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

In this paper, a new numerical matrix-collocation technique is considered to solve functional integrodifferential 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.

___

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.

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.

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.

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.

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.

N. Kurt and M. Sezer, “Polynomial solution of high-order linear Fredholm integrodifferential equations with constant coefficients,” J. Franklin Inst., vol. 345, pp. 839–850, 2008.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

M. Çetin and M. Sezer, “C. Güler, Lucas polynomial approach for system of highorder linear differential equations and residual error estimation,” Math. Prob. Eng., vol. 2015, Article ID: 625984, 14 pages, 2015.

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.

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.

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.

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.

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.

T. Koshy, “Fibonacci and Lucas Numbers with Applications,” Wiley, New York, USA, 2001.

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.

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.

E. W. Weisstein, “Lucas Polynomial, MathWorld: A Wolfram Web Resource,” http://mathworld.wolfram.com/LucasPolyn omial.html.

F.A. Oliveira, “Collocation and residual correction,” Numer. Math., vol. 36, pp. 27– 31, 1980.

İ. Çelik, “Collocation method and residual correction using Chebyshev series,” Appl. Math. Comput., vol. 174, pp. 910–920, 2006.

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.

J. Zhao, Y. Cao, and Y. Xu, “Sinc numerical solution for pantograph Volterra delayintegro- differential equation,” Int. J. Comput. Math., vol. 94, no. 5, pp. 853–865, 2017.

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.
Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi-Cover
  • ISSN: 1301-4048
  • Yayın Aralığı: Yılda 6 Sayı
  • Başlangıç: 1997
  • Yayıncı: Sakarya Üniversitesi Fen Bilimleri Enstitüsü