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Fuzzy quasi-interpolation solution for Fredholm fuzzy integral equations of second kind

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Abstract

In this paper, we intend to offer a new approach to solve Fredholm fuzzy integral equation of the second kind by the concept of source distance and fuzzy quasi-interpolation. The error estimation of the proposed method is proved in terms of the fuzzy modulus of continuity which will be introduced in this paper. Finally some examples have been given to emphasize acceptable accuracy of our method.

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References

  • Abbasbandy S, Babolian E (1998) Interpolation of fuzzy data by natural splines. J Appl Math Comput 5:457–463

    MathSciNet  MATH  Google Scholar 

  • Abbasbandy S (2001) Interpolation of fuzzy data by complete splines. Kor J Comput Appl Math 8:587–594

    MathSciNet  MATH  Google Scholar 

  • Abbasbandy S, Amirfakhrian M (2006) A new approach to universal approximation of fuzzy functions on a discrete set of points. Appl Math Model 30:1525–1534

    Article  MATH  Google Scholar 

  • Abbasbandy S, Amirfakhrian M (2006) Numerical approximation of fuzzy functions by fuzzy polynomials. Appl Math Comput 174:1001–1006

    MathSciNet  MATH  Google Scholar 

  • Abbasbