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

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Abstract

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.

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References

  • Allahviranloo T, Khezerloo M, Ghanbari M, Khezerloo S (2010) The homotopy perturbation method for fuzzy Volterra integral equations. Int J Comput Cognit 8:31–37

    Google Scholar 

  • Diamond P (2002) Theory and applications of fuzzy Volterra integral equations. IEEE Trans Fuzzy Syst 10:97–102

    Article  Google Scholar 

  • Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626

    Article  MathSciNet  Google Scholar 

  • Eman AH, Ayad AW (2013) Homotopy analysis method for solving nonlinear fuzzy integral equations. Int J Appl Math 28:2051–5227

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Goestscel R, Voxman W (1986) Elementary Fuzzy calculus. Fuzzy Sets Syst 18:31–34

    Article  MathSciNet  Google Scholar 

  • Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317

    Article  MathSciNet  Google Scholar 

  • Kaleva O (2006) A note on fuzzy differential equations. Nonlinear Anal 64:895–900

    Article  MathSciNet  Google Scholar 

  • Lakshmikantham V, Mohapatra RN (2003) Theory of fuzzy differential equations and inclusions. Taylor and Francis, London

    Book  Google Scholar 

  • Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst 105:133–138

    Article  MathSciNet  Google Scholar 

  • Mosleh M, Otadi M (2013) Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis. J Adv Inf Technol 4:148–155

    Google Scholar 

  • 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 5:387

    Article  Google Scholar 

  • Otadi M, Mosleh M (2015) Numerical solution of fuzzy Volterra integral equation of the first kind. Mat Inverse Probl 2:1–15

    MATH  Google Scholar 

  • Park JY, Han HK (1999) Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets Syst 105:481–488

    Article  MathSciNet  Google Scholar 

  • Puri ML, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114:409–422

    Article  MathSciNet  Google Scholar 

  • Salahshour S, Allahviranloo T (2013) Application of fuzzy differential transform method for solving fuzzy Volterra integral equations. Appl Math Model 37:1016–1027

    Article  MathSciNet  Google Scholar 

  • Salehi P, Nejatiyan M (2011) Numerical method for nonlinear fuzzy Volterra integral equations of the second kind. Int J Ind Math 3:169–179

    Google Scholar 

  • Shafiee M, Abbasbandy S, Allahviranloo T (2011) Predictor–corrector method for nonlinear fuzzy Volterra integral equations. Aust J Basic Appl Sci 5:2865–2874

    Google Scholar 

  • Sikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 159:319–330

    Article  MathSciNet  Google Scholar 

  • 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 225:631–637

    MathSciNet  MATH  Google Scholar 

  • Wazwaz AM (2011) Linear and nonlinear integral equations: methods and applications. Higher education, Beijing. Springer, Berlin

    Book  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  Google Scholar 

  • 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. https://doi.org/10.1109/ICMLC.2009.5212439

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Correspondence to S. M. Karbassi.

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Saberirad, F., Karbassi, S.M. & Heydari, M. Numerical solution of nonlinear fuzzy Volterra integral equations of the second kind for changing sign kernels. Soft Comput 23, 11181–11197 (2019). https://doi.org/10.1007/s00500-018-3668-x

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