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The spectral method for solving systems of Volterra integral equations. (English) Zbl 1295.65128
Summary: This paper presents a high accurate and stable Legendre-collocation method for solving systems of Volterra integral equations (SVIEs) of the second kind. The method transforms the linear SVIEs into the associated matrix equation. In the nonlinear case, after applying our method we solve a system of nonlinear algebraic equations. Also, sufficient conditions for the existence and uniqueness of the linear SVIEs, in which the coefficient of the main term is a singular (or nonsingular) matrix, have been formulated. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods. All of the numerical computations have been performed on a PC using several programs written in MAPLE 13.

MSC:
65R20 Numerical methods for integral equations
45F05 Systems of nonsingular linear integral equations
45G15 Systems of nonlinear integral equations
45D05 Volterra integral equations
Software:
Maple
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References:
[1] Aminikhah, H., Biazar, J.: A new analytical method for solving systems of Volterra integral equations. Int. J. Comput. Math. 87, 1142–1157 (2010) · Zbl 1197.65218 · doi:10.1080/00207160903128497
[2] Berenguer, M.I., Gamez, D., Garralda-Guillem, A.I., Ruiz Galan, M., Serrano Perez, M.C.: Biorthogonal systems for solving Volterra integral equation systems of the second kind. J. Comput. Appl. Math. 235, 1873–1875 (2011) · Zbl 1215.65192 · doi:10.1016/j.cam.2010.07.011
[3] Biazar, J.: Solving system of integral equations by Adomian decomposition method. Ph.D. thesis, Teaching Training University, Iran (2002) · Zbl 1012.65146
[4] 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
[5] Biazar, J., Ebrahimi, H.: Chebyshev wavelets approach for nonlinear systems of Volterra integral equations. Comput. Math. Appl. 63, 608–616 (2012) · Zbl 1238.65122 · doi:10.1016/j.camwa.2011.09.059
[6] Biazar, J., Ghazvini, H.: He’s homotopy perturbation method for solving system of Volterra integral equations of the second kind. Chaos Solitons Fractals 39, 770–777 (2009) · Zbl 1197.65219 · doi:10.1016/j.chaos.2007.01.108
[7] Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2000)
[8] Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004) · Zbl 1059.65122
[9] Bulatov, M.V.: Regularization of singular systems of Volterra integral equations. Comput. Math. Math. Phys. 42, 315–320 (2002) · Zbl 1058.65144
[10] Bulatov, M.V.: Transformation of differential-algebraic systems of equations. Comput. Math. Math. Phys. 34, 301–331 (1994)
[11] Bulatov, M.V., Chistyakov, V.F.: The properties of differential-algebraic systems and their integral analogs. Preprint, Memorial University of Newfoundland (1997)
[12] Bulatov, M.V., Lima, P.M.: Two-dimensional integral-algebraic systems: analysis and computational methods. J. Comput. Appl. Math. 236, 132–140 (2011) · Zbl 1229.65232 · doi:10.1016/j.cam.2011.06.001
[13] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988) · Zbl 0658.76001
[14] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer, New York (2006) · Zbl 1093.76002
[15] Chen, Y.P., Tang, T.: Spectral methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009) · Zbl 1186.65161 · doi:10.1016/j.cam.2009.08.057
[16] Chistyakov, V.F.: Differential-Algebraic Operators with Finite Dimensional Kernel. Nauka, Novosibirk (1996) (in Russian) · Zbl 0999.34002
[17] Delves, L.M., Mohamed, J.L.: Computational Methods for Integral Equations. Cambridge University Press, London (1985) · Zbl 0592.65093
[18] Elnagar, G.N., Kazemi, M.: Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations. J. Comput. Appl. Math. 76, 147–158 (1996) · Zbl 0873.65122 · doi:10.1016/S0377-0427(96)00098-2
