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Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational tau method and error estimation. (English) Zbl 1170.65101
The authors investigate the solutions of a system of nonlinear Volterra integro-differential equations with an error estimation for a class using the operational tau method based on the paper of {\it M. K. El-Daou} and {\it E. L. Ortiz} [J. Math. Anal. Appl. 326, No. 1, 622--631 (2007; Zbl 1119.65065)].

MSC:
65R20Integral equations (numerical methods)
45G15Systems of nonlinear integral equations
45J05Integro-ordinary differential equations
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References:
[1] Ortiz, E. L.; Samara, H.: An operational approach to the tau method for the numerical solution of nonlinear differential equations, Computing 27, 15-25 (1981) · Zbl 0449.65053 · doi:10.1007/BF02243435
[2] Ortiz, E. L.: The tau method, SIAM J. Numer. anal. 6, 480-492 (1969) · Zbl 0195.45701 · doi:10.1137/0706044
[3] Liu, M. K.; Pan, C. K.: The automatic solution to systems of ordinary differential equations by the tau method, Comput. math. Appl. 17, 197-210 (1999) · Zbl 0945.65073 · doi:10.1016/S0898-1221(99)00275-8
[4] Ortiz, E. L.; Pun, K. S.: Numerical solution of nonlinear partial differential equations with the tau method, J. comput. Appl. math. 12, 511-516 (1985) · Zbl 0579.65124 · doi:10.1016/0377-0427(85)90044-5
[5] Ortiz, E. L.; Aliabadi, M. H.: Numerical treatment of moving and free boundary value problems with the tau method, Comput. math. Appl. 35, No. 8, 53-61 (1998) · Zbl 0999.65110 · doi:10.1016/S0898-1221(98)00044-3
[6] Pour-Mahmoud, J.; Rahimi-Ardabili, M. Y.; Shahmorad, S.: Numerical solution of the system of Fredholm integro-differential equations by the tau method, Appl. math. Comput. 168, 465-478 (2005) · Zbl 1082.65600 · doi:10.1016/j.amc.2004.09.026
[7] Hosseini, S. M.; Shamorad, S.: A matrix formulation of the tau method and Volterra linear integro-differential, Korean J. Comput. appl. Math. 9, No. 2, 497-507 (2002) · Zbl 1005.65148
[8] Ortiz, E. L.: On the numerical solution of nonlinear and functional differential equations with the tau method, Lecture notes in mathematics 679, 127-139 (1978) · Zbl 0387.65053
[9] Ebadi, G.; Rahimi, M. Y.; Shahmorad, S.: Numerical solution of the nonlinear Volterra integro-differential equations by the tau method, Appl. math. Comput. 188, 1580-1586 (2007) · Zbl 1119.65123 · doi:10.1016/j.amc.2006.11.024
[10] Shahmorad, S.: Numerical solution of the general form linear Fredholm--Volterra integro-differential equations by the tau method with an error estimation, Appl. math. Comput. 167, 1418-1429 (2005) · Zbl 1082.65602 · doi:10.1016/j.amc.2004.08.045
[11] Hosseini, S. M.; Shahmorad, S.: Numerical solution of a class of integro-differential equations by the tau method with an error estimation, Appl. math. Comput. 136, 550-570 (2003) · Zbl 1027.65182 · doi:10.1016/S0096-3003(02)00081-4
[12] Dehghan, M.; Saadatmandi, A.: A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification, Comput. math. Appl. 52, 933-940 (2006) · Zbl 1125.65340 · doi:10.1016/j.camwa.2006.04.017
[13] El-Daou, M. K.; Ortiz, E. L.: The weighting subspaces of the tau method and orthogonal collocation, J. math. Anal. appl. 326, 622-631 (2007) · Zbl 1119.65065 · doi:10.1016/j.jmaa.2006.03.019
[14] Ebadi, G.; Rahimi-Ardabili, M. Y.; Shahmorad, S.: Numerical solution of the system of nonlinear Fredholm integro-differential equations by the operational tau method with an error estimation, Sci. iran. 14, No. 6, 546-554 (2007) · Zbl 1178.65146
[15] J. Biazar, H. Ghazvini, M. Eslami, He’s homotopy perturbation method for systems of integro-differential equations, Chaos Solitons Fractals. doi:10.1016/j.chaos.2007.06.001 · Zbl 1197.65106