zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind. (English) Zbl 1192.65154
Summary: A computational method for solving Fredholm integral equations of the first kind is presented. The method utilizes Chebyshev wavelets constructed on the unit interval as basis in the Galerkin method and reduces the solving of the integral equation to the solving of a system of algebraic equations. The properties of Chebyshev wavelets are used to make the wavelet coefficient matrices sparse which eventually leads to the sparsity of the coefficient matrix of the obtained system. Finally, numerical examples are presented to show the validity and efficiency of the technique.

65R20Integral equations (numerical methods)
65T60Wavelets (numerical methods)
45B05Fredholm integral equations
Full Text: DOI EuDML
[1] M. Razzaghi and Y. Ordokhani, “Solution of nonlinear Volterra-Hammerstein integral equations via rationalized Haar functions,” Mathematical Problems in Engineering, vol. 7, no. 2, pp. 205-219, 2001. · Zbl 0990.65152 · doi:10.1155/S1024123X01001612 · eudml:49314
[2] M. Razzaghi and Y. Ordokhani, “A rationalized Haar functions method for nonlinear Fredholm-Hammerstein integral equations,” International Journal of Computer Mathematics, vol. 79, no. 3, pp. 333-343, 2002. · Zbl 0995.65145 · doi:10.1080/00207160211932
[3] A. Alipanah and M. Dehghan, “Numerical solution of the nonlinear Fredholm integral equations by positive definite functions,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1754-1761, 2007. · Zbl 1122.65408 · doi:10.1016/j.amc.2007.02.063
[4] C. Hsiao, “Hybrid function method for solving Fredholm and Volterra integral equations of the second kind,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 59-68, 2009. · Zbl 1167.65473 · doi:10.1016/j.cam.2008.10.060
[5] A. Akyüz-Da\cscıo\uglu, “Chebyshev polynomial solutions of systems of linear integral equations,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 221-232, 2004. · Zbl 1049.65149 · doi:10.1016/S0096-3003(03)00334-5
[6] S. Yousefi and A. Banifatemi, “Numerical solution of Fredholm integral equations by using CAS wavelets,” Applied Mathematics and Computation, vol. 183, no. 1, pp. 458-463, 2006. · Zbl 1109.65121 · doi:10.1016/j.amc.2006.05.081
[7] S. Yousefi and M. Razzaghi, “Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations,” Mathematics and Computers in Simulation, vol. 70, no. 1, pp. 1-8, 2005. · Zbl 1205.65342 · doi:10.1016/j.matcom.2005.02.035
[8] M. Lakestani, M. Razzaghi, and M. Dehghan, “Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets,” Mathematical Problems in Engineering, vol. 2005, no. 1, pp. 113-121, 2005. · Zbl 1073.65568 · doi:10.1155/MPE.2005.113 · eudml:51729
[9] H. Adibi and P. Assari, “Using CAS wavelets for numerical solution of Volterra integral equations of the second kind,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 16, no. 5, pp. 673-685, 2009. · Zbl 1186.65156 · http://dcdis001.watam.org/volumes/contents2009/v16n5a.html
[10] U. Lepik and E. Tamme, “Application of the Haar wavelets for solution of linear integral equations,” in Dynamical Systems and Applications, pp. 395-407, 2005.
[11] E. Babolian and L. M. Delves, “An augmented Galerkin method for first kind Fredholm equations,” Journal of the Institute of Mathematics and Its Applications, vol. 24, no. 2, pp. 157-174, 1979. · Zbl 0428.65065 · doi:10.1093/imamat/24.2.157
[12] G. Hanna, J. Roumeliotis, and A. Kucera, “Collocation and Fredholm integral equations of the first kind,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 5, article 131, pp. 1-8, 2005. · Zbl 1082.65142 · eudml:125684
[13] B. A. Lewis, “On the numerical solution of Fredholm integral equations of the first kind,” Journal of the Institute of Mathematics and Its Applications, vol. 16, no. 2, pp. 207-220, 1975. · Zbl 0308.65075 · doi:10.1093/imamat/16.2.207
[14] K. Maleknejad, R. Mollapourasl, and K. Nouri, “Convergence of numerical solution of the Fredholm integral equation of the first kind with degenerate kernel,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1000-1007, 2006. · Zbl 1105.65127 · doi:10.1016/j.amc.2006.01.074
[15] X. Shang and D. Han, “Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 440-444, 2007. · Zbl 1193.65231 · doi:10.1016/j.amc.2007.02.108
[16] M. T. Rashed, “Numerical solutions of the integral equations of the first kind,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 413-420, 2003. · Zbl 1032.65147 · doi:10.1016/S0096-3003(02)00497-6
[17] B. K. Alpert, “A class of bases in L2 for the sparse representation of integral operators,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 246-262, 1993. · Zbl 0764.42017 · doi:10.1137/0524016
[18] M. Razzaghi and S. Yousefi, “The Legendre wavelets operational matrix of integration,” International Journal of Systems Science, vol. 32, no. 4, pp. 495-502, 2001. · Zbl 1006.65151 · doi:10.1080/002077201300080910
[19] E. Babolian and F. Fattahzadeh, “Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 1016-1022, 2007. · Zbl 1114.65366 · doi:10.1016/j.amc.2006.10.073
[20] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2003.
[21] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999. · Zbl 0924.28001
[22] I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992. · Zbl 0788.42013 · doi:10.1137/0523059