A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays

Year 2018, Volume: 22 Issue: 6, 1659 - 1668, 01.12.2018

Sevin Gümgüm , Nurcan Baykuş Savaşaneril Ömür Kıvanç Kürkçü , Mehmet Sezer

https://doi.org/10.16984/saufenbilder.384592

Cited By: 15

https://izlik.org/JA62SU34YF

Abstract

In this paper, a new numerical matrix-collocation technique is considered
to solve functional integro-differential equations involving variable delays
under the initial conditions. This technique is based essentially on Lucas
polynomials together with standard and Chebyshev-Lobatto collocation points.
Some descriptive examples are performed to observe the practicability of the
technique and the residual error analysis is employed to improve the obtained
solutions. Also, the numerical results obtained by using these collocation
points are compared in tables and figures.

Keywords

Functional equations , Matrix technique , Lucas polynomials , Residual error analysis.

References

• [1] V.V. Vlasov and R. Perez Ortiz, “Spectral Analysis of Integro-Differential Equations in Viscoelasticity and Thermal Physics”, Math. Notes, vol. 98, no. 4, pp. 689–693, 2015. [2] M. Dehghan and F. Shakeri, “Solution of an integro-differential equation arising in oscillating magnetic field using He’s homotopy perturbation method”, Prog. Electromagnet. Res. PIER, vol. 78, pp. 361–376, 2008. [3] Y. Chu, F. You, J. M. Wassick, and A. Agarwal, “Integrated planning and scheduling under production uncertainties: Bi-level model formulation and hybrid solution method,” Computers and Chemical Engineering, vol. 72, pp. 255–272, 2015. [4] F. S. Chan, V. Kumar, and M. Tiwari, “Optimizing the Performance of an Integrated Process Planning and Scheduling Problem: An AIS-FLC based Approach,” 2006 IEEE Conference on Cybernetics and Intelligent Systems, pp. 1–8, 2006. [5] W. Tan and B. Khoshnevis, “Integration of process planning and scheduling— a review,” Journal of Intelligent Manufacturing, vol. 11, no. 1, pp. 51–63, 2000. [6] N. Kurt and M. Sezer, “Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients,” J. Franklin Inst., vol. 345, pp. 839–850, 2008. [7] N. Kurt and M. Çevik, “Polynomial solution of the single degree of freedom system by Taylor matrix method,” Mech. Res. Commun., vol. 35, pp. 530–536, 2008. [8] N. Baykuş and M. Sezer, “Solution of High-Order Linear Fredholm Integro-Differential Equations with Piecewise Intervals,” Numer. Methods Partial Differ. Equ., vol. 27, pp. 1327–1339, 2011. [9] M. Sezer and A. Akyüz-Daşcıoğlu, “A Taylor method for numerical solution of generalized pantograph equations with linear functional argument,” J. Comput. Appl. Math., vol. 200, pp. 217–225, 2007. [10] N. Baykuş Savaşaneril and M. Sezer, “Laguerre Polynomial Solution of High-Order Linear Fredholm Integro-Differential Equations”, New Trends in Math. Sci., vol. 4, no. 2, 273–284, 2016. [11] B. Gürbüz, M. Sezer and C. Güler, “Laguerre collocation method for solving Fredholm-integro-differential equations with functional arguments,” J. Appl. Math., vol. 2014, Article ID: 682398, 12 pages, 2014. [12] N. Baykuş Savaşaneril and M. Sezer, “Hybrid Taylor–Lucas Collocation Method for Numerical Solutional of High-Order Pantograph Type Delay Differential Equations with Variables Delays,” Appl. Math. Inf. Sci., vol. 11, no. 6, pp. 1795–1801, 2017. [13] Ö.K. Kürkçü, E. Aslan and M. Sezer, “A numerical approach with error estimation to solve general integro-differential–difference equations using Dickson polynomials,” Appl. Math. Comput., vol. 276, pp. 324–339, 2016.¬¬ [14] Ö.K. Kürkçü, E. Aslan and M. Sezer, “A novel collocation method based on residual error analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials,” Sains Malays., vol. 46, pp. 335–347, 2017. [15] Ö.K. Kürkçü, E. Aslan and M. Sezer, “A numerical method for solving some model problems arising in science and convergence analysis based on residual function,” Appl. Numer. Math., vol. 121, pp. 134–148, 2017. [16] C. Oğuz and M. Sezer, “Chelyshkov collocation method for a class of mixed functional integro-differential equations,” Appl. Math. Comput., vol. 259, pp. 943–954, 2015. [17] M. Çetin and M. Sezer, “C. Güler, Lucas polynomial approach for system of high-order linear differential equations and residual error estimation,” Math. Prob. Eng., vol. 2015, Article ID: 625984, 14 pages, 2015. [18] N. Şahin, Ş. Yüzbaşı and M. Sezer, “A Bessel polynomial approach for solving general linear Fredholm integro-differential equations,” Int. J. Comput. Math., vol. 88, no. 14, pp. 3093–3111, 2011. [19] K. Erdem, S. Yalçınbaş and M. Sezer, “A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro differential-difference equations,” J. Difference Equ. Appl., vol. 19, no. 10, pp. 1619–1631, 2013. [20] 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., vol. 37, no. 6, pp. 4283–4294, 2013. [21] H-E.D. Gherjalar and H. Mohammodikia, “Numerical solution of functional integral and integro-differential equations by using B-Splines,” Appl. Math., vol. 3, pp. 1940–1944, 2012. [22] S. Yu. Reutskiy, “The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type,” J. Comput. Appl. Math., vol. 296, pp. 724–738, 2016. [23] T. Koshy, “Fibonacci and Lucas Numbers with Applications,” Wiley, New York, USA, 2001. [24] P. Filipponi and A.F. Horadam, “Second derivative sequences of Fibonacci and Lucas polynomials,” The Fibonacci Quarterly, vol. 31, no. 3, pp. 194–204, 1993. [25] A. Constandache, A. Das and F. Toppan, “Lucas polynomials and a standard Lax representation for the polytropic gas dynamics,” Lett. Math. Phys., vol. 60, no. 3, pp. 197–209, 2002. [26] E. W. Weisstein, “Lucas Polynomial, MathWorld: A Wolfram Web Resource,” http://mathworld.wolfram.com/LucasPolynomial.html. [27] F.A. Oliveira, “Collocation and residual correction,” Numer. Math., vol. 36, pp. 27–31, 1980. [28] İ. Çelik, “Collocation method and residual correction using Chebyshev series,” Appl. Math. Comput., vol. 174, pp. 910–920, 2006. [29] M.A. Ramadan, K.R. Raslan, T.S.E. Danaf and M.A.A.E. Salam, “An exponential Chebyshev second kind approximation for solving high-order ordinary differential equations in unbounded domains, with application to Dawson’s integral,” J. Egyptian Math. Soc., vol. 25, pp. 197–205, 2017. [30] J. Zhao, Y. Cao and Y. Xu, “Sinc numerical solution for pantograph Volterra delay-integro-differential equation,” Int. J. Comput. Math., vol. 94, no. 5, pp. 853–865, 2017. [31] J.G. Dix, “Asymptotic behavior of solutions to a first-order differential equation with variable delays,” Comput. Math. Appl., vol. 50, pp. 1791–1800, 2005.