Abstract
A numerical method based on the use of Taylor polynomials is proposed to construct a collocation solution \(u\in S_{m-1}^{(-1)}(\Pi _{N})\) for approximating the solution of delay integral equations. It is shown that this method is convergent. Some numerical examples are given to show the validity of the presented method.
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Ali, I., Brunner, H., Tang, T.: Spectral methods for pantograph-type differential and integral equations with multiple delays. Front. Math. China 4, 49–61 (2009)
Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10, 643–647 (1943)
Bellour, A., Rawashdeh, E.: Numerical solution of first kind integral equations by using Taylor polynomials. J. Inequal. Spec. Funct. 1, 23–29 (2010)
Bélair, J.: Population models with state-dependent delays. In: Arino, O., Axelrod, D.E., Kimmel, M. (eds.) Mathematical Population Dynamics, pp. 165–176. Marcel Dekker, New York (1991)
Bownds, J.M., Cushing, J.M., Schutte, R.: Existence, uniqueness and extendibility of solutions of Volterra integral systems with multiple variable lags. Funcial. Ekvac. 19, 101–111 (1976)
Brunner, H.: The discretization of neutral functional integro-differential equations by collocation methods. J. Anal. Appl. 18, 393–406 (1999)
Brunner, H.: Iterated collocation methods for Volterra integral equations with delay arguments. Math. Comput. 62, 581–599 (1994)
Brunner, H.: Collocation methods for Volterra integral and related functional differential equations. Cambridge university press, Cambridge (2004)
Brunner, H.: Collocation and continuous implicit Runge-Kutta methods for a class of delay Volterra integral equations. J. Comput. Appl. Math. 53, 61–72 (1994)
Brunner, H., Yatsenko, Y.: Spline collocation methods for nonlinear Volterra integral equations with unknown delay. J. Comput. Appl. Math. 71, 67–81 (1996)
Cahlon, B., Nachman, L.J.: Numerical solutions of Volterra integral equations with a solution dependant delay. J. Comput. Appl. Math. 112, 541–562 (1985)
Caliò, F., Marchetti, E., Pavani, R.: About the deficient spline collocation method for particular differential and integral equations with delay. Rend. Sem. Mat. Univ. Pol. Torino 61, 287–300 (2003)
Caliò, F., Marchetti, E., Pavani, R., Micula, G.: About some Volterra problems solved by a particular spline collocation. Studia Univ. Babes-Bolyai 48, 135–158 (2003)
Cooke, K.L.: An epidemic equation with immigration. Math. Biosci. 29, 135–158 (1976)
Darania, P., Ivaz, K.: Numerical solution of nonlinear Volterra-Fredholm integro-differential equations. Comput. Math. Appl. 56(9), 2197–2209 (2008)
Horvat, V.: On collocation methods for Volterra integral equations with delay arguments. Math. Commun. 4, 93–109 (1999)
Hu, Q.: Multilevel correction for discrete collocation solutions of Volterra integral equations with delay arguments. Appl. Numer. Math. 31, 159–171 (1999)
Maleknejad, K., Mahmoudi, Y.: Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations. Appl. Math. Comput. 2, 641–653 (2003)
Sezer, M., Gülsu, M.: Polynomial solution of the most general linear Fredholm-Volterra integrodifferential-difference equations by means of Taylor collocation method. Appl. Math. Comput. 185, 646-657 (2007)
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Bellour, A., Bousselsal, M. A Taylor collocation method for solving delay integral equations. Numer Algor 65, 843–857 (2014). https://doi.org/10.1007/s11075-013-9717-8
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DOI: https://doi.org/10.1007/s11075-013-9717-8