A new collocation method based on Euler polynomials for solution of generalized pantograph equations

Yıl 2016, Cilt: 4 Sayı: 4, 285 - 294, 31.12.2016

Birol Ibis Mustafa Bayram

Öz

In this paper, a new collocation method based on Euler
polynomials is improved for the numerical solution of generalized pantograph
equations. This method transforms the generalized pantograph equations into the
matrix equation with the help of Euler polynomials and collocation points. This
matrix equation corresponds to a system of linear algebraic equations with the
unknown Euler coefficients. By solving this system, the unknown Euler coefficients
of the solution are found. Some numerical examples are given and comparisons
with other methods are made in order to demonstrate the applicability and
validity of the proposed method.

Anahtar Kelimeler

Euler collocation method (ECM), collocation points, Euler polynomial

Kaynakça

• J. R. Ockendon and A. B. Tayler. The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. London A, 332 ,447-468(1971).
• W.G. Ajello, H.I. Freedman and J. Wu, A model of stage structured population growth with density depended time delay, SIAMJ.Appl.Math., 52 ,855-869(1992).
• Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic, New York, NY, USA, 1993.
• D. Li and M. Liu, Runge-Kutta methods for the multi-pantograph delay equation, Appl. Math. Comput.,163(1),383-395(2005).
• M. Liu, Z. Yang and Y. Xu, The stability of modified Runge-Kutta methods for the pantograph equation, Math. Comput., 75, 1201-1216(2006).
• Y. Keskin, A. Kurnaz, M. Kiris and G. Oturanc, Approximate solutions of generalized pantograph equations by the differential transform method, Int. J. Nonlinear Sci. Numer. Simul., 8, 159-164(2007).
• M. Sezer, S.Yalcinbas and N. Sahin, Approximate solution of multi-pantograph equation with variable coefficients, J. Comput. Appl. Math., 214 ,406-416(2008).
• Z. Yu, Variational iteration method for solving the multi-pantograph delay equation, Phys.Lett. A, 372, 6475-6479(2008).
• A. Saadatmandi and M. Dehghan, Variational iteration method for solving a generalized pantograph equation, Comput.Math.Appl., 58(11-12) ,2190-2196(2009).
• X. Feng, An analytic study on the multi-pantograph delay equations with variable coefficients, Bull. Math. Soc. Sci. Math. Roumanie, 56(2) , 205-215(2013).
• C. Bota and B. Caruntu, ϵ-Approximate polynomial solutions for the multi-pantograph equation with variable coefficients, Appl. Math. Comput.,219(4) ,1785-1792(2012).
• S. Sedaghat, Y. Ordokhani and M. Dehghan,Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials, Commun. Nonlinear Sci., 17(12) , 4815-4830(2012).
• S. Yuzbasi, An efficient algorithm for solving multipantograph equation systems, Comput. Math. Appl., 64(4),589–603(2012).
• S. Yalcinbas, M. Aynigul and M. Sezer, A collocation method using Hermite polynomials for approximate solution of pantograph equations, J. Frankl. Ins., 348(6),1128-1139(2011).
• O.R. Isik, Z. Guney and M. Sezer, Bernstein series solutions of pantograph equations using polynomial interpolation, J. Differ. Eq. Appl. ,18(3),357-374(2012).
• Y. Ozturk, M. Gulsu, Approximate solution of linear generalized pantograph equations with variable coefficients on Chebyshev-Gauss grid, J. Adv. Res.Sci. Comput.,4,36-51(2012).
• T. Akkaya, S. Yalcinbas and M. Sezer, Numeric solutions for the pantograph type delay differential equation using First Boubaker polynomials, Appl. Math. Comput., 219(17),9484-9492(2013).
• E. Tohidi, A.H. Bhrawy and K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation , Appl. Math. Model., 37(6),4283-4294(2013).
• E.H. Doha, A.H. Bhrawy, D. Baleanu and R.M. Hafez, A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations, Appl. Numer. Math.,77,43-54(2014).
• M. Gulsu, M. Sezer and Z. Guney, Approximate solution of general high-order linear nonhomogeneous difference equations by means of Taylor collocation method, Appl. Math. Comput.,173(2), 683-693(2006).
• S. Esmaeili, M. Shamsi and Y. Luchko, Numerical solution of fractional differential equations with a collocation method based on Muntz polynomials,Comput. Math. Appl.,62(3), 918-929(2011).
• M. Gulsu, B. Gurbuz, Y. Ozturk and M. Sezer, Laguerre polynomial approach for solving linear delay difference equations, Appl. Math. Comput., 217(15), 6765-6776(2011).
• A.H. Bhrawy and A.S. Alofi, A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations, Commun. Nonlinear Sci., 17(1) , 62-70(2012).
• K. Maleknejad, B. Basirat and E. Hashemizadeh, A Bernstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations,Math. Comput. Model., 55(3-4) , 1363-1372(2012).
• M. M. Khader,N. H. Sweilam and A. M. S. Mahdy, Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM, Appl. Math. Inf. Sci.,7(5),2011-2018(2013).
• E. Celik, M. Bayram, The numerical solution of physical problems modeled, as a systems of differential-algebraic equations (DAEs), Journal of The Franklin Institute-Engineering And Applied Mathematics, 342(1), 1-6 (2005).