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The use of airfoil and Chebyshev polynomials methods for solving fuzzy Fredholm integro-differential equations with Cauchy kernel

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Abstract

In this paper the airfoil and the Chebyshev polynomials methods to solve the fuzzy Fredholm integro-differential equation with Cauchy kernel under generalized \(H\)-differentiability are discussed. The existence and uniqueness of the solution and convergence of the proposed methods are proved in details. Finally some examples show the accuracy of these methods.

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References

  • Abbasbandy S, Allahviramloo T (2002) Numerical solution of fuzzy differential equations by taylor method. Comput Methods Appl Math 2:113–124

    Article  MathSciNet  Google Scholar 

  • Alikhani R, Bahrami F, Jabbari A (2012) Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations. Nonlinear Anal 75:1810–1821

    Article  MATH  MathSciNet  Google Scholar 

  • Allahviranloo T (2005) The Adomian decomposition method for fuzzy system of linear equations. Appl Math Comput 163:553–563

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Allahviranloo T, Amirteimoori A, Khezerloo M, Khezerloo S (2011a) A new method for solving fuzzy Volterra integro-differential equations. Austral J Basic Appl Sci 5:154–164

    Google Scholar 

  • Allahviranloo T, Khalilzadeh N, Khezerloo S (2011b) Solving linear Fredholm fuzzy integral equations of the second kind by modified trapezoidal method. J Appl Math Islamic Azad Univ Lahijan 7:25–37

    Google Scholar 

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

    Article  MATH  Google Scholar 

  • Attari H, Yazdani A (2011) A computational method for fuzzy Volterra–Fredholm integral equations. Fuzzy Inf Eng 2:147–156

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Bede B, Gal SG (2005) Generalizations of differentiability of fuzzy-number valued function with application to fuzzy differential equations. Fuzzy Sets Syst 151:581–599

    Article  MATH  MathSciNet  Google Scholar 

  • Behzadi ShS (2011) Solving fuzzy nonlinear Volterra–Fredholm integral equations by using homotopy analysis and adomian decomposition methods. J Fuzzy Set Valued Anal 2011:1–13

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Chalco-Cano Y, Romn-Flores H (2006) On new solutions of fuzzy differential equations. Chaos Solit Fract 38:1016–1043

    Google Scholar 

  • Chang SSL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Man Cybern 2:30–34

    Article  MATH  MathSciNet  Google Scholar 

  • Congxin W, Ma M (1991) On embedding problem of fuzzy number spaces. Fuzzy Sets Syst 44:33–38

    Article  MATH  Google Scholar 

  • Delves LM, Mohamed JL (1983) Computation method for integral equations. Cambridge University Press, Cambridge

    Google Scholar 

  • Desmarais RN, Bland SR (1995) Tables of properties of airfoil polynomials. Publication, Nasa Reference 1343

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Dubois D, Prade H (1980) Theory and application, fuzzy sets and systems. Academic Press, London

    Google Scholar 

  • Dubois D, Prade H (1982) Towards fuzzy differential calculus. Fuzzy Sets Syst 8:1–7

    Article  MATH  MathSciNet  Google Scholar 

  • Dubois D, Prade H, Testemale C (1988) Weighted fuzzy pattern matching. Fuzzy Sets Syst 28:313–331

    Article  MATH  MathSciNet  Google Scholar 

  • Friedman M, Ma M, Kandel A (1996) Numerical methods for calculating the fuzzy integral. Fuzzy Sets Syst 83:57–62

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Haji ghasemi S (2011) Fuzzy Fredholm–Volterra integral equations and existence and uniqueness of solution of them. Austral J Basic Appl Sci 5:1–8

    Google Scholar 

  • Hassan AAM, Amer SM (2007) Existence of solution for a class of Quasi-linear singular imtegro-differential equations. Serdica Math J 33:253–266

    MATH  MathSciNet  Google Scholar 

  • Jafarzadeh Y (2012) Numerical solution for fuzzy Fredholm integral equations with upper-bound on error by splines interpolation. Fuzzy Inf Eng 3:339–347

    Article  MathSciNet  Google Scholar 

  • Jahantigh M, Allahviranloo T, Otadi M (2008) Numerical solution of fuzzy integral equation. Appl Math Sci 2:33–46

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kauffman A, Gupta MM (1991) Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York

    Google Scholar 

  • Khorasany M, Khezerloo S, Yildirim A (2011) Numerical method for solving fuzzy abel integral equations. World Appl Sci J 13:2350–2354

    Google Scholar 

  • Mikaeilvand N, Khakrangin S, Allahviranloo T (2011) Solving fuzzy Volterra integro-differential equation by fuzzy differential transform method. Mathematics Conference, France

  • Mizumoto M, Tanaka K (1976) The four operations of arithmetic on fuzzy numbers. Syst Comput Controls 7:73–81

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Mordeson J, Newman W (1995) Fuzzy integral equations. Inf Sci 87:215–229

    Google Scholar 

  • Mosleh M, Otadi M (2011) Numerical solution of fuzzy integral equations using Bernstein polynomials. Austral J Basic Appl Sci 5:724–728

    Google Scholar 

  • Muskhelishvili NI (1977) Singular integral equations: boundary problems of functions theory and their applications to mathematical physics. Noordhoff, Leyden

    Book  Google Scholar 

  • Nguyen HT (1978) A note on the extension principle for fuzzy sets. J Math Anal Appl 64:369–380

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Park JY, Jeong JU (2000) The approximate solutions of fuzzy functional integral equations. Fuzzy Sets Syst 110:79–90

    Article  MATH  MathSciNet  Google Scholar 

  • Park JY, Lee SY, Jeong JU (1995) Existence of solutions of fuzzy integral equations in Banach spaces. Fuzzy Sets Syst 72:373–378

    Article  MATH  Google Scholar 

  • Park JY, Lee SY, Jeong JU (2000) On the existence and uniqueness of solutions of fuzzy Volterra–Fredholm integral equations. Fuzzy Sets Syst 115:425–431

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Sugeno M (1974) Theory of fuzzy integrals and its application, Ph.D. thesis, Tokyo Institute of Technology

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

    Article  MATH  MathSciNet  Google Scholar 

  • Zadeh LA (1975) The concept of linguistic variable and its application to approximate reasoning. Inf Sci 8:199–249

    Article  MATH  MathSciNet  Google Scholar 

  • Zeinali M, Shahmorad S, Mirnia K (2013) Fuzzy integro-dierential equations: discrete solution and error estimation. Iran J Fuzzy Syst 10(1):107–122

    MATH  MathSciNet  Google Scholar 

  • Ziari S, Ezzati R, Abbasbandy S (2012) Numerical solution of linear fuzzy Fredholm integral equations of the second kind using fuzzy Haar wavelet. Commun Comput Inf Sci 299(3):79–89

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

    T. Allahviranloo & Sh. S. Behzadi

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  1. T. Allahviranloo
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  2. Sh. S. Behzadi
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Correspondence to Sh. S. Behzadi.

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Communicated by F. Marcelloni.

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Allahviranloo, T., Behzadi, S.S. The use of airfoil and Chebyshev polynomials methods for solving fuzzy Fredholm integro-differential equations with Cauchy kernel. Soft Comput 18, 1885–1897 (2014). https://doi.org/10.1007/s00500-013-1173-9

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