Geodesic
Sparse representation of system of Fredholm integro-differential equations by using Alpert multiwavelets
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 9, pp. 1468-1483 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A numerical technique is presented for the solution of system of Fredholm integro-differential equations. The method consists of expanding the required approximate solution as the elements of Alpert multiwavelet functions (see Alpert B. et al. J. Comput. Phys. 2002, vol. 182, pp. 149–190). Using the operational matrix of integration and wavelet transform matrix, we reduce the problem to a set of algebraic equations. This system is large. We use thresholding to obtain a new sparse system; consequently, GMRES method is used to solve this new system. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results. 
@article{ZVMMF_2015_55_9_a6,
     author = {Behzad Nemati Saray and Mehrad Lakestani and Mohsen Razzaghi},
     title = {Sparse representation of system of {Fredholm} integro-differential equations by using {Alpert} multiwavelets},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1468--1483},
     year = {2015},
     volume = {55},
     number = {9},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_9_a6/}
}
TY  - JOUR
AU  - Behzad Nemati Saray
AU  - Mehrad Lakestani
AU  - Mohsen Razzaghi
TI  - Sparse representation of system of Fredholm integro-differential equations by using Alpert multiwavelets
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2015
SP  - 1468
EP  - 1483
VL  - 55
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_9_a6/
LA  - ru
ID  - ZVMMF_2015_55_9_a6
ER  - 
%0 Journal Article
%A Behzad Nemati Saray
%A Mehrad Lakestani
%A Mohsen Razzaghi
%T Sparse representation of system of Fredholm integro-differential equations by using Alpert multiwavelets
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2015
%P 1468-1483
%V 55
%N 9
%U http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_9_a6/
%G ru
%F ZVMMF_2015_55_9_a6
Behzad Nemati Saray; Mehrad Lakestani; Mohsen Razzaghi. Sparse representation of system of Fredholm integro-differential equations by using Alpert multiwavelets. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 9, pp. 1468-1483. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_9_a6/

[1] J. Pour-Mahmoud, M. Y. Rahimi-Ardabili, S. Shahmorad, “Numerical solution of the system of Fredholm integro-differential equations by the Tau method”, Appl. Math. Comput., 168 (2005), 465–478 | DOI | MR | Zbl

[2] J. Rashidinia, M. Zarebnia, “Convergence of approximate solution of system of Fredholm integral equations”, J. Math. Anal. Appl., 333 (2007), 1216–1227 | DOI | MR | Zbl

[3] A. Davari, M. Khanian, “Solution of system of Fredholm integro-differential equations by Adomian decomposition method”, Austral. J. Basic Appl. Sci., 5:12 (2011), 2356–2361 | MR

[4] P. Oja, D. Saveljeva, “Cubic spline collocation for Volterra integral equations”, Computing, 69 (2001), 319–337 | DOI | MR

[5] B. Zhang, T. Lin, Y. Lin, M. Rao, “Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equation”, Appl. Math., 45 (2000), 241–263 | DOI | MR | Zbl

[6] E. L. Ortiz, L. Samara, “An operational approach to the tau method for the numerical solution of nonlinear differential equations”, Computing, 27 (1981), 15–25 | DOI | MR | Zbl

[7] M. Lakestani, M. Razzaghi, M. Dehghan, “Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations”, Math. Probl. Eng., 2006, 96184, 1–12 | DOI | MR | Zbl

[8] M. Lakestani, B. N. Saray, M. Dehghan, “Numerical solution for the weakly singular Fredholm integrodifferential equations using Legendre multiwavelets”, J. Comput. Appl. Math., 235 (2011), 3291–3303 | DOI | MR | Zbl

[9] M. Razzaghi, S. Yousefi, “Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations”, Math. Comput. Simul., 70 (2005), 1–8 | DOI | MR | Zbl

[10] M. Lakestani, M. Dehghan, “Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions”, Int. J. Comput. Math., 87:6 (2010), 1389–1394 | DOI | MR | Zbl

[11] Y. Ren, Â. Zhang, H. Qiao, “A simple Taylor-series expansion method for a class of second kind integral equations”, J. Comput. Appl. Math., 110 (1999), 15–24 | DOI | MR | Zbl

[12] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, PA, 1985 | MR | Zbl

[13] A. J. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, 1999 | MR | Zbl

[14] S. Abbasbandy, A. Taati, “Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation”, J. Comput. Appl. Math., 231 (2009), 106–113 | DOI | MR | Zbl

[15] A. Khani, M. M. Moghadam, S. Shahmorad, “Numerical solution of special class of system of nonlinear Volterra integro-differential equations by a simple high accuracy method”, Bull. Iran. Math. Soc., 34:2 (2008), 141–152 | MR | Zbl

[16] G. Ebadi, M. Y. Rahimi, S. Shahmorad, “Numerical solution of the system of nonlinear Fredholm integro-differential equations by the operational Tau method with an error estimation”, Sci. Iran., 14 (2007), 546–554 | MR | Zbl

[17] M. Zarebnia, M. G. Ali Abadi, “Numerical solution of system of nonlinear second-order integro-differential equations”, Comput. Math. Appl., 60 (2010), 591–601 | DOI | MR | Zbl

[18] R. Dai, J. E. Cochran Jr., “Wavelet collocation method for optimal control problems”, J. Optim. Theory Appl., 143 (2009), 265–287 | DOI | MR

[19] I. Daubechies, “Orthonormal bases of compactly supported wavelets”, Commun. Pure Appl. Math., 41 (1988), 909–996 | DOI | MR | Zbl

[20] Â. Alpert, G. Beylkin, D. Gines, L. Vozovoi, “Adaptive solution of partial differential equations in multiwavelet bases”, J. Comput. Phys., 182 (2002), 149–190 | DOI | MR | Zbl

[21] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992 | MR | Zbl

[22] M. Shamsi, M. Razzaghi, “Solution of Hallen's integral equation using multiwavelets”, Comput. Phys. Commun., 168 (2005), 187–197 | DOI | Zbl

[23] E. G. Quak, N. Weyrich, “Wavelet on the interval”, Approximation Theory, Wavelets, and Applications, ed. S. P. Singh (Toim), Kluwer, 1995, 247–283 | DOI | MR | Zbl

[24] M. Shamsi, M. Razzaghi, “Numerical solution of the controlled Duffing oscillator by the interpolating scaling functions”, Electromagn. Waves Appl., 18:5 (2004), 691–705 | DOI | MR

[25] M. Lakestani, B. N. Saray, “Numerical solution of telegraph equation using interpolating scaling functions”, Comput. Math. Appl., 60 (2010), 1964–1972 | DOI | MR | Zbl

[26] G. Hanwei, L. Kecheng, H. Jianguo, Y. Jiaxian, L. Peiguo, “A novel wavelet transform matrix for efficient solutions of electromagnetic integral equations”, Proceedings of 1999 International Conference on Computational Electromagnetics and Its Applications, ICCEA'99 (1999)

[27] M. Dehghan, Â. N. Saray, Ì. Lakestani, “Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers–Huxley equation”, Math. Comput. Model., 55 (2012), 1129–1142 | DOI | MR | Zbl

[28] Y. Saad, M. H. Schultz, “GMRES: A generalized minimal residual method for solving nonsymmetric linear systems”, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869 | DOI | MR | Zbl

[29] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003 | MR | Zbl

[30] J. C. Goswami, A. K. Chan, Ñ. K. Chui, “On solving first-kind integral equations using wavelets on bounded interval”, IEEE Trans. Antennas Propag., 43:6 (1995), 614–622 | DOI | MR | Zbl