A numerical approach with residual error estimation for solution of high-order linear differential-difference equations by using Gegenbauer polynomials

The main aim of this study is to apply the Gegenbauer polynomials for the solution of high-order lineardifferential difference equations with functional arguments under initial-boundary conditions.The technique wehave used is essentially based on the truncated Gegenbauer series and its matrix representations along withcollocation points. Also, by using the Mean-Value Theorem and residual function, an efficient error estimationtechnique is proposed and some illustrative examples are presented to demonstrate the validity and applicabilityof the method.

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[1] Iserles, A. On Neutral FunctionalDifferential Equations with Proportional Delays. J. Math. Anal. Appl. 1997; 207, 73-95.
[2] Wang, W.S.; Li, S. F. On the one-leg Qmethods for solving nonlinear neutral functional differential equations. Appl. Math. Comput. 2007; 193(1), 285-301.
[3] Kuang, Y.; Feldstein, A. Monotonic and Oscillatory solutions of a linear neutral delay equations with infinite lag, SIAM J. Math. Anal. 1990; 21(6), 1633-1641.
[4] Cahlon, B.; Schmidt, D. Stability criteria for certain high even order delay differential equations. J. Math. Anal. Appl. 2007; 334, 859- 875.
[5] El-Khatib, M.A. Convergenue of the spline function for functional differential equation of neutral type. Intern. J. Comput. Math. 2003; 80(11), 1437-1447.
[6] Rashed, M.T. Numerical solution of functional differential,integral and integrodifferential equations. Appl. Math. Comput. 2004; 156, 485-492.
[7] Cheng, K.; Chen, Z.; Zhang, Q. An approximate solution for a neutral functional differential equation with proportional delays. Appl. Math. Comput. 2015; 260, 27-34.
[8] Bhrawy, A.H.; Assas, L.M.; Tohidi, E.; Alghamdi, M.A. A Legendre-Gauss collocation method for neutral functional differential equations with proportional delays. Advances in Difference Equations. 2013; 2013, 63.
[9] Heydari, M.; Loghmani, G.B.; Hosseini, S.M. Operational matrices of Chebyshev cardinal functions and their aplication for solving delay differential equations arising in elektrodynamics with error estimation. Appl. Math. Model. 2013; 37, 7789-7809.
[10] Işik, O.R.; Güney, Z.; Sezer, M. Bernstein series solutions of pantograph equations using polynomial interpolation. Journal of Difference Equations and Applications. 2012; 18(3), 357- 374.
[11] Gülsu, M.; Öztürk, Y.; Sezer, M. A new Chebyshev polnomial approximation for solving delay differential equations. Journal of Difference Equations and Applications. 2012; 18(6), 1043-1065.
[12] Gürbüz, B.; Sezer, M.; Güler, C. Laguerre collocation method for solving Fredholm integro- differential equations with functional arguments. J. Appl. Math. 2014; (2014) Article ID 682398,12 pages.
[13] Yüzbaşı, Ş.; Gök, E.; Sezer, M. Laguerre matrix method with the residual error estimation for solutions of a class of delay differential equations. Math. Meth. Appl. Sci. 2014; 37(4), 453-463.
[14] Şahin, N.; Yüzbaşı, Ş.; Sezer, M. A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations.
[15] Sezer, M.; Gülsu, M. Solving high-order linear differential equations by a Legendre matrix method based on hybrid Legendre and Taylor polynomials. Numerical Methods for Partial Differential Equations. 2010; 26, 647- 661.
[16] Gülsu, M.; Sezer, M. A Taylor collocation method for solving high-order linear pantograph equations with linear functional argument. Numerical Methods for Partial Differential Equations. 2011; 27, 1628-1638.
[17] Gülsu, M.; Sezer, M. A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials. International Journal of Computer Mathematics. 2004; 82(5), 629-642.
[18] Kim, D.S.; Kim, T.; Rim, S.H. Some identities involving Gegenbauer polynomials. Advance in Difference Equations. 2012; 2012: 219.
[19] Khan, S.; Al-Gonah, A.A.; Yasmin, G. Generalized and mixed type Gegenbauer polynomials. J. Math. Anal. Appl. 2012; 390, 197-207.
[20] Lewanowicz, S. Results on the associated Jacobi and Gegenbauer polynomials. Journal of Computational and Applied Mathematics. 1993; 49, 137-143.
[21] Kürkçü, Ö.K.; Aslan, E.; Sezer, M. A numerical approach with error estimation to solve general integro-differential–difference equations using Dickson polynomials. Applied Mathematics and Computation. 2016; 276, 324- 339.