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

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Abstract

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.

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References

  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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). https://doi.org/10.1177/10775463211016967

    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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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  1. Department of Mathematics, LTM, University of Batna 2, Mostefa Ben Boulaïd, Algeria

    Meriem Araour & Abdelaziz Mennouni

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  1. Meriem Araour
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  2. Abdelaziz Mennouni
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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). https://doi.org/10.1007/s40819-022-01245-0

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