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A neural network approach for solving Fredholm integral equations of the second kind

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Abstract

In this paper, a novel method based on feed-forward neural networks is presented for solving Fredholm integral equations of the second kind. In the present approach, we first approximate the unknown function based on neural networks, then substitute the approximate function in the appropriate error function of the integral equation, and finally train the network with as few neurons as necessary to achieve the desired accuracy. This novel method, in comparison with Harr function and Bernstein polynomials methods, shows that the use of neural networks provides solutions with very good generalizations and higher accuracy. The proposed method is illustrated by several examples.

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References

  1. Bôcher M (1914) Integral equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  2. Doncheski MA, Robinett RW (2002) Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals. Ann Phys 299:208–227

    Article  MathSciNet  MATH  Google Scholar 

  3. Effati S, Pakdaman M (2010) Artificial neural network approach for solving fuzzy differential equations. Information Sciences 180:1434–1457

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Golbabai A, Seifollahi S (2006) Numerical solution of the second kind integral equations using radial basis function networks. Appl Math Comput 181:903–907

    Article  MathSciNet  MATH  Google Scholar 

  6. Golbabai A, Seifollahi S (2006) An iterative solution for the second kind integral equations using radial basis functions. Appl Math Comput 181:903–907

    Article  MathSciNet  MATH  Google Scholar 

  7. Hagan MT, Menhaj MB (1994) Training feedforward neural network with the Marquardt algorithm. IEEE Trans Neural Netw 5(6):989–993

    Google Scholar 

  8. Haykin S (1999) Neural Networks: A Comprehensive Foundation, 2nd edn. Prentice-Hall, New York

    MATH  Google Scholar 

  9. Hornick K, Stinchcombe M, White H (1989) Multi-layer feed forward networks are universal approximators. Neural Networks 2(5):359–366

    Article  Google Scholar 

  10. Hristev AM (1998) Artificial neural networks. The GNU Public License, ver. 2. 1st edn

  11. Jerri AJ (1932) Introduction to integral equations with applications. Clarkson University Press, Potsdam

    Google Scholar 

  12. Kecman V (2001) Learning and soft computing, Massachusetts institute of technology. MIT Press, New York

    Google Scholar 

  13. Kincaid D, Cheney W (1991) Numerical analysis. Brooks/Cole Publishing Company, Pacific Grove

    MATH  Google Scholar 

  14. Krose B, Vander Smagt P (1996) An introduction to neural networks. Amsterdam University Press, Amsterdam

    Google Scholar 

  15. Lagaris IE, Likas A (1998) Artificial neural networks for solving ordinary and partial differential equations, IEEE. Trans Neural Networks 9(5):987–1000

    Google Scholar 

  16. Lapedes A, Farber R (1988) How neural nets work? In: Anderson DZ (eds), Neural information processing systems. AIP, New York, pp 442–456

  17. Luenberger DG (1984) Linear and nonlinear programming. 2nd edn. Addison-wesley, Reading

    MATH  Google Scholar 

  18. Reihani MH, Abadi Z (2007) Rationalized Haar functions method for solving Fredholm and Volterra integral equations. J Comp Appl Math 200:12–20

    Article  MathSciNet  MATH  Google Scholar 

  19. Shirin A, Islam MS (2010) Numerical solutions of fredholm integral equations using bernstein polynomials. J Sci Res 2(2):264–272

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

    Sohrab Effati & Reza Buzhabadi

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  1. Sohrab Effati
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  2. Reza Buzhabadi
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Correspondence to Sohrab Effati.

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Effati, S., Buzhabadi, R. A neural network approach for solving Fredholm integral equations of the second kind. Neural Comput & Applic 21, 843–852 (2012). https://doi.org/10.1007/s00521-010-0489-y

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  • DOI: https://doi.org/10.1007/s00521-010-0489-y

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