Soft Computing

, Volume 23, Issue 21, pp 11181–11197 | Cite as

Numerical solution of nonlinear fuzzy Volterra integral equations of the second kind for changing sign kernels

  • F. Saberirad
  • S. M. KarbassiEmail author
  • M. Heydari
Methodologies and Application


In this research, a numerical iterative method based on the trapezoidal quadrature rule to solve the nonlinear fuzzy Volterra integral equations of the second kind (NFVIEs-2) with changing sign kernels is proposed. Moreover, the convergence analysis of this method is investigated in detail. Several numerical examples are given, and the numerical results are reported to show the validity and efficiency of the proposed method.


Nonlinear fuzzy Volterra integral equations (NFVIEs) Trapezoidal quadrature formula Convergence analysis 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human participants

This article does not contain any studies with animals or with human participants.


  1. Allahviranloo T, Khezerloo M, Ghanbari M, Khezerloo S (2010) The homotopy perturbation method for fuzzy Volterra integral equations. Int J Comput Cognit 8:31–37Google Scholar
  2. Diamond P (2002) Theory and applications of fuzzy Volterra integral equations. IEEE Trans Fuzzy Syst 10:97–102CrossRefGoogle Scholar
  3. Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626MathSciNetCrossRefGoogle Scholar
  4. Eman AH, Ayad AW (2013) Homotopy analysis method for solving nonlinear fuzzy integral equations. Int J Appl Math 28:2051–5227Google Scholar
  5. Friedman M, Ma M, Kandel A (1999) Numerical solution of fuzzy differential and integral equations. Fuzzy Set Syst 106:35–48MathSciNetCrossRefGoogle Scholar
  6. Goestscel R, Voxman W (1986) Elementary Fuzzy calculus. Fuzzy Sets Syst 18:31–34MathSciNetCrossRefGoogle Scholar
  7. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317MathSciNetCrossRefGoogle Scholar
  8. Kaleva O (2006) A note on fuzzy differential equations. Nonlinear Anal 64:895–900MathSciNetCrossRefGoogle Scholar
  9. Lakshmikantham V, Mohapatra RN (2003) Theory of fuzzy differential equations and inclusions. Taylor and Francis, LondonCrossRefGoogle Scholar
  10. Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst 105:133–138MathSciNetCrossRefGoogle Scholar
  11. Mosleh M, Otadi M (2013) Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis. J Adv Inf Technol 4:148–155Google Scholar
  12. Narayanamoorthy S, Sathiyapriya SP (2016) Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind. SpringerPlus 5:387CrossRefGoogle Scholar
  13. Otadi M, Mosleh M (2015) Numerical solution of fuzzy Volterra integral equation of the first kind. Mat Inverse Probl 2:1–15zbMATHGoogle Scholar
  14. Park JY, Han HK (1999) Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets Syst 105:481–488MathSciNetCrossRefGoogle Scholar
  15. Puri ML, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114:409–422MathSciNetCrossRefGoogle Scholar
  16. Salahshour S, Allahviranloo T (2013) Application of fuzzy differential transform method for solving fuzzy Volterra integral equations. Appl Math Model 37:1016–1027MathSciNetCrossRefGoogle Scholar
  17. Salehi P, Nejatiyan M (2011) Numerical method for nonlinear fuzzy Volterra integral equations of the second kind. Int J Ind Math 3:169–179Google Scholar
  18. Shafiee M, Abbasbandy S, Allahviranloo T (2011) Predictor–corrector method for nonlinear fuzzy Volterra integral equations. Aust J Basic Appl Sci 5:2865–2874Google Scholar
  19. Sikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 159:319–330MathSciNetCrossRefGoogle Scholar
  20. Wang K, Wang Q, Guan K (2013) Iterative method and convergence analysis for a kind of mixed nonlinear Volterra–Fredholm integral equation. Appl Math Comput 225:631–637MathSciNetzbMATHGoogle Scholar
  21. Wazwaz AM (2011) Linear and nonlinear integral equations: methods and applications. Higher education, Beijing. Springer, BerlinCrossRefGoogle Scholar
  22. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefGoogle Scholar
  23. Zhang DK, Liu XJ, Zhou CJ, Qiu JQ (2009) Numerical solutions of fuzzy Volterra integral equations by Characterization theorem. In: Proceedings of the eighth international conference on machine learning and cybernetics. Baoding.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsYazd UniversityYazdIran

Personalised recommendations