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An efficient algorithm for solving multi-pantograph equation systems. (English) Zbl 1252.65136
Summary: We present a numerical approach for solving the system of multi-pantograph equations with mixed conditions. This system is usually difficult to solve analytically. By expanding the approximate solutions by means of the Bessel functions of first kind with unknown coefficients, the proposed approach consists of reducing the problem to a linear algebraic equation system. The unknown coefficients of the Bessel functions of first kind are computed using the matrix operations of derivatives together with the collocation method. An error estimation is given. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. All of the numerical computations have been performed on a computer with the aid of a program written in Matlab.

##### MSC:
 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 33C10 Bessel and Airy functions, cylinder functions, ${}_0F_1$
Matlab
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##### References:
 [1] Yu, Z. -H.: Variational iteration method for solving the multi-pantograph delay equation, Phys. lett. A 372, 6475-6479 (2008) · Zbl 1225.34024 · doi:10.1016/j.physleta.2008.09.013 [2] Liu, M. Z.; Li, D.: Properties of analytic solution and numerical solution of multi-pantograph equation, Appl. math. Comput. 155, 853-871 (2004) · Zbl 1059.65060 · doi:10.1016/j.amc.2003.07.017 [3] Sezer, M.; Yalçinbaş, S.; Şahin, N.: Approximate solution of multi-pantograph equation with variable coefficients, J. comput. Appl. math. 214, 406-416 (2008) · Zbl 1135.65345 · doi:10.1016/j.cam.2007.03.024 [4] Du, P.; Geng, F.: A new method of solving singular multi-pantograph delay differential equation in reproducing kernel space, Appl. math. Sci. 2, No. 27, 1299-1305 (2008) · Zbl 1162.34346 · http://www.m-hikari.com/ams/ams-password-2008/ams-password25-28-2008/dupingAMS25-28-2008.pdf [5] Evans, D. J.; Raslan, K. R.: The Adomian decomposition method for solving delay differential equation, Int. J. Comput. math. 82, No. 1, 49-54 (2005) · Zbl 1069.65074 · doi:10.1080/00207160412331286815 [6] Yusufoğlu, Elçin: An efficient algorithm for solving generalized pantograph equations with linear functional argument, Appl. math. Comput. 217, No. 7, 3591-3595 (2010) · Zbl 1204.65083 · doi:10.1016/j.amc.2010.09.005 [7] Sezer, M.; Yalçinbaş, S.; Gülsu, M.: A Taylor polynomial approach for solving generalized pantograph equations with nonhomogeneous term, Int. J. Comput. math. 85, No. 7, 1055-1063 (2008) · Zbl 1145.65048 · doi:10.1080/00207160701466784 [8] Saadatmandi, A.; Dehghan, M.: Variational iteration method for solving a generalized pantograph equation, Comput. math. Appl. 58, No. 11--12, 2190-2196 (2009) · Zbl 1189.65172 · doi:10.1016/j.camwa.2009.03.017 [9] Ş. Yüzbaşı, N. Şahin, M. Sezer, A Bessel collocation method for numerical solution of generalized pantograph equations, Numer. Methods Partial Differential Equations (2011), (doi:10.1002/num.20660) (in press). [10] Mikaeilvand, N.; Hossieni, L.: The Taylor method for numerical solution of fuzzy generalized pantograph equations with linear functional argument, Int. J. Ind. math. 2, No. 2, 115-127 (2010) [11] Sezer, M.; Akyüz-Daşcıoğlu, 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 · doi:10.1016/j.cam.2005.12.015 [12] Brunner, H.; Huang, Q.; Xie, H.: Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. Numer. anal. 48, 1944-1967 (2010) · Zbl 1219.65076 · doi:10.1137/090771922 [13] Yüzbaşı, Ş.; Şahin, N.; Sezer, M.: Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases, Comput. math. Appl. 61, No. 10, 3079-3096 (2011) · Zbl 1222.65154 · doi:10.1016/j.camwa.2011.03.097 [14] Yüzbaşı, Ş.; Şahin, N.; Sezer, M.: Bessel matrix method for solving high-order linear Fredholm integro-differential equations, J. adv. Res. appl. Math. 3, No. 2, 23-47 (2011) [15] Şahin, N.; Yüzbaşı, Ş.; Gülsu, M.: A collocation approach for solving systems of linear Volterra integral equations with variable coefficients, Comput. math. Appl. 62, No. 2, 755-769 (2011) · Zbl 1228.65248 · doi:10.1016/j.camwa.2011.05.057 [16] Yüzbaşı, Ş.; Sezer, M.: A collocation approach to solve a class of Lane--Emden type equations, J. adv. Res. appl. Math. 3, No. 2, 58-73 (2011) [17] Yüzbaşı, Ş.; Şahin, N.; Sezer, M.: A Bessel polynomial approach for solving linear neutral delay differential equations with variable coefficients, J. adv. Res. differ. Equ. 3, No. 1, 81-101 (2011) [18] Watson, G. N.: A treatise on the theory of Bessel functions, (1966) · Zbl 0174.36202 [19] Arfken, G. B.; Weber, H. J.; Harris, F. E.: Mathematical methods for physicists, (2005) · Zbl 1066.00001 [20] Shahmorad, S.: Numerical solution of the general form linear Fredholm--Volterra integro-differential equations by the tau method with an error estimation, Appl. math. Comput. 167, 1418-1429 (2005) · Zbl 1082.65602 · doi:10.1016/j.amc.2004.08.045