LUCAS POLYNOMIAL SOLUTION FOR NEUTRAL DIFFERENTIAL EQUATIONS WITH PROPORTIONAL DELAYS

Year 2020, Volume: 10 Issue: 1, 259 - 269, 01.01.2020

S. Gumgum N. B. Savasaneril O. K. Kurkcu M. Sezer

Abstract

This paper proposes a combined operational matrix approach based on Lucas and Taylor polynomials for the solution of neutral type dierential equations with proportional delays. The advantage of the proposed method is the ease of its application. The method facilitates the solution of the given problem by reducing it to a matrix equation. Illustrative examples are validated by means of absolute errors. Residual error estimation is presented to improve the solutions. Presented in graphs and tables the results are compared with the existing methods in literature.

Keywords

Neutral dierential equations, Lucas and Taylor polynomials, collocation and matrix methods.

References

• Abolhasani, M., Ghaneai, H. and Heydari, M., (2010), Modified homotopy perturbation method for solving delay differential equations, Appl. Sci. Reports, 16(2), pp. 89–92.
• Bayku¸s-Sava¸saneril, N. and Sezer, M., (2017), Hybrid Taylor-Lucas Collocation Method for Numerical Solution of High-Order Pantograph Type Delay Differential Equations with Variables Delays, Appl. Math. Inf. Sci., 11(6), pp. 1795–1801.
• Bellen, A. and Zennaro, M., (2003), Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, NY, USA.
• Bhrawy, A. H., Assas, L. M., Tohidi, E. and Alghamdi, M. A., (2013), A Legendre-Gauss colloca- tion method for neutral functional-differential equations with proportional delays, Adv. Differ. Eq., 2013(63), 16 pages.
• Bhrawy, A. H., Alghamdi, M. A. and Baleanu, D., (2013), Numerical Solution of a Class of Functional- Differential Equations Using Jacobi Pseudospectral Method, Abstr. Appl. Anal., 2013, Article ID 513808, 9 pages.
• Biazar, J. and Ghanbari, B., (2012), The homotopy perturbation method for solving neutral functional-differential equations with proportional delays, J. King Saud Uni. Sci., 24, pp. 33–37.
• Cˇaruntu, B. and Bota, C., (2014), Analytical Approximate Solutions for a General Class of Nonlinear Delay Differential Equations, Sci. World J., 2014, Article ID 631416, 6 pages.
• Chen, X. and Wang, L., (2010), The variational iteration method for solving a neutral functional- differential equation with proportional delays, Comput. Math. Appl., 59, pp. 2696–2702.
• Cheng, X., Chen, Z. and Zhang, Q., (2015), An approximate solution for a neutral functional- differential equation with proportional delays, Appl. Math. Comput., 260, pp. 27–34.
• C¸ etin, M., Sezer, M. and G¨uler, C., (2015), Lucas polynomial approach for system of high-order linear differential equations and residual error estimation, Math. Prob. Eng., 2015, 14 pages.
• Ghaneai, H., Hosseini, M. M. and Mohyud-Din, S. T., (2012), Modified variational iteration method for solving a neutral functional-differential equation with proportional delays, Int. J. Numer. Meth. H., 22(8), pp. 1086–1095.
• Ghomanjani, F. and Farahi, M. H., (2012), The Bezier Control Points Method for Solving Delay Differential Equation, Intell. Control Automation, 3, pp. 188–196.
• G¨umg¨um, S., Bayku¸s Sava¸saneril, N., K¨urk¸c¨u, ¨O. K. and Sezer, M., (2018), A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays, Sakarya Uni. J. Sci., 22(6), 10 pages.
• Ibis, B. and Bayram, M., (2016), Numerical solution of the neutral functional-differential equations with proportional delays via collocation method based on Hermite polynomials, Commun. Math. Model. Appl., 1(3), pp. 22–30.
• K¨urk¸c¨u, ¨O. K., Aslan, E., Sezer, M. and ˙Ilhan, ¨O., (2018), A numerical approach technique for solving generalized delay integro-differential equations with functional bounds by means of Dickson polynomials, Int. J. Comput. Methods, 15(5), No: 1850039, 24 pages.
• Lucas, E., (1878), Theorie de fonctions numeriques simplement periodiques, Amer. J. Math., 1, pp. 184–240; 289–321.
• Lv, X. and Gao, Y., (2013), The RKHSM for solving neutral functional-differential equations with proportional delays, Math. Meth. Appl. Sci., 36, pp. 642–649.
• Rebenda, J., ˇSmarda, Z. and Khan, Y., (2015), A Taylor Method Approach for Solving of Nonlinear Systems of Functional Differential Equations with Delay, arXiv:1506.0564v1,[math.CA].
• Sakar, M. G., (2017), Numerical solution of neutral functional-differential equations with proportional delays, Int. J. Opt. Control: Theo. Appl., 7(2), pp. 186–194.
• Wang, W. and Li, S., (2007), On the one-leg-methods for solving nonlinear neutral functional differ- ential equations, Appl. Math. Comput. 193(1), pp. 285–301.
• Y¨uzba¸sı, S¸. and Sezer, M., (2015), Shifted Legendre approximation with the residual correction to solve pantograph-delay type differential equations, Appl. Math. Model., 39, pp. 6529–6542.