Skip to main content
SpringerLink
Log in

Block-Pulse Functions in the Method of Successive Approximations for Nonlinear Fuzzy Fredholm Integral Equations

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we propose a method of successive approximations for nonlinear fuzzy Fredholm integral equations of the second kind. The main approximation tool is based on fuzzy block-pulse functions. The error estimation of the proposed method is established. A number of illustrative examples that demonstrate accuracy and convergence is given as well.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbasbandy, S., Allahviranloo, T.: The Adomian decomposition method applied to the fuzzy system of Fredholm integral equations of the second kind. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 14, 101–110 (2006)

    Article  MathSciNet  Google Scholar 

  2. Abbasbandy, S., Babolian, E., Alavi, M.: Numerical method for solving linear Fredholm fuzzy integral equations of the second kind. Chaos Solitons Fractals 31(1), 138–146 (2007)

    Article  MathSciNet  Google Scholar 

  3. Babolian, E., Sadeghi Goghary, H., Abbasbandy, S.: Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method. Appl. Math. Comput. 161, 733–744 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Baghmisheh, M., Ezzati, R.: Numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using hybrid of block-pulse functions and Taylor series. Adv. Differ. Equ. NY. 2015, 51 (2015). https://doi.org/10.1186/s13662-015-0389-7

    Article  MATH  Google Scholar 

  5. Baghmisheh, M., Ezzati, R.: Error estimation and numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using triangular functions. J. Intell. Fuzzy Syst. 30(2), 639–649 (2016)

    Article  Google Scholar 

  6. Balachandran, K., Prakash, P.: Existence of solutions of nonlinear fuzzy Volterra- Fredholm integral equations. Indian J. Pure Appl. Math. 33, 329–343 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Balachandran, K., Kanagarajan, K.: Existence of solutions of general nonlinear fuzzy Volterra–Fredholm integral equations. J. Appl. Math. Stoch. Anal. 3, 333–343 (2005)

    Article  MathSciNet  Google Scholar 

  8. Balasubramaniam, P., Muralisankar, S.: Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integro-differential equations. Appl. Math. Lett. 14, 455–462 (2001)

    Article  MathSciNet  Google Scholar 

  9. Bede, B., Gal, S.G.: Quadrature rules for integrals of fuzzy-number-valued functions. Fuzzy Sets Syst. 145, 359–380 (2004)

    Article  MathSciNet  Google Scholar 

  10. Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer, Berlin (2013)

    Book  Google Scholar 

  11. Bica, A.M.: Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations. Inf. Sci. 178, 1279–1292 (2008)

    Article  MathSciNet  Google Scholar 

  12. Bica, A.M., Popescu, C.: Approximating the solution of nonlinear Hammerstein fuzzy integral equations. Fuzzy Sets Syst. 245, 1–17 (2014)

    Article  MathSciNet  Google Scholar 

  13. Ezzati, R., Ziari, S.: Numerical solution and error estimation of fuzzy Fredholm integral equation using fuzzy Bernstein polynomials. Aust. J. Basic Appl. Sci. 5(9), 2072–2082 (2011)

    Google Scholar 

  14. Ezzati, R., Ziari, S.: Numerical solution of nonlinear fuzzy Fredholm integral equations using iterative method. Appl. Math. Comput. 225, 33–42 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Fang, J.-X., Xue, Q.-Y.: Some properties of the space fuzzy-valued continuous functions on a compact set. Fuzzy Sets Syst. 160, 1620–1631 (2009)

    Article  MathSciNet  Google Scholar 

  16. Fariborzi Araghi, M.A., Parandin, N.: Numerical solution of fuzzy Fredholm integral equations by the Lagrange interpolation based on the extension principle. Soft Comput. 15, 2449–2456 (2011)

    Article  Google Scholar 

  17. Friedman, M., Ma, M., Kandel, A.: Numerical solutions of fuzzy differential and integral equations. Fuzzy Sets Syst. 106, 35–48 (1999)

    Article  MathSciNet  Google Scholar 

  18. Friedman, M., Ma, M., Kandel, A.: Solutions to fuzzy integral equations with arbitrary kernels. Int. J. Approx. Reason. 20, 249–262 (1999)

