Yıl 2018, Cilt 22, Sayı 6, Sayfalar 1659 - 1668 2018-12-01

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

Sevin Gümgüm [1] , Nurcan Baykuş Savaşaneril [2] , Ömür Kıvanç Kürkçü [3] , Mehmet Sezer [4]

134 186

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.

Functional equations, Matrix technique, Lucas polynomials, Residual error analysis.
  • [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.
Birincil Dil en
Konular Matematik
Yayımlanma Tarihi December 2018
Dergi Bölümü Araştırma Makalesi
Yazarlar

Orcid: 0000-0002-0594-2377
Yazar: Sevin Gümgüm (Sorumlu Yazar)
Kurum: İZMİR EKONOMİ ÜNİVERSİTESİ
Ülke: Turkey


Orcid: 0000-0002-3098-2936
Yazar: Nurcan Baykuş Savaşaneril
Kurum: DOKUZ EYLÜL ÜNİVERSİTESİ
Ülke: Turkey


Orcid: 0000-0002-3987-7171
Yazar: Ömür Kıvanç Kürkçü
Kurum: İZMİR EKONOMİ ÜNİVERSİTESİ
Ülke: Turkey


Orcid: 0000-0002-7744-2574
Yazar: Mehmet Sezer
Kurum: CELÂL BAYAR ÜNİVERSİTESİ
Ülke: Turkey


Bibtex @araştırma makalesi { saufenbilder384592, journal = {Sakarya University Journal of Science}, issn = {1301-4048}, eissn = {2147-835X}, address = {Sakarya Üniversitesi}, year = {2018}, volume = {22}, pages = {1659 - 1668}, doi = {10.16984/saufenbilder.384592}, title = {A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays}, key = {cite}, author = {Kürkçü, Ömür Kıvanç and Gümgüm, Sevin and Baykuş Savaşaneril, Nurcan and Sezer, Mehmet} }
APA Gümgüm, S , Baykuş Savaşaneril, N , Kürkçü, Ö , 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 University Journal of Science, 22 (6), 1659-1668. DOI: 10.16984/saufenbilder.384592
MLA Gümgüm, S , Baykuş Savaşaneril, N , Kürkçü, Ö , Sezer, M . "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 University Journal of Science 22 (2018): 1659-1668 <http://www.saujs.sakarya.edu.tr/issue/31266/384592>
Chicago Gümgüm, S , Baykuş Savaşaneril, N , Kürkçü, Ö , Sezer, M . "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 University Journal of Science 22 (2018): 1659-1668
RIS TY - JOUR T1 - A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays AU - Sevin Gümgüm , Nurcan Baykuş Savaşaneril , Ömür Kıvanç Kürkçü , Mehmet Sezer Y1 - 2018 PY - 2018 N1 - doi: 10.16984/saufenbilder.384592 DO - 10.16984/saufenbilder.384592 T2 - Sakarya University Journal of Science JF - Journal JO - JOR SP - 1659 EP - 1668 VL - 22 IS - 6 SN - 1301-4048-2147-835X M3 - doi: 10.16984/saufenbilder.384592 UR - http://dx.doi.org/10.16984/saufenbilder.384592 Y2 - 2018 ER -
EndNote %0 Sakarya University Journal of Science A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays %A Sevin Gümgüm , Nurcan Baykuş Savaşaneril , Ömür Kıvanç Kürkçü , Mehmet Sezer %T A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays %D 2018 %J Sakarya University Journal of Science %P 1301-4048-2147-835X %V 22 %N 6 %R doi: 10.16984/saufenbilder.384592 %U 10.16984/saufenbilder.384592
ISNAD Gümgüm, Sevin , Baykuş Savaşaneril, Nurcan , Kürkçü, Ömür Kıvanç , Sezer, Mehmet . "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 University Journal of Science 22 / 6 (Aralık 2018): 1659-1668. http://dx.doi.org/10.16984/saufenbilder.384592