Skip to main content
Springer Nature Link
Log in

A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains

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

Abstract

In this article, we study a kernel-based method to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. The radial kernels are utilized as a basis in the discrete collocation method to reduce the solution of linear integral equations to that of a linear system of algebraic equations. Integrals appeared in the scheme are approximately computed by the Gauss–Legendre and Monte Carlo quadrature rules. The method does not require any background mesh or cell structures, so it is mesh free and accordingly independent of the domain geometry. Thus, for the three-dimensional linear Fredholm integral equation, an irregular domain can be considered. The convergence analysis is also given for the method. Finally, numerical examples are presented to show the efficiency and accuracy of the technique.

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  • Aronszajn N (1950) Theory of reproducing kernels. Trans Am Math Soc 68:337–404

    Article  MathSciNet  Google Scholar 

  • Assari P, Adibi H, Dehghan M (2013a) A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method. Appl Math Mod 37:9269–9294

    Article  MathSciNet  Google Scholar 

  • Assari P, Adibi H, Dehghan M (2013b) A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis. J Comput Appl Math 239:72–92

    Article  MathSciNet  Google Scholar 

  • Atkinson K (1997) The numerical solution of integral equations of the second kind, vol 4. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Atkinson K, Potra F (1989) The discrete Galerkin method for linear integral equations. IMA J Numer Anal 9:385–403

    Article  MathSciNet  Google Scholar 

  • Atkinson K, Graham I, Sloan I (1983) Piecewise continuous collocation for integral equations. SIAM J Numer Anal 20:172–186

    Article  MathSciNet  Google Scholar 

  • Boersma J, Danicki E (1993) On the solution of an integral equation arising in potential problems for circular and elliptic disks. SIAM J Appl Math 53:931–941

    Article  MathSciNet  Google Scholar 

  • Bremer J, Rokhlin V, Sammis I (2010) Universal quadratures for boundary integral equations on two-dimensional domains with corners. J Comput Phys 229:8259–8280

    Article  MathSciNet  Google Scholar 

  • Cheng G, Shcherbakov V (2018) Anisotropic radial basis function methods for continental size ice sheet simulations. J Comput Phys 372:161–177

    Article  MathSciNet  Google Scholar 

  • Farengo R, Lee YC, Guzdar PN (1983) An electromagnetic integral equation: application to microtearing modes. Phys Fluids 26:3515–3523

    Article  Google Scholar 

  • Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. Interdisciplinary mathematical sciences. World Scientific Publishing Company, Singapore

    Book  Google Scholar 

  • Foy BH, Perré P, Turner I (2017) The meshfree finite volume method with application to multi-phase porous media models. J Comput Phys 333:369–386

    Article  MathSciNet  Google Scholar 

  • Golbabai A, Seifollahi S (2006) Numerical solution of the second kind integral equations using radial basis function networks. Appl Math Comput 174:877–883

    MathSciNet  MATH  Google Scholar 

  • Golbabai A, Seifollahi S (2007) Radial basis function networks in the numerical solution of linear integro-differential equations. Appl Math Comput 188:427–432

    MathSciNet  MATH  Google Scholar 

  • Golbabai A, Mammadov M, Seifollahi S (2009) Solving a system of nonlinear integral equations by an RBF network. Comput Math Appl 57:1651–1658

    Article  MathSciNet  Google Scholar 

  • Han G, Wang J (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 

  • Hatamzadeh-Varmazyar S, Masouri Z (2011) Numerical method for analysis of one- and two-dimensional electromagnetic scattering based on using linear Fredholm integral equation models. Math Comput Mod 54:2199–2210

    Article  MathSciNet  Google Scholar 

  • Jerri AJ (1999) Introduction to integral equations with applications. Wiley, Toronto

    MATH  Google Scholar 

  • Kansa EJ (1990a) Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics-I. Comput Math Appl 19:127–145

    Article  MathSciNet  Google Scholar 

  • Kansa EJ (1990b) Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics-II. Comput Math Appl 19:147–161

    Article  MathSciNet  Google Scholar 

  • Kazemi Z, Hematiyan MR, Vaghefi R (2017) Meshfree radial point interpolation method for analysis of viscoplastic problems. Eng Anal Bound Elem 82:172–184

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  • Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55:1–34

    Article  Google Scholar 

  • Li XF, Rong EQ (2002) Solution of a class of two-dimensional integral equations. J Comput Appl Math 145:335–343

    Article  MathSciNet  Google Scholar 

  • Liu GR, Gu YT (2005) An introduction to meshfree methods and their programming. Springer, Dordrech

    Google Scholar 

  • Manzhirov AV (1985) On a method of solving two-dimensional integral equations of axisymmetric contact problems for bodies with complex rheology. J Appl Math Mech 49:777–782

    Article  MathSciNet  Google Scholar 

  • Micchelli CA (1986) Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr Approx 2:11–22

    Article  MathSciNet  Google Scholar 

  • Mirkin MV, Bard AJ (1992) Multidimensional integral equations: a new approach to solving microelectrode diffusion problems. J Electroanal Chem 323:29–51

    Article  Google Scholar 

  • Parand K, Rad JA (2012) Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions. Appl Math Comput 218:5292–5309

    MathSciNet  MATH  Google Scholar 

  • Quarteroni A, Sacco R, Saleri F (2007) Numerical mathematics, 2nd edn. Texts in applied mathematics. Springer, New York

    Book  Google Scholar 

  • Radlow J (1964) A two-dimensional singular integral equation of diffraction theory. Bull Am Math Soc 70:596–599

    Article  MathSciNet  Google Scholar 

  • Rastkar S, Zahedi M, Korolev I, Agarwal A (2017) A meshfree approach for homogenization of mechanical properties of heterogeneous materials. Eng Anal Bound Elem 75:79–88

    Article  MathSciNet  Google Scholar 

  • Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 39:811–841

    Article  MathSciNet  Google Scholar 

  • Stroud AH (1971) Approximation calculation of multiple integrals. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  • Wendland H (2005) Scattered data approximation, vol 17. Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran

    Hamid Esmaeili

  2. Department of Mathematics, Faculty of Mathematics Science and Statistics, Malayer University, Malayer, Iran

    Davoud Moazami

Authors
  1. Hamid Esmaeili
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Davoud Moazami
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Hamid Esmaeili.

Additional information

Communicated by Hui Liang.

Publisher's Note

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

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Esmaeili, H., Moazami, D. A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. Comp. Appl. Math. 38, 181 (2019). https://doi.org/10.1007/s40314-019-0959-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-019-0959-5

Keywords

Mathematics Subject Classification