    Article  MathSciNet  Google Scholar 

  19. Gal, S.G.: Approximation theory in fuzzy setting. In: Anastassiou, G.A. (ed.) Handbook of Analytic-Computational Methods in Applied Mathematics, pp. 617–666. Chapman & Hall/CRC Press, Boca Raton (2000)

    Google Scholar 

  20. Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18, 31–43 (1986)

    Article  MathSciNet  Google Scholar 

  21. Jiang, Z.H., Schanfelberger, W.: Block-Pulse Functions and Their Applications in Control Systems. Springer, Berlin (1992)

    Book  Google Scholar 

  22. Kaleva, O.: The calculus of fuzzy valued functions. Appl. Math. Lctt. 3(2), 55–59 (1990)

    Article  MathSciNet  Google Scholar 

  23. Nanda, S.: On sequences of fuzzy numbers. Fuzzy Sets Syst. 33(1), 123–126 (1989)

    Article  MathSciNet  Google Scholar 

  24. Nieto, J.J., Rodriguez-Lopez, R.: Bounded solutions for fuzzy differential and integral equations. Chaos Soliton Fractals 27(5), 1376–1386 (2006)

    Article  MathSciNet  Google Scholar 

  25. Parandin, N., Fariborzi Araghi, M.A.: The numerical solution of linear fuzzy Fredholm integral equations of the second kind by using finite and divided differences methods. Soft Comput. 15, 729–741 (2010)

    Article  Google Scholar 

  26. Park, J.Y., Han, H.K.: Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets Syst. 105, 481–488 (1999)

    Article  MathSciNet  Google Scholar 

  27. Park, J.Y., Jeong, J.U.: On the existence and uniqueness of solutions of fuzzy Volttera–Fredholm integral equations. Fuzzy Sets Syst. 115, 425–431 (2000)

    Article  Google Scholar 

  28. Subrahmanyam, P.V., Sudarsanam, S.K.: A note on fuzzy Volterra integral equations. Fuzzy Sets Syst. 81, 237–240 (1996)

    Article  MathSciNet  Google Scholar 

  29. Wu, C., Gong, Z.: On Henstock integral of fuzzy-number-valued functions (I). Fuzzy Sets Syst. 120, 523–532 (2001)

    Article  MathSciNet  Google Scholar 

  30. Ziari S., Ezzati R., Abbasbandy S.: Numerical solution of linear fuzzy fredholm integral equations of the second kind using fuzzy haar wavelet. In: Greco S., Bouchon-Meunier B., Coletti G., Fedrizzi M., Matarazzo B., Yager R.R. (eds.) Advances in computational intelligence. IPMU 2012. Communications in Computer and Information Science, vol 299, pp. 79–89. Springer, Berlin, Heidelberg (2012)

  31. Ziari, S., Bica, A.M.: New error estimate in the iterative numerical method for non-linear fuzzy Hammerstein–Fredholm integral equations. Fuzzy Sets Syst. 295, 136–152 (2016)

    Article  Google Scholar 

  32. Ziari, S.: Towards the accuracy of iterative numerical methods for fuzzy Hammerstein-Fredholm integral equations. Fuzzy Sets Syst. (2018). https://doi.org/10.1016/j.fss.2018.09.006

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of Irina Perfilieva has been partially supported by the Grant Agency of the Czech Republic (GACR), Project no. 18-06915S.

Author information

Authors and Affiliations

  1. Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

    Shokrollah Ziari

  2. Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Ostrava, Czech Republic

    Irina Perfilieva

  3. Department of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran

    Saeid Abbasbandy

Authors
  1. Shokrollah Ziari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irina Perfilieva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saeid Abbasbandy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shokrollah Ziari.

Additional information

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

Ziari, S., Perfilieva, I. & Abbasbandy, S. Block-Pulse Functions in the Method of Successive Approximations for Nonlinear Fuzzy Fredholm Integral Equations. Differ Equ Dyn Syst 30, 995–1009 (2022). https://doi.org/10.1007/s12591-019-00482-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-019-00482-y

Keywords

Search

Navigation