Skip to main content
Springer Nature Link
Log in

LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper is devoted to extend an iterative algorithm for sparse least-squares problems for solving one- and two-dimensional linear Fredholm integral equations. We consider the operator form of these equations and then develop the LSMR method for solving them in an appropriate manner. The proposed method is based on bidiagonalization process of Golub–Kahan and reducing the linear operator \({\mathcal {L}}\) to the lower bidiagonal matrix form. Convergence property of the numerical solution associated with the suggested scheme is also provided. The method is fast and very accurate in solving two-dimensional equations. Other advantage of the proposed method is its high accuracy for solving Fredholm integral equations with non-smooth solutions. Solving such problems with other numerical methods, for instance, spectral methods, are not accurate as our proposed LSMR method. Numerical results confirm this claim.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Allahviranloo T, Ghanbari M (2011) Discrete homotopy analysis method for the nonlinear Fredholm integral equations. Ain Shams Eng J 2:133–140

    Article  Google Scholar 

  • Alipanah A, Esmaeili S (2011) Numerical solution of the two-dimensional Fredholm integral equations by Gaussian radial basis function. J Comput Appl Math 235:5342–5347

    Article  MathSciNet  Google Scholar 

  • Avazzadeh Z, Heydari M (2012) Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind. Comput Appl Math 39:127–142

    Article  MathSciNet  Google Scholar 

  • Babolian E, Bazm S, Lima P (2011) Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions. Commun Nonlinear Sci Numer Simul 16:1164–1175

    Article  MathSciNet  Google Scholar 

  • Babolian E, Maleknejad K, Mordad M, Rahimi B (2011) A numerical method for solving Fredholm–Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix. J Comput Appl Math 235:3965–3971

    Article  MathSciNet  Google Scholar 

  • Babolian E, Masouri Z, Hatamzadeh-Varmazyar S (2009) A direct method for numerical solving integral equations system using orthogonal triangular functions. Int J Ind Math 1(2):135–145

    MATH  Google Scholar 

  • Babolian E, Biazar J, Vahidi AR (2004) The decomposition method applied to systems of Fredholm integral equations of the second kind. Appl Math Comput 148:443–452

    MathSciNet  MATH  Google Scholar 

  • Bazm S, Babolian E (2012) Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules. Commun Nonlinear Sci Numer Simul 17:1215–1223

    Article  MathSciNet  Google Scholar 

  • Beyrami H, Lotfi T, Mahdiani K (2016) A new efficient method with error analysis for solving the second kind Fredholm integral equation with Cauchy kernel. J Comput Appl Math 300:385–399

    Article  MathSciNet  Google Scholar 

  • Chan R, Ng MK (1996) Conjugate gradient methods for Toeplitz systems. SIAM Rev 38:427–482

    Article  MathSciNet  Google Scholar 

  • Chin-Lung Fong D, Saunders M (2011) LSMR: An iterative algorithm for sparse least-squares problems. SIAM J Sci Cmput 33:2950–2971

    Article  MathSciNet  Google Scholar 

  • Chokri C (2013) On the numerical solution of Volterra–Fredholm integral equations with Abel kernel using Legendre polynomials. Int J Adv Sci Tech Res 3:1–9

    Google Scholar 

  • Engl HW, Neubauer A (1985) An improved version of Marti’s method for solving ill-posed linear integral equations. Math Comput 45:401–416

    MathSciNet  MATH  Google Scholar 

  • Frammartino C (2013) Numerical treatment of boundary value problems by solving a system of second kind Fredholm integral equations. Calcolo 50:123–140

    Article  MathSciNet  Google Scholar 

  • Han GQ, Wang RF (2002) Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations. J Comput Appl Math 139:49–63

    Article  MathSciNet  Google Scholar 

  • Han GH, Jiong W (2001) Extrapolation of Nystrom solution for two dimensional nonlinear Fredholm integral equations. J Comput Appl Math 134:259–268

    Article  MathSciNet  Google Scholar 

  • Hanson RJ, Phillips JL (1978) Numerical solution of two-dimensional integral equations using linear elements. SIAM J Numer Anal 15:113–121

    Article  MathSciNet  Google Scholar 

  • Hetmaniok E, Nowak I, Slota D, Witula R (2013) A study of the convergence and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations. Appl Math Lett 26:165–169

