Taylor collocation method for high-order neutral delay Volterra integro-differential equations

Main Article Content

Hafida Laib
Azzeddine Bellour
https://orcid.org/0000-0002-3644-0804
Aissa Boulmerka
https://orcid.org/0000-0003-4920-1850

Abstract

In this paper, the Taylor collocation method is applied to numerically solve a kth-order neutral linear Volterra integro-differential equation with constant delay and variable coefficients.
We also provide a rigorous analysis to estimate the difference between the exact and approximate solution and their derivatives up to order k-1.
Numerical examples are included to prove the performance of the presented convergent algorithm.

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How to Cite
[1]
Laib, H., Bellour, A. and Boulmerka, A. 2022. Taylor collocation method for high-order neutral delay Volterra integro-differential equations. Journal of Innovative Applied Mathematics and Computational Sciences. 2, 1 (May 2022), 53–77. DOI:https://doi.org/10.58205/jiamcs.v2i1.10.
Section
Research Articles

References

R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second edition, Marcel Dekker, Inc., New York, 2000.

E. Babolian and A. S. Shamloo, Numerical solution of Volterra integral and integrodifferential equations of convolution type by using operational matrices of piecewise constant orthogonal functions, J. Comput. Appl. Math. 214(2) (2008), 495–508.

A. Bellour and M. Bousselsal, A Taylor collocation method for solving delay integral equations, Numer. Algor. 65(4) (2014), 843–857.

A. Bellour and M. Bousselsal, Numerical solution of delay integro-differential equations by using Taylor collocation method, Math. Meth. Appl. Sci. 37 (2014), 1491–1506 .

A. Bellour, M. Bousselsal and H. Laib, Numerical Solution of Second-Order Linear Delay Differential and Integro-Differential Equations by Using Taylor Collocation Method, International Journal of Computational Methods, 17 (09):1950070 (2020), 1–31.

H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge university press, Cambridge (2004).

H. Brunner, Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations, Math. Comput. 42 (1984), 95–109.

H. Brunner, Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal. 6 (1986), 221–339.

H. Brunner, On the numerical solution of nonlinear Volterra integro-differential equations, BIT. 13 (1973), 381–390.

H. Brunner, A. Pedas and G. Vainikko, Piece-wise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal. 39 (2001), 957–982.

T. A. Burton, Volterra Integral and Differential Equations, Academic Press, New York (1983).

P. Darania, Multistep collocation method for nonlinear delay integral equations, Sahand Communications in Mathematical Analysis (SCMA). 3 (2016), 47–65.

P. Darania and S. Pishbin, High-order collocation methods for nonlinear delay integral equation, J. Comput. Appl. Math. 326 (2017), 284–295.

G. Ebadi, M. Rahimi-Ardabili and S. Shahmorad, Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method, Appl. Math. Comput. 188 (2007), 1580–1586.

H. M. El-Hawary and A. El-ShamiK, Spline Collocation Methods for Solving Second Order Neutral Delay Differential Equations, Int. J. Open Problems Compt. Math. 2(4) (2009), 536–545.

M. Ghasemi, M. Kajani and E. Babolian, Application of He’s homotopy perturbation method to nonlinear integro-differential equation, Appl. Math. Comput. 188 (2007), 538–548.

E. Hairer, C. Lubich and S. P. Nfrsett, Order of convergence of one-step methods for Volterra integral equations of the second kind, SIAM J. Numer. Anal. 20 (1983), 569–579.

S. N. Jator and J. Li, A self starting linear multistep method for a direct solution of the general second order initial value problems, Intern J. Comp. Math. 86(5) (2009), 827–836.

H. Laib, A. Bellour and M. Bousselsal, Numerical solution of high-order linear Volterra integro-differential equations by using Taylor collocation method, Int. J. Comput. Math. 96(5) (2019), 1066–1085.

H. Laib, A. Bellour and A. Boulmerka, Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic, Int. J. Comput. Math. 99(4) (2021), 852–876.

O. Lepik, Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comput. 176 (2006), 324–333.

K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear Volterra Fredholm integro-differential equations, Appl. Math. Comput. 145 (2003), 641–653.

E. Rawashdeh, D. Mcdowell and L. Rakesh, Polynomial spline collocation methods for second-order Volterra integro-differential equations, Int. J. Math. Math. Sci. 56 (2004), 3011–3022.

J. Saberi-Nadja and A. Ghorbani, He’s homotopy perturbation method: an effective tool for solving non-linear integral and integro-differential equations, Comput. Math. Appl. 58 (2009), 2379–2390.

H. Y. Seong and Z. Abdul Majid, Solving second order delay differential equations using direct two-point block method. Ain Shams Engineering Journal, 8 (2017), 59-66.

Y. Wei and Y. Chen, Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation, Appl. Numer. Math. 81 (2014), 15–29.

Y. Wei and Y. Chen, Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations, Numer. Math. Theory Methods Appl. 4 (2011), 339–358.

Y. Wei and Y. Chen, Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv. Appl. Math. Mech. 4 (2012), 1–20.

A. F. Yeniçerioˇglu, The behavior of solutions of second order delay differential equations, J. Math. Anal. Appl. 332 (2007), 1278–1290.