A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. (English) Zbl 1273.34082

Summary: This paper presents a direct solution technique for solving the generalized pantograph equation with variable coefficients subject to initial conditions, using a collocation method based on Bernoulli operational matrix of derivatives. Only a small dimension of the Bernoulli operational matrix is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for the generalized pantograph equation.


34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L03 Numerical methods for functional-differential equations
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[1] Ockendon, J. R.; Tayler, A. B., The dynamics of a current collection system for an electric locomotive Proc, Roy. Soc. London A, 322, 447-468 (1971)
[2] 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
[3] Buhmann, M. D.; Iserles, A., Stability of the discretized pantograph differential equation, Math. Comput., 60, 575-589 (1993) · Zbl 0774.34057
[4] Fox, L.; Mayers, D. F.; Ockendon, J. A.; Tayler, A. B., On a functional differential equation, J. Inst. Math. Appl., 8, 271-307 (1971) · Zbl 0251.34045
[5] Derfel, G. A.; Vogl, F., On the asymptotics of solutions of a class of linear functional-differential equations, Eur. J. Appl. Math., 7, 511-518 (1996) · Zbl 0859.34049
[6] 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
[7] Sezer, M., A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Educ. Sci. Technol., 27, 821-834 (1996) · Zbl 0887.65084
[8] Sezer, M.; Gulsu, M., A new polynomial approach for solving difference and Fredholm integro-diffrential equation with mixed argument, Appl. Math. Comput., 171, 332-344 (2005) · Zbl 1084.65133
[9] 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
[10] Shakeri, F.; Dehghan, M., Solution of the delay differential equations via homotopy perturbation method, Math. Comput. Model., 48, 486-498 (2008) · Zbl 1145.34353
[11] Sezer, M.; Yalcinbas, S.; Sahin, N., Approximate solution of multi-pantograph equation with variable coeffcients, J. Comput. Appl. Math., 214, 406-416 (2008) · Zbl 1135.65345
[12] Li, D.; Liu, M. Z., Runge-Kutta methods for the multi-pantograph delay equation, App. Math. Comput., 163, 383-395 (2005) · Zbl 1070.65060
[13] Yu, Z. H., Variational iteration method for solving the multi-pantograph delay equation, Phys. Lett. A, 372, 6475-6479 (2008) · Zbl 1225.34024
[14] Saadatmandi, A.; Dehghan, M., Variational iteration method for solving a generalized pantograph equation, Comput. Math. Appl., 58, 2190-2196 (2009) · Zbl 1189.65172
[15] Yuzbasi, S.; Sahin, N.; Sezer, M., A Bessel collocation method for numerical solution of generalized pantograph equations, Numer. Methods. Partial. Differ. Equ. (2012) · Zbl 1257.65035
[16] 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
[17] Koblitz, N., p-Adic Numbers, p-Adic Analysis and Zeta-Functions (1984), Springer-Verlag: Springer-Verlag New York
[18] Lang, S., Introduction to Modular Forms (1976), Springer-Verlag
[19] Boas, R. P.; Buck, R. C., Polynomial Expansions of Analytic Functions (1964), Springer-Verlag: Springer-Verlag New York · Zbl 0116.28105
[20] Fairlie, D. B.; Veselov, A. P., Faulhaber and Bernoulli polynomials and solitons, Phys. D, 152-153, 47-50 (2001) · Zbl 0984.37083
[22] Doyon, B.; Lepowsky, J.; Milas, A., Twisted vertex operators and Bernoulli polynomials, Commun. Contemp. Math., 8, 247-307 (2006) · Zbl 1119.17011
[23] Derfel, G.; Iserles, A., The pantograph equation in the complex plane, J. Math. Anal. Appl., 213, 117-132 (1997) · Zbl 0891.34072
[24] 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
[25] 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
[26] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59, 1326-1336 (2010) · Zbl 1189.65151
[27] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35, 5662-5672 (2011) · Zbl 1228.65126
[28] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62, 2364-2373 (2011) · Zbl 1231.65126
[29] Doha, E. H.; Bhrawy, A. H.; Ezzeldeen, S. S., A new jacobi operational matrix: an application for solving fractional differential equation, Appl. Math. Model., 36, 4931-4943 (2012) · Zbl 1252.34019
[31] Bhrawy, A. H.; Alofi, A. S.; Ezzeldeen, S. S., A quadrature tau method for fractional differential equations with variable coefficients, Appl. Math. Lett., 24, 2146-2152 (2011) · Zbl 1269.65068
[32] Bhrawy, A. H.; Alofi, A. S., The operational matrix of fractional integration for shifted Chebyshev polynomials, Appl. Math. Lett., 26, 25-31 (2013) · Zbl 1255.65147
[33] Srivastava, H. M.; Choi, J., Series Associated with the Zeta and Related Functions (2001), Kluwer Academic: Kluwer Academic Dordreeht · Zbl 1014.33001
[34] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions, vol. III (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0064.06302
[35] Luke, Y. L., The Special Functions and Their Approximations, vol. I (1969), Academic Press: Academic Press New York
[36] Magnus, W.; Oberhettinger, F.; Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged Edition (1966), Springer-Verlag: Springer-Verlag New York
[37] Mashayekhi, S.; Ordokhani, Y.; Razzaghi, M., Hybrid functions approach for nonlinear constrained optimal control problems, Commun. Nonlinear. Sci. Numer. Simulat., 17, 1831-1843 (2012) · Zbl 1239.49043
[38] Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables (1972), National Bureau of Standards: National Bureau of Standards Wiley, New York · Zbl 0543.33001
[39] Rao, G. P.; Palanisamy, K. R., Walsh stretch matrices and functional differential equation, IEEE Trans. Autom. Control, 27, 272-276 (1982) · Zbl 0488.34060
[40] Hwang, C.; Shih, Y. P., Laguerre series solution of a functional differential equation, Int. J. Syst. Sci., 13, 7, 783-788 (1982) · Zbl 0483.93056
[41] 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
[42] 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)
[44] Reid, W. T., Riccati Differential Equations (1972), Academic Press: Academic Press New York · Zbl 0254.34003
[45] Dehghan, M.; Taleei, A., A compact split-step finite difference method for solving the nonlinear Schrdinger equations with constant and variable coefficients, Comput. Phys. Commun., 181, 43-51 (2010) · Zbl 1206.65207
[46] Carinena, J. F.; Marmo, G.; Perelomov, A. M.; Ranada, M. F., Related operators and exact solutions of Schrdinger equations, Int. J. Modern Phys. A, 13, 4913-4929 (1998) · Zbl 0927.34065
[47] Scott, M. R., Invariant Imbedding and Its Applications to Ordinary Differential Equations (1973), Addison-Wesley
[48] Adomian, G., Solving Frontier Problems of Fhysics: Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecth · Zbl 0802.65122
[49] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput., 172, 485-490 (2006) · Zbl 1088.65063
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