    Article  MathSciNet  Google Scholar 

  • Islam S, Aziz I, Din Z (2015) Meshless methods for multivariate highly oscillatory Fredholm integral equations. Eng Anal Bound Elements 53:100–112

    Article  MathSciNet  Google Scholar 

  • Kammerer WJ, Nashed MZ (1972) Iterative methods for best approximate solutions of integral equations of the first and second kinds. J Math Anal Appl 40:547–573

    Article  MathSciNet  Google Scholar 

  • Karimi S, Jozi M (2015) A new iterative method for solving linear Fredholm integral equations using the least squares method. Appl Comput 250:744–758

    MathSciNet  MATH  Google Scholar 

  • Kirsch A (2011) An introduction to the mathematical theory of inverse problems. Springer, New York

    Book  Google Scholar 

  • Kress R (1989) Linear integral equations. Springer, Berlin

    Book  Google Scholar 

  • Liang F, Lin FR (2010) A fast numerical solution method for two dimensional Fredholm integral equations of the second kind based on piecewise polynomial interpolation. Appl Math Comput 216:3073–3088

    MathSciNet  MATH  Google Scholar 

  • Marti JT (1978) An algorithm for computing minimum norm solutions of Fredholm integral equations of the first kind. SIAM J Numer Anal 15:1071–1076

    Article  MathSciNet  Google Scholar 

  • Paige CC, Saunders MA (1982) LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans Math Softw 8:43–71

    Article  MathSciNet  Google Scholar 

  • Paige CC, Saunders MA (1982) Algorithm 583; LSQR: sparse linear equations and sparse least squares problems. ACM Trans Math Softw 8:195–209

    Article  Google Scholar 

  • Rahimi MY, Shahmorad S, Talati F, Tari A (2010) An operational method for the numerical solution of two-dimensional linear Fredholm integral equations with an error estimation. Bult Iran Math Soc 36:119–132

    MathSciNet  MATH  Google Scholar 

  • Rajan D, Chaudhuri S (2003) Simultaneous estimation of super-resolved sence and depth map from low resolution defocused observations. IEEE Trans Pattern Anal Mach Intell 25:1102–1117

    Article  Google Scholar 

  • Sahu PK, Ray SS (2016) A new approach based on semi-orthogonal B-spline wavelets for the numerical solutions of the system of nonlinear Fredholm integral equations of second kind. Comput Appl Math 33:859–872

    Article  MathSciNet  Google Scholar 

  • Sahu PK, Ray SS (2015) Hybrid Legendre Block-Pulse functions for the numerical solutions of system of nonlinear Fredholm Hammerstein integral equations. Appl Math Comput 270:871–878

    MathSciNet  MATH  Google Scholar 

  • Tang T, Xu X, Chen J (2008) On spectral methods for Volterra type integral equations and the convergence analysis. J Comput Math 26:825–837

    MathSciNet  MATH  Google Scholar 

  • Tari A, Shahmorad S (2008) A computational method for solving two-dimensional linear Fredholm integral equations of the second kind. ANZIAM J 49:543–549

    Article  MathSciNet  Google Scholar 

  • Tohidi E (2014) Taylor matrix method for solving linear two-dimensional Fredholm integral equations with piecewise intervals. Comput Appl Math 34:1117–1130

    Article  MathSciNet  Google Scholar 

  • Xie WJ, Lin FR (2009) A fast numerical solution method for two dimensional Fredholm integral equations of the second kind. Appl Numer Math 59:1709–1719

    Article  MathSciNet  Google Scholar 

  • Yousefi SA, Lotfi A, Dehghan M (2009) He’s variational iteration method for solving nonlinear mixed Volterra–Fredholm integral equations. Comput Math Appl 58:2172–2176

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very much indebted to the referees for their constructive comments and suggestions which greatly improved the original manuscript of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Science and Computer, Kharazmi University, Tehran, Iran

    Z. Asgari & E. Babolian

  2. Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    F. Toutounian

  3. Department of Mathematics, Kosar University of Bojnord, P. O. Box 9415615458, Bojnord, Iran

    E. Tohidi

Authors
  1. Z. Asgari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Toutounian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Babolian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Tohidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Toutounian.

Additional information

Communicated by Hui Liang.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Asgari, Z., Toutounian, F., Babolian, E. et al. LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations. Comp. Appl. Math. 38, 135 (2019). https://doi.org/10.1007/s40314-019-0903-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-019-0903-8

Keywords

Mathematics Subject Classification

Search

Navigation

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy