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Numerical solving initial value problem for Fredholm type linear integro-differential equation system. (English) Zbl 1168.45300
Summary: We introduce a numerical method for solving initial value problems for a system of linear integro-differential equations. The main idea is based on the interpolations of unknown functions at distinct interpolation points. We next use Clenshaw-Curtis quadrature formulae required in the approximation of the integral equations. The technique is very effective and simple. In the end, to show the efficiency of this method, we present some numerical examples.
MSC:
45B05Fredholm integral equations
References:
[1]Atkinson, K. E.: The numerical solution of the second kind, (1997)
[2]Avudainayagam, A.; Vani, C.: Wavelet-Galerkin method for integro-differential equations, Appl. numer. Math. 32, 247-254 (2000) · Zbl 0955.65100 · doi:10.1016/S0168-9274(99)00026-4
[3]Akyüz-Daşçıo&gbreve, A.; Lu: Chebyshev polynomial solutions of systems of linear integral equations, Appl. math. Comput. 151, No. 1, 221-232 (2004)
[4]Akyüz-Daşçıo&gbreve, A.; Lu; Sezer, M.: Chebyshev polynomial solutions of systems of higher-order linear Fredholm – Volterra integro-differential equations, J. franklin inst. 342, No. 6, 688-701 (2005)
[5]Babolian, E.; Biazar, J.: Solution of a system of nonlinear Volterra equations of the second kind, Far east J. Math. sci. 2, No. 6, 935-945 (2000) · Zbl 0979.65123
[6]Babolian, E.; Biazar, J.; Vahidi, A. R.: On the decomposition method for system of linear equations and system of linear Volterra integral equations, Appl. math. Comput. 147, No. 1, 19-27 (2004) · Zbl 1032.65027 · doi:10.1016/S0096-3003(02)00644-6
[7]Babolian, E.; Biazar, J.; Vahidi, A. R.: The decomposition method applied to systems of Fredholm integral equations of the second kind, Appl. math. Comput. 148, No. 2, 443-452 (2004) · Zbl 1042.65104 · doi:10.1016/S0096-3003(02)00859-7
[8]Berenguer, M. I.; Fortes, M. A.; Guillem, A. I. Garralda; Galan, M. Ruiz: Linear Volterra integro-differential equation and Schauder bases, Appl. math. Comput. 159, No. 2, 495-507 (2004) · Zbl 1068.65143 · doi:10.1016/j.amc.2003.08.132
[9]Biazar, J.; Babolian, E.; Islam, R.: Solution of a system of Volterra integral equations of the first kind by Adomian method, Appl. math. Comput. 139, 249-258 (2003) · Zbl 1027.65180 · doi:10.1016/S0096-3003(02)00173-X
[10]Clenshaw, C. W.; Curtis, A. R.: A method for numerical integration on an automatic computer, Numer. math. 2, 197-205 (1960) · Zbl 0093.14006 · doi:10.1007/BF01386223
[11]Davis, P. J.: Interpolation and approximation, (1975) · Zbl 0329.41010
[12]Delves, L. M.; Mohamed, J. L.: Computational methods for integral equations, (1985)
[13]Fox, L.; Parker, I. B.: Chebyshev polynomials in numerical analysis, Oxford mathematical handbooks, (1968) · Zbl 0153.17502
[14]C.D. Green, Integral Equation Methods, Nelson, NY, 1969. · Zbl 0179.44301
[15]M.A. Golberg, Numerical Solution of Integral Equations, NY, 1990.
[16]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, 559-570 (2003) · Zbl 1027.65182 · doi:10.1016/S0096-3003(02)00081-4
[17]Hosseini, S. M.; Shahmorad, S.: Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases, J. appl. Math. modell. 27, 145-154 (2003) · Zbl 1047.65114 · doi:10.1016/S0307-904X(02)00099-9
[18]Hosseini, S. M.; Shahmorad, S.: A matrix formulation of the tau method for Fredholm and Volterra linear integro-differential equations, Korean J. Comput. appl. Math. 9, No. 2, 497-507 (2002) · Zbl 1005.65148
[19]Jackiewicz, Z.; Rahman, M.; Welfert, B. D.: Numerical solution of a Fredholm integro-differential equation modelling neural networks, Appl. numer. Math. 195, 523-536 (2008) · Zbl 1132.65116 · doi:10.1016/j.amc.2007.05.031
[20]Kalitvin, A. S.: On two problems for the barbashin integro-differential equation, J. math. Sci. New York 126, No. 6, 1600-1606 (2005) · Zbl 1078.45006 · doi:10.1007/s10958-005-0049-7
[21]R.P. Kanwal, Linear Integral Equations Theory and Technique, Boston, 1996.
