# Article ID: MTJPAM-D-22-00026

## Title: Solutions of high-order linear Volterra integro-differential equations via Lucas polynomials

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-22-00026; Volume 5 / Issue 1 / Year 2023, Pages 22-33

Document Type: Research Paper

aDokuz Eylul University, Bergama Vocational School, Izmir, Turkey

bDokuz Eylul University, Izmir Vocational School, Izmir, Turkey

Received: 15 August 2022, Accepted: 29 November 2022, Published: 28 December 2022

Corresponding Author: Deniz Elmacı (Email address: deniz.elmaci@deu.edu.tr)

Abstract

It is often not possible to find the analytical solution of every type of equation encountered in physics and engineering applications. For this reason, approximate solution methods are needed and errors occur in the solutions obtained by such methods. Therefore, besides being practical and useful, solution methods that give the best approach are sought. For this purpose a matrix method called the Lucas collocation method is presented for numerically solving high-order linear Volterra integro-differential equations under mixed conditions in this paper. Numerical results were compared and interpreted with tables and graphs and the solution was shown to be consistent. The outcomes demonstrate the effectiveness and precision of the current work. On the computer, a MATLAB program was used to run all of the numerical calculations.

Keywords: Volterra integro-differential equations, Lucas series and polynomials, Lucas matrix method, collocation points

References:
1. I. S. Ali, Haar wavelet collocation technique for solving linear volterra integro differential equations, NeuroQuantology 18 (7), 39–44, 2020.
2. N. Baykuş Savaşaneril and M. Sezer, 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), 1795–1801, 2017; doi:10.18576/amis/110627.
3. M. Bicknell, A primer for the Fibonacci numbers VII, Fibonacci Quart. 8, 407–420, 1970.
4. H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge University Press, 2004.
5. B. Bülbül, M. Gülsu and M. Sezer, A new Taylor collocation method for nonlinear Fredholm‐Volterra integro‐differential equations, Numer. Methods Partial Differ. Equations 26 (5), 1006–1020, 2010.
6. L. M. Delves and J. L. Mohamed, Computational methods for integral equations, Cambridge University Press, Cambridge, 1985.
7. D. Elmacı and N. Baykuş Savaşaneril, Euler polynomials method for solving linear integro differential equations, New Trends in Mathematical Sciences 9 (3), 21–34, 2021.
8. D. Elmacı, N. Baykuş Savaşaneril, F. Dal and M. Sezer, On the application of Euler’s method to linear integro differential equations and comparison with existing methods, Turkish J. Math. 46 (1), 99–122, 2022.
9. K. Erdem Biçer and H. G. Dağ, Boole approximation method with residual error function to solve linear Volterra integro-differential equations, Celal Bayar University Journal of Science 17 (1), 59–66, 2021.
10. K. Erdem Biçer and M. Sezer, Bernoulli matrix-collocation method for solving general functional integro-differential equations with hybrid delays, J. Inequal. Spec. Funct. 8 (3), 85–99, 2017.
11. S. Gümgüm, N. Baykuş Savaşaneril, Ö. K. Kürkçü and M. Sezer, 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, 2018; doi:10.16984/saufenbilder.384592.
12. S. Gümgüm, N. Baykuş Savaşaneril, Ö. K. Kürkçü and M. Sezer, Lucas polynomial solution of nonlinear differential equations with variable delay, Hacet. J. Math. Stat. 49 (2), 553–564, 2020.
13. N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (8), 839–850, 2008.
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 46, 335–347, 2017.
15. K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear Volterra–Fredholm integro-differential equations, Appl. Math. Comput. 145 (2-3), 641–653, 2003.
16. T. Mollaoğlu, Volterra tipi gecikmeli fonksiyonel integro-diferansiyel denklemler için Gegenbauer polinom yaklaşımı, Master thesis, Manisa Celal Bayar University, Manisa, 2017.
17. M. T. Rashed, Numerical solution of functional differential, integral and integro-differential equations, Appl. Numer. Math. 156, 485–492, 2004.
18. A. M. Wazwaz, The variational iteration method for solving linear and nonlinear Volterra integral and integro-differential equations, Int. J. Comput. Math. 87 (5), 1131–1141, 2010.
19. S. Yalçinbaş and K. Erdem, Approximate solutions of nonlinear Volterra integral equation systems, Internat. J. Modern Phys. B 24 (32), 6235–6258, 2010.
20. S. Yalçinbaş and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2-3), 291–308, 2000.
21. Ş. Yüzbaşı and I. Nurbol, An operational matrix method for solving linear Fredholm‒Volterra integro-differential equations, Turkish J. Math. 42 (1), 243–256, 2018.
22. Ş. Yüzbaşı, N. Şahin and M. Sezer, Bessel polynomial solutions of high-order linear Volterra integro-differential equations, Comput. Math. Appl. 62 (4), 1940–1956, 2011.
23. Ş. Yüzbaşı, N. Şahin and A. Yildirim, A collocation approach for solving high-order linear Fredholm–Volterra integro-differential equations, Math. Comput. Modelling 55 (3-4), 547–563, 2012.
24. M. Zarebnia, Sinc numerical solution for the Volterra integro-differential equation, Commun. Nonlinear Sci. Numer. Simul. 15, 700–706, 2010.
25. H. Zuoshang, Boundness of solutions to functional integro-differential equations, Proc. Amer. Math. Soc. 114 (2), 617–625, 1992.