Skip to main content

A New Procedures for Solving Two Classes of Fuzzy Singular Integro-Differential Equations: Airfoil Collocation Methods


This paper gives and justifies a practical approach for solving fuzzy singular integro-differential equations. First, by using different techniques, we show that solutions to two types of fuzzy singular integro-differential equations exist and are unique: Picard’s theorem for logarithmic kernels and Arzelà–Ascoli theorem for Cauchy ones. Then, utilizing airfoil polynomials, we provide a collocation method to solve the current problems numerically. We also look at the approximate equations’ solutions, and we introduce the concept of error analysis. Using new procedures, we obtain two systems of linear equations. These are the problems to be examined. Eventually, we exhibit the precision of the proposed approach via numerical examples.

This is a preview of subscription content, access via your institution.

Data Availibility

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.


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

    MathSciNet  Article  Google Scholar 

  2. Allahviranloo, T., Salehi, P., Nejatiyan, M.: Existence and uniqueness of the solution of nonlinear fuzzy Volterra integral equations. Iran. J. Fuzzy Syst. 12, 75–86 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  4. Desmarais, R. N., Bland, S. R.: Tables of properties of airfoil polynomials, Nasa reference publication 1343, (1995)

  5. Dobritoiu, M.: The study of the solution of a Fredholm-Volterra integral equation by Picard operators, Studia Universitatis Babeş-Bolyai. Mathematica 64, 551–563 (2019)

    MathSciNet  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  7. Ganji, R.M., Jafari, H., Kgarose, M., Mohammadi, A.: Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials. Alexandria Eng. J. 60, 4563–4571 (2021)

    Article  Google Scholar 

  8. Ganji, R.M., Jafari, H., Moshokoa, S.P., Nkomo, N.S.: A mathematical model and numerical solution for brain tumor derived using fractional operator. Res. Phys. 28, 104671 (2021)

    Google Scholar 

  9. Gerami, N., Fayek, S.A.R.: Computational method for fuzzy arithmetic operations on triangular fuzzy numbers by extension principle. E Internat. J. Approx. Reason. 106, 172–193 (2019)

    MathSciNet  Article  Google Scholar 

  10. Gumah, G., Al-Omari, S., Baleanu, D.: Soft computing technique for a system of fuzzy Volterra integro-differential equations in a Hilbert space. Appl. Numer. Math. 152, 310–322 (2020)

    MathSciNet  Article  Google Scholar 

  11. Jafari, H., Ganji, R.M., Nkomo, N.S., Lv, Y.P.: A numerical study of fractional order population dynamics model. Res. Phys. 27, 104456 (2021)

    Google Scholar 

  12. Jafari, H., Ganji, R.M., Sayevand, K., Baleanu, D.: A numerical approach for solving fractional optimal control problems with mittag-leffler kernel. J. Vib. Control (2021).

    Article  Google Scholar 

  13. Jafari, H., Ghorbani, M., Ebadattalab, M., Moallem, R., Baleanu, D.: Optimal Homotopy asymptotic method—A tool for solving fuzzy differential equations. J. Comput. Complex. Appl. 2, 112–123 (2016)

    Google Scholar 

  14. Luplescu, V., O’Regan, D.: A new derivative concept for set-valued and fuzzy-valued functions. Differential and integral calculus in quasilinear metric spaces. Fuzzy Sets Syst. 404, 75–110 (2021)

    MathSciNet  Article  Google Scholar 

  15. Mennouni, A.: The iterated projection method for integro-differential equations with Cauchy kernel. J. Appl. Math. Inf. Sci. 31, 661–667 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Mennouni, A.: A projection method for solving Cauchy singular integro-differential equations. Appl. Math. Lett. 25, 986–989 (2012)

    MathSciNet  Article  Google Scholar 

  17. Mennouni, A.: Airfoil polynomials for solving integro-differential equations with logarithmic kernel. Appl. Math. Comput. 218, 11947–11951 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Mennouni, A.: Improvement by projection for integro-differential equations, Mathematical Methods in the Applied Sciences, (2020)

  19. Molabahrami, A., Shidfar, A., Ghyasi, A.: An analytical method for solving linear Fredholm fuzzy integral equations of the second kind. Comput. Math. Appl. 61, 2754–2761 (2011)

    MathSciNet  Article  Google Scholar 

  20. Mosleh, M., Otadi, M.: Existence of solution of nonlinear Fuzzy Fredholm Integro-differential equations. Fuzzy Inf. Eng. 8, 17–30 (2016)

    MathSciNet  Article  Google Scholar 

  21. 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)

    MathSciNet  Article  Google Scholar 

  22. Sadatrasoul, S.M., Ezzati, R.: Numerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula. J. Comput. Appl. Math. 292, 430–446 (2016)

    MathSciNet  Article  Google Scholar 

  23. Sahu, P.K., Saha Ray, S.: A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein–Volterra delay integral equations. Fuzzy Sets Syst. 309, 131–144 (2017)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  25. Yang, H., Gong, Z.: Ill-posedness for fuzzy Fredholm integral equations of the first kind and regularization methods. Fuzzy Sets Syst. 358, 132–149 (2019)

    MathSciNet  Article  Google Scholar 

  26. Zeinali, M., Shahmorad, S.: An equivalence lemma for a class of fuzzy implicit integro-differential equations. J. Comput. Appl. Math. 327, 388–399 (2018)

  27. Ziari, S.: Towards the accuracy of iterative numerical methods for fuzzy Hammerstein–Fredholm integral equations. Fuzzy Sets Syst. 375, 161–178 (2019)

    MathSciNet  Article  Google Scholar 

Download references


The authors thank the anonymous referees for their constructive criticism and suggestions.


No funding.

Author information

Authors and Affiliations



All authors contributed equally to this manuscript.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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

Verify currency and authenticity via CrossMark

Cite this article

Araour, M., Mennouni, A. A New Procedures for Solving Two Classes of Fuzzy Singular Integro-Differential Equations: Airfoil Collocation Methods. Int. J. Appl. Comput. Math 8, 35 (2022).

Download citation

  • Accepted:

  • Published:

  • DOI:


  • Fuzzy logarithmic integro-differential equations
  • Cauchy kernel
  • Collocation method
  • Airfoil polynomials

Mathematics Subject Classification

  • 45E05
  • 45J05
  • 03E72