[19] Hadizadeh, M., Ghoreishi, F., Pishbin, S.: Jacobi spectral solution for integral algebraic equations of index-2. Appl. Numer. Math. 61, 131–148 (2011) · Zbl 1206.65260 · doi:10.1016/j.apnum.2010.08.009
[20] Han, H., Liu, Y., Lau, T., He, X.: New algorithm for the system of nonlinear weakly singular Volterra integral equations of the second kind and integro-differential equations. J. Inf. Comput. Sci. 7, 1229–1235 (2010)
[21] Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007) · Zbl 1111.65093
[22] Jiang, Y.-J.: On spectral methods for Volterra-type integro-differential equations. J. Comput. Appl. Math. 230, 333–340 (2009) · Zbl 1202.65170 · doi:10.1016/j.cam.2008.12.001
[23] Katani, R., Shahmorad, S.: Block by block method for the systems of nonlinear Volterra integral equations. Appl. Math. Model. 34, 400–406 (2010) · Zbl 1185.65237 · doi:10.1016/j.apm.2009.04.013
[24] Kauthen, J.P.: The numerical solution of integral-algebraic equations of index-1 by polynomial spline collocation methods. Math. Comput. 70, 1503–1514 (2001) · Zbl 0979.65122 · doi:10.1090/S0025-5718-00-01257-6
[25] Kress, R.: Linear Integral Equations. Springer, Berlin (1989) · Zbl 0671.45001
[26] Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985) · Zbl 0566.65094
[27] Maleknejad, K., Salimi Shamloo, A.: Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices. Appl. Math. Comput. 195, 500–505 (2008) · Zbl 1132.65117 · doi:10.1016/j.amc.2007.05.001
[28] Nadjafi, J.S., Samadi, O.R.N., Tohidi, E.: Numerical solution of two dimensional integral equations via aspectral Galerkin method. J. Appl. Math. Biol. 1, 343–359 (2011) · Zbl 1270.65079
[29] Rabbani, M., Maleknejad, K., Aghazadeh, N.: Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method. Appl. Math. Comput. 187, 1143–1146 (2007) · Zbl 1114.65371 · doi:10.1016/j.amc.2006.09.012
[30] Saberi-Nadjafi, J., Tamamgar, M.: The variational iteration method: a highly promising method for solving the system of integro differential equations. Comput. Math. Appl. 56, 346–351 (2008) · Zbl 1155.65399 · doi:10.1016/j.camwa.2007.12.014
[31] Sahin, N., Yuzbasi, S., Gulsu, M.: A collocation approach for solving systems of linear Volterra integral equations with variable coefficients. Comput. Math. Appl. 62, 755–769 (2011) · Zbl 1228.65248 · doi:10.1016/j.camwa.2011.05.057
[32] Sorkun, H.H., Yalkinbas, S.: Approximate solutions of linear Volterra integral equation systems with variable coefficients. Appl. Math. Model. 34, 3451–3464 (2010) · Zbl 1201.45001 · doi:10.1016/j.apm.2010.02.034
[33] Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008) · Zbl 1174.65058
[34] Tohidi, E.: Legendre approximation for solving linear HPDEs and comparison with Taylor and Bernoulli matrix methods. Appl. Math. 3, 410–416 (2012). doi: 10.4236/am.2012.35063 · doi:10.4236/am.2012.35063
[35] Tohidi, E., Samadi, O.R.N.: Optimal control of nonlinear Volterra integral equations via Legendre polynomials. IMA J. Math. Control Inf. (2012). doi: 10.1093/imamci/DNS014 · Zbl 1275.49056
[36] Tohidi, E., Samadi, O.R.N., Farahi, M.H.: Legendre approximation for solving a class of nonlinear optimal control problems. J. Math. Finance 3, 8–13 (2011). doi: 10.4236/jmf.2011.11002 · doi:10.4236/jmf.2011.11002
[37] Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4, 181–193 (2009) · Zbl 1396.65165 · doi:10.1007/s11464-009-0002-z
[38] Yusufoglu, E.: Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules. Int. J. Comput. Math. 84, 33–140 (2007) · Zbl 1119.65073 · doi:10.1080/00207160601138830
[39] Yuzbasi, S., Sahin, N., Sezer, M.: A Bessel collocation method for numerical solution of generalized pantograph equations. Numer. Methods Partial Differ. Equ. (2010). doi: 10.1002/num.20660
[40] Zarebnia, M., Rashidinia, J.: Approximate solution of systems of Volterra integral equations with error analysis. Int. J. Comput. Math. 87, 3052–3062 (2010) · Zbl 1206.65265 · doi:10.1080/00207160902906463
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