, M. Z.; Li, D., Properties of analytic solution and numerical solution and multi-pantograph equation, Appl. Math. Comput., 155, 853-871, (2004) · Zbl 1059.65060
[28] Muroya, Y.; Ishiwata, E.; Brunner, H., On the attainable order of collocation methods for pantograph integro-differential equations, J. Comput. Appl. Math., 152, 347-366, (2003) · Zbl 1023.65146
[29] Ockendon, J. R.; Tayler, A. B., The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 322, 447-468, (1971)
[30] Ozturk, Y.; Gulsu, M., Approximate solution of linear generalized pantograph equations with variable coefficients on Chebyshev-Gauss grid, J. Adv. Res. Sci. Comput., 4, 36-51, (2012)
[31] Parand, K.; Razzaghi, M., Rational Chebyshev tau method for solving higher-order ordinary differential equations, Int. J. Comput. Math., 81, 73-80, (2004) · Zbl 1047.65052
[32] Parand, K.; Razzaghi, M., Rational Legendre approximation for solving some physical problems on semi-infinite intervals, Phys. Scr., 64, 353-357, (2004) · Zbl 1063.65146
[33] Saadatmandi, A.; Dehghan, M., Variational iteration method for solving a generalized pantograph equation, Comput. Math. Appl., 58, 2190-2196, (2009) · Zbl 1189.65172
[34] Sezer, M., A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials, Internat. J. Math. Ed. Sci. Tech., 27, 821-834, (1996) · Zbl 0887.65084
[35] Sezer, M.; Akyuz-Das-cioglu, A., A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, J. Comput. Appl. Math., 200, 217-225, (2007) · Zbl 1112.34063
[36] Sezer, M.; Yalcinbas, S.; Gulsu, M., A Taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term, Int. J. Comput. Math., 85, 1055-1063, (2008) · Zbl 1145.65048
[37] Sezer, M.; Yalcinbas, S.; Sahin, N., Approximate solution of multi-pantograph equation with variable coefficients, J. Comput. Appl. Math., 214, 406-416, (2008) · Zbl 1135.65345
[38] Shakeri, F.; Dehghan, M., Solution of the delay differential equations via homotopy perturbation method, Math. Comput. Modelling, 48, 486-498, (2008) · Zbl 1145.34353
[39] Tohidi, E.; Bhrawy, A. H.; Erfani, K., A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model., 37, 4283-4294, (2013) · Zbl 1273.34082
[40] Wang, Z. Q.; Guo, B. Y., Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comp., 77, 181-199, (2008)
[41] Yalinbas, S.; Aynigul, M.; Sezer, M., A collocation method using Hermite polynomials for approximate solution of pantograph equations, J. Franklin Inst., 348, 1128-1139, (2011) · Zbl 1221.65187
[42] Yi, Y. G.; Guo, B. Y., Generalized Jacobi rational spectral method on the half line, Adv. Comput. Math., 37, 1-37, (2012) · Zbl 1259.65156
[43] Yu, Z. H., Variational iteration method for solving the multi-pantograph delay equation, Phys. Lett. A, 372, 6475-6479, (2008) · Zbl 1225.34024
[44] Yuzbasi, S.; Sahin, N.; Sezer, M., A Bessel collocation method for numerical solution of generalized pantograph equations, Numer. Methods Partial Differential Equations, 28, 4, 1105-1123, (2012) · Zbl 1257.65035
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.