Pell-Lucas Collocation Method for Solving High-Order Functional Differential Equations with Hybrid Delays

Abstract

In this study, the Pell-Lucas collocation method has been presented to solve high-order linear functional differential equations with hybrid delays under mixed conditions. The proposed method is based on the matrix forms of Pell-Lucas polynomials and their derivatives, along with the collocation points. The used technique reduces the problem to a matrix equation corresponding to a set of algebraic equations with the unknown Pell-Lucas coefficients. In addition, an error analysis based on residual function is performed and some numerical examples are presented to show the efficiency and accuracy of the method.

Keywords

• Pell-Lucas polynomials,collocation method,functional differential-equations,residual error analysis,matrix method

References

1. 1. Reutskiy, S. Yu. A new collocation method for approximate solution of the pantograph functional differential equations with proportional delay, Applied Mathematics and Computation, 2015, 266, 642-655.
2. 2. El-Khatib, M. A. Convergenue of the spline function for functional differential equation of neutral type, International J0urnal of Computer Mathematics. 2003, 80(11), 1437-1447.
3. 3. Rashed, M. T. Numerical solution of functional differential, integral and integro- differential equations, Applied Mathematics and Computation, 2004, 156, 485-492.
4. 4. 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. 5. Heydari, M. Loghmani, G. B. Hosseini, S. M. Operational matrices of Chebyshev cardinal functions and their aplication for solving delay differential equations arising in electrodynamics with error estimation, Applied Mathematical.Modeling, 2013, 37, 7789-7809.
5. 6. Sedaghat, S. Ordokhani, Y. Dehghan, M. Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials, Common Nonlinear Science Numerical Simulation, 2012, 17, 4815-4830.
6. 7. Akyüz, A. Sezer, A Chebyshev Collocation method for the solution of linear integro- differential equations, International Journal of Computer Mathematics, 1999, 72 (4) 491-507.
7. 8. Gürbüz, B. Gülsu, M. Sezer, M. Numerical approach of high-order linear delay-difference equations with variable coefficients in terms of Laguerre polynomials, Mathematical and Computational Applications, 2011, 16, 267-278.
8. 9. Wang, W. S. Li, S. F. On the one-leg Q-methods for solving nonlinear neutral functional differential equations, Applied Mathematics and Computation, 2007, 193 (1),285-301.

9. 10. Cheng, X. Chen, Z. Zhang, Q. An approximate solution for a neutral functional- differential equation with proportional delays, Applied Mathematics and Computation, 2015, 260 27-34.
10. 11. Kürkçü, Ö.K. Aslan, E. Sezer, M. A Novel Collocation Method Based on Residual Error Analysis for Solving Integro-Differential Equations Using Hybrid Dickson and Taylor Polynomials, Sains Malaysiana, 2017, 46, 2335–347.
11. 12. Dai, C. Zhang, J. Jacobian elliptic function method for nonlinear differential difference equations, Chaos, Solitons & Fractals, 2006, 27, 1042-1047.
12. 13. Çelik, İ. Collocation method and residual correction using Chebyshev series, Applied Mathematics and Computation, 2006, 174, 910–920.
13. 14. Wei, Y. Chen, Y. Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation, Applied Numeric Mathematics, 2014, 81, 15–29.
14. 15. Wang, K. Wang, Q. Lagrange collocation method for solving Volterra–Fredholm integral equations, Applied Mathematics and Computation, 2013, 219 10434–10440.
15. 16. Sezer, M. Daşçıoğlu, A. Taylor polynomial solutions of general linear differantial-difference equations with variable coefficients. Applied Mathematics and Computation, 2006, 174, 1526-1538.
16. 17. Sezer, M., Taylor polynomial solution of Volterra integral equations, International Journal of Mathematical Education in Science and Technology, 1994, 25, 625–633.
17. 18. Sezer, M. and Kaynak, M. Chebyshev polynomial solutions of linear differential equations, International Journal of Mathematical Education in Science and Technology, 1996, 27, 607–618.
18. 19. A.F.Horadam and J.M. Mahon, Pell and Pell-Lucas Polynomials, Fibonacci Quartery, 23(1), 17-20, (1985).
19. 20. P. Filipponi, A. F. Horadam, Second derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quarterly, 31(3) (1993), 194-204.