On integral equations with Weakly Singular kernel by using
Taylor series and Legendre polynomials

• Authors: Esmail Babolian , Danial Hamedzadeh , Hossein Jafari , Asghar Arzhang Hajikandi , Dumitru Baleanu
• Languages of publication: EN

Abstracts

EN

This paper is concerned with the numerical solution
for a class of weakly singular Fredholm integral equations
of the second kind. The Taylor series of the unknown
function, is used to remove the singularity and the truncated
Taylor series to second order of k(x, y) about the
point (x0, y0) is used. The integrals that appear in this
method are computed exactly and some of these integrals
are computed with the Cauchy principal value without using
numerical quadratures. The solution in the Legendre
polynomial form generates a system of linear algebraic
equations, this system is solved numerically. Through numerical
examples, performance of the present method is
discussed concerning the accuracy of the method.

Keywords

EN

weakly singular; Fredholm integral equations; Taylor series; Galerkin method; Legendre functions

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2015
• Volume: 13
• Issue: 1

Dates

online

10 - 11 - 2015

received

11 - 8 - 2015

accepted

16 - 9 - 2015

Contributors

author

Esmail Babolian

• Department of
  Mathematics, Sciences And Research Branch, Islamic Azad University,
  Tehran, Iran

author

Danial Hamedzadeh

• Department of Mathematical Sciences, University of
  South Africa, UNISA 0003, South Africa

author

Hossein Jafari

• Department of Mathematics, University of Mazandaran,
  Babolsar, Iran

author

Asghar Arzhang Hajikandi

• Department of
  Mathematics, Sciences And Research Branch, Islamic Azad University,
  Tehran, Iran

author

Dumitru Baleanu

• Department of Mathematics and Computer Sciences,
  Faculty of Art and Science, Balgat 06530, Ankara, Turkey
• Institute of Space Sciences, Magurele-Bucharest,
  Romania

References

• [1] Y. Ren, B. Zhang, H. Qiao, J. Comput. Appl. Math. 110, 15 (1999)
• [2] C. Schneider, Integr. Equat. Oper. Th. 2, 62 (1979)[Crossref]
• [3] C. Schneider, Math. Comp. 36, 207 (1981)
• [4] S. Xu, X. Ling,C. Cattani, G.N. Xie, X.J. Yang, Y. Zhao, Math.Probl. Eng. 2014, 914725 (2014)
• [5] A.A. Badr, J. Comput. Appl. Math. 134, 191 (2001)
• [6] R. Estrada, Ram P. Kanwal, Singular Integral Equations(Birkhauser, Boston, 2000)
• [7] F.G. Tricomi, Integral Equations (Dover, New York, 1985)
• [8] N.I. Muskhelishvili, Singular Integral Equations, 2nd edition (P.Noordhoff, N.V. Groningen, Holland, 1953)
• [9] E. Babolian, A. Arzhang Hajikandi, J. Comput. Appl. Math. 235,1148 (2011)
• [10] M. Lakestani, B. Nemati Saray, M. Dehghan, J. Comput. Appl.Math. 235, 3291 (2011)
• [11] W. Jiang, M. Cui, Appl. Math. Comp. 202, 666 (2008)
• [12] A. Pedas, E. Tamme, Appl. Numeric. Math. 61, 738 (2011)
• [13] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions(Dover, New York, 1972)
• [14] M. Abramowitz, I.A. Stegun, Pocketbook of Mathematical Functions(Verlag Harri Deutsch, Germany, 1984)
• [15] G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists,5th edition (Academic Press, New York, 2001)
• [16] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products,7th edition (Academic Press, Oxford, 2007)
• [17] Philip J. Davis, Interpolation and Approximation (Dover Publications,New York, 1975)
• [18] C. Allouch, P. Sablonnière, D. Sbibih, M. Tahrichi, J. Comput.Appl. Math. 233, 2855 (2010)

Identifiers

DOI

10.1515/phys-2015-0037