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Approximate Solution of Two-dimensional Fredholm Integral Equation of the First Kind Using Wavelet Base Method

  • Fariba FattahzadehEmail author
Original Paper
  • 24 Downloads

Abstract

A numerical direct method for solving two-dimensional linear and nonlinear Fredholm integral equations of the first kind based on Haar wavelet is introduced. The main characteristic of the method is that, unlike several other methods, it does not involve numerical integration, which leads to higher accuracy and quick computations as well. Further more an estimation of error bound for the present method is proved. Finally several test problems have been solved and compared with existing recent methods in order to demonstrate the effectiveness and applicability of the proposed technique.

Keywords

Linear and nonlinear two-dimensional integral equations (2DIE) Haar wavelet Regularity method 

Mathematics Subject Classification

45B05 45G99 65R20 

Notes

References

  1. 1.
    Lu, Y., shen, L., Xu, Y.: Integral equation models for image restoration: high accuracy methods and fast algorithms. Inverse Probl. 26(045006), 32 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Babolian, E., Fattahzadeh, F.: Numerical computation method in solving integral equations by using chebyshev wavelet operational matrix of integration. Appl. Math. Comput. 188, 1016–1022 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Babolian, E., Fattahzadeh, F., Golpar Raboky, E.: A chebyshev approximation for solving nonlinear integral equations of Hammerstem type. Appl. Math. Comput. 189, 641–646 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fattahzadeh, F., Golpar Raboky, E.: Multivariable and scattered data interpolation for solving multivatiable integral equations. J. Prime Res. Math. 8, 51–60 (2013)zbMATHGoogle Scholar
  5. 5.
    Babolian, E., Abbadbandy, S., Fattahzadeh, F.: A numerical method for solving a class of functional and two dimensional integral equations. Appl. Math. Comput. 198, 35–43 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Baker, C.: The Numerical Treatment of Integral Equations. Clarendon Press, Oxford (1978)Google Scholar
  7. 7.
    Atkinson, K.-E.: The Numerical Solution of Integral Equations of Second Kind. Cambridge University Press, Cambridge (2011)Google Scholar
  8. 8.
    Xie, W., Lin, F.-R.: A fast numerical solution method for two dimensional Fredholm integral equations of the second kind. Appl. Numer. Math. 59, 1709–1719 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bazm, S., Babolian, E.: Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using gauss product quadrature rules. Commun. Nonlinear Sci. Numer. Simult. 17, 1215–1223 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Han, G., Wang, R.: Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations. J. Comput. Appl. Math. 139, 49–63 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Malenknejad, K., Jafari Behbahanim, Z.: Application of two-dimensional triangular functions for solving nonlinear class of mixed Volterra-Fredholm integral equations. Math. Comput. Mode 55, 1833–1844 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Babolian, E., Maleknejad, K., Roodaki, M., Almasieh, H.: Two dimensional triangular functions and their applications to nonlinear 2D Volterra-Fredholm equations. Comput. Math. Appl. 60, 1711–1722 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nemati, S., Lima, P., Ordokhani, Y.: Numerical solution of a class of two-dimensional nonlinear volterra integral equations using legender polynomials. J. Comput. Appl. Math. 242, 53–69 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Assari, P., Adibi, H., Dehghal, M.: A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domins using radial basis functions with error analysis. J. Comput. Appl. Math. 239, 72–92 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Reihani, M.H., Abadi, Z.: Rationalized Haar functions method for solving Fredholm and Volterra integral equations. J. Comput. Appl. Math. 200, 12–20 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Maleknejad, K., Rostami, Y., Kalalagh, H.: Numerical solution for first kind integral equations by using sinc collocation method. Int. J. Appl. Phys. Math. 6(3), 120–128 (2016)CrossRefGoogle Scholar
  17. 17.
    Maleknejad, K., Saeedipoor, E., Dehbozorgi, R.: Legendre wavelet direct method for the numerical solution of Fredholm Integral equation of the first kind. In: Proceedings of the world congress on Engineering, vol. 1, WCE 2016, London, UK (2016)Google Scholar
  18. 18.
    Torabi, S.M., Tari, A.: Numerical solution of two-dimensional IE of the first kind by multi-step method. Comput. Method Differ. Equ. 4(2), 128–138 (2016)zbMATHGoogle Scholar
  19. 19.
    Tahami, M., Askari Hemmat, A., Yousefi, S.A.: Numerical solution of two-dimensional First kind Fredholm IE using linear Legendre wavelet. Int. J. Wavel. Multiresolution Inf. Process. 14(01), 1650004 (2016)CrossRefGoogle Scholar
  20. 20.
    Alturk, A.: On multidimensional Fredholm IE of the first kind. J. Inequal. Spec. Funct. 8(4), 65–95 (2017)MathSciNetGoogle Scholar
  21. 21.
    Aziz, I., Siraj-ul-Islam, K.F.: A new method based Haar wavelet numerical solution of two-dimensional nonlinear integral equations. J. Comput. Appl. Math. 272, 70–80 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lee, B., Tamg, Y.S.: Application of the discrete wavelet transform to the monitoring of tools failure in end miling using the spindle motor current. Int. J. Adv. Manuf. Technol. 15(4), 238–243 (1999)CrossRefGoogle Scholar
  23. 23.
    Lin, E., Zhou, X.: Coiflet Interpolation and approximation solutions of elliptic partial differential equations. Number Methods Partial Differ. Equ. 13, 303–320 (1997)CrossRefGoogle Scholar
  24. 24.
    Jonca, K.K.: Numerical solution of a nonlinear Fredholm integral equation of the first kinf, Ph.D. thesis, Mathematics, Montana state University (1988)Google Scholar
  25. 25.
    Tari, A., shahmorad, S.: On the existence and uniqueness of solution of the two dimensional linear integral equations of the second kind. In: 40th Annul Iranian Mathematics conference. 17–20 August 2009. Sharif University of Technology, Tehran, Iran, pp. 462–467 (2006)Google Scholar
  26. 26.
    Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-posed Problems. Springer, Dordrecht (1995)CrossRefGoogle Scholar
  27. 27.
    Bertero, M., Brianzi, P., Pike, E.: Super resolution is confocal scanning mecroscopy. Inverse Probl. 3, 195–212 (1987)CrossRefGoogle Scholar
  28. 28.
    Ziyaee, F., Tari, A.: Regularization method for the two-dimensional Fredholm integral equations of the first kind. Int. J. Nonlinear Sci. 18(3), 189–194 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Alturk, A.: The regularization-homotopy method for the two-dimensional Fredholm integral equations of the first kind. Math. Comput. Appl. 21, 9 (2016).  https://doi.org/10.3390/mca21020009 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lin, E., Al-Jarrah, Y.: Wavelet based method for numerical solutions of two dimensional integral equations. Mathematica Aeterna 4(8), 839–853 (2014)Google Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsCentral Tehran Branch, Islamic Azad UniversityTehranIran

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