[22]Karamete, A.; Sezer, M.: A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. math. 79, No. 9, 987-1000 (2002) · Zbl 1006.65144 · doi:10.1080/00207160212116
[23]Liz, E.; Nieto, J. J.: Boundary value problems for second order integro-differential equations of Fredholm type, J. comput. Appl. math. 72, 215-225 (1996) · Zbl 0857.45006 · doi:10.1016/0377-0427(95)00273-1
[24]Maleknejad, K.; Mirzaee, F.; Abbasbandy, S.: Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. math. Comput. 155, No. 2, 317-328 (2004) · Zbl 1056.65144 · doi:10.1016/S0096-3003(03)00778-1
[25]Maleknejad, K.; Mirzaee, F.: Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method, Int. J. Comput. math. 80, No. 11, 1397-1405 (2003) · Zbl 1045.65115 · doi:10.1080/0020716031000148214
[26]Maleknejad, M.; Kajani, K. M. Tavassoli: Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. math. Comput. 159, No. 3, 603-612 (2004) · Zbl 1063.65145 · doi:10.1016/j.amc.2003.10.046
[27]Maleknejad, K.; Kajani, M. Tavassoli; Mahmoudi, Y.: Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets, Kybernetes 32, No. 9 – 10, 1530-1539 (2003) · Zbl 1059.65127 · doi:10.1108/03684920310493413
[28]Maleknejad, K.; Mahmoudi, Y.: Taylor polynomial solution of high-order nonlinear Volterra – Fredholm integro-differential equations, Appl. math. Comput. 145, 641-653 (2003) · Zbl 1032.65144 · doi:10.1016/S0096-3003(03)00152-8
[29]Maleknejad, K.; Shahrezaee, M.: Using Runge – Kutta method for numerical solution of the system of Volterra integral equation, Appl. math. Comput. 149, No. 2, 399-410 (2004) · Zbl 1038.65148 · doi:10.1016/S0096-3003(03)00148-6
[30]Maleknejad, K.; Lotfi, T.: Numerical expansion methods for solving integral equations by interpolation and Gauss quadrature rules, Appl. math. Comput. 168, 111-124 (2005) · Zbl 1082.65598 · doi:10.1016/j.amc.2004.08.048
[31]Maleknejad, K.; Derili, H.: Numerical solution of integral equations by using combination of spline-collocation method and Lagrange interpolation, Appl. math. Comput. 175, 1235-1244 (2006) · Zbl 1093.65125 · doi:10.1016/j.amc.2005.08.034
[32]Pour-Mahmoud, J.; Rahimi-Ardabili, M. Y.; Shahmorad, S.: Numerical solution of Volterra linear integro-differential equations by the tau method with the Chebyshev and lejandre bases, Appl. math. Comput. 170, 314-338 (2005) · Zbl 1080.65127 · doi:10.1016/j.amc.2004.11.039
[33]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, No. 1, 465-478 (2005) · Zbl 1082.65600 · doi:10.1016/j.amc.2004.09.026
[34]Rashed, M. T.: Numerical solution of special type of integro-differential equations, Appl. math. Comput. 143, 73-88 (2003) · Zbl 1025.65063 · doi:10.1016/S0096-3003(02)00347-8
[35]Rashed, M. T.: Lagrange interpolation to compute the numerical solutions of differential, integral and integro-differential equations, Appl. math. Comput. 151, 869-878 (2004) · Zbl 1048.65133 · doi:10.1016/S0096-3003(03)00543-5
[36]Rashed, M. T.: Numerical solution of functional differential, integral and integro-differential equations, Appl. math. Comput. 156, 485-492 (2004) · Zbl 1061.65146 · doi:10.1016/j.amc.2003.08.021
[37]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, No. 2, 1418-1429 (2005) · Zbl 1082.65602 · doi:10.1016/j.amc.2004.08.045
[38]Sezer, M.; Do&gbreve, S.; An: Chebyshev series solutions of Fredholm integral equations, Int. J. Math. educ. Sci. technol. 27, No. 5, 649-657 (1996)
[39]Yalçımbaş, S.; Sezer, M.: The approximate solution of high-order linear Volterra – Fredholm integro-differential equations in terms of Taylor polynomials, Appl. math. Comput. 112, 291-308 (2000) · Zbl 1023.65147 · doi:10.1016/S0096-3003(99)00059-4
[40]Zhao, J.; Corless, R. M.: Compact finite difference method for integro-differential equations, Appl. math. Comput. 177, 271-288 (2006) · Zbl 1102.65144 · doi:10.1016/j.amc.2005.11.007
[41]Wang, W.: An algorithm for solving the high-order nonlinear Volterra – Fredholm integro-differential equation with mechanization, Appl. math. Comput. 172, 1-23 (2006) · Zbl 1088.65118 · doi:10.1016/j.amc.2005.01.116
[42]Wazwaz, A. -M.: First course in integral equations, (1997) · Zbl 0924.45001
[43]Wazwaz, A. -M.: The existence of noise terms for systems of inhomogeneous differential and integral equations, Appl. math. Comput. 146, No. 1, 81-92 (2003) · Zbl 1032.65114 · doi:10.1016/S0096-3003(02)00527-1
[44]Wazwaz, A. -M.: The modified decomposition method for analytic treatment of non-linear integral equations and systems of non-linear integral equations, Int. J. Comput. math. 82, No. 9, 1107-1115 (2005) · Zbl 1075.65155 · doi:10.1080/00207160500113041