Numerical Solution of High-Order Linear Fredholm Integro-Differential Equations by Lucas Collocation Method

Abstract

In this paper, a useful matrix approach for high-order linear Fredholm integro-differential equations with initial boundary conditions expressed as Lucas polynomials is proposed. Using a matrix equation
which is equivalent to a set of linear algebraic equations the method transforms to integro-differential equation. When compared to other methods that have been proposed in the literature, the numerical results from the suggested technique reveal that it is effective and promising. And also, error estimation of the scheme was derived. These results were compared with the exact solutions and the other numerical methods to the tested problems.

Keywords

• Fredholm integro-differential equations
• Initial value problem
• Lucas matrix method
• Lucas polynomials and series

References

1. S. Yalcinbas and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput., 112 (2000), 291–308.
2. K. Maleknejad and Y. Mahmoudi, Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions, Appl Math Comput, 149 (2004), 799–806.
3. M. T. Rashed, Numerical solution of functional differential, integral and integro-differential equations, Appl Numer Math, 156 (2004), 485–492.
4. W. Wang, An algorithm for solving the higher-order nonlinear Volterra-Fredholm integro-differential equation with mechanization, Appl Math Comput, 172,(2006), 1–23.
5. S. M. Hosseini and S. Shahmorad, Numerical solution of a class of integro-differential equations by the Tau method with an error estimation}, Appl Math Comput, 136 (2003), 559–570.
6. S. Nas, S.Yalçınbas, and M. Sezer, A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations, Int J Math Educ Sci Technol, 31 (2000), 213–225.
7. R. Farnoosh and M. Ebrahimi, Monte Carlo method for solving Fredholm integral equations, Appl Math Comput, 195 (2008), 309–315.
8. M. Sezer and M. Gulsu, Polynomial solution of the most general linear Fredholm integro-differential difference equation by means of Taylor matrix method, Int J Complex Variables, 50 (2005), 367–382.

9. N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J Franklin Inst 345 (2008), 839–850.
10. Ş. Yüzbaşı, et. al. A collocation approach for solving high-order linear Fredholm–Volterra integro-differential equations, Mathematical and Computer Modelling, 55.3-4, (2012), 547-563.
11. N. Şahin, Ş. Yüzbaşi, and M. Sezer, A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations, International Journal of Computer Mathematics, 88.14, (2011), 3093-3111.
12. D. Elmacı, and N. Baykuş Savaşaneril, Euler polynomials method for solving linear integro differential equations, New Trends in Mathematical Sciences, 9.3 (2021), 21-34.
13. D. Elmacı, et al. On the application of Euler's method to linear integro differential equations and comparison with existing methods, Turkish Journal of Mathematics, 46.1 (2022), 99-122.
14. Ş. Yüzbaşi, and I. Nurbol, An operational matrix method for solving linear Fredholm‒Volterra integro-differential equations, Turkish Journal of Mathematics, 42.1, (2018), 243-256.
15. Gümgüm S., Baykuş Savaşaneril N., Kürkçü Ö.K., 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, vol. 22.6, (2018), 1659-1668.
16. Baykuş Savaşaneril N., Sezer M., Hybrid Taylor-Lucas Collocation Method for Numerical Solution of High-Order Pantograph Type Delay Differential Equations with Variables Delays, Appl. Math. Inf. Sci. 11, No. 6, (2017), 1795-1801 .
17. Gümgüm, S., Savaşaneril, N. B., Kürkçü, Ö. K., Sezer, M., Lucas polynomial solution of nonlinear differential equations with variable delay, Hacettepe Journal of Mathematics and Statistics, (2019), 1-12.
18. K. Erdem, S. Yalçinbaş, M. Sezer, A Bernoulli approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations, Journal of Difference Equations and Applications, 19.10, (2013), 1619-1631.
19. H. Gül Dağ, K. Erdem Biçer, Boole collocation method based on residual correction for solving linear Fredholm integro-differential equation, Journal of Science and Arts, 20.3, (2020), 597-610.
20. Ş. Yüzbaşi, An exponential method to solve linear Fredholm-Volterra integro-differential equations and residual improvement, Turkish Journal of Mathematics, 42.5 (2018), 2546-2562.
21. 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, (2017), 335-347.
22. N. Baykus, and M. Sezer, Solution of high‐order linear Fredholm integro‐differential equations with piecewise intervals, Numerical Methods for Partial Differential Equations, 27.5 (2011), 1327-1339.
23. S. Yalçinbaş, M. Sezer, and H. H. Sorkun, Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Applied Mathematics and Computation, 210.2, (2009), 334-349.
24. E. Çimen, and K. Enterili, Fredholm İntegro Diferansiyel Denklemin Sayısal Çözümü için Alternatif Bir Yöntem, Erzincan University Journal of Science and Technology, 13.1 (2020), 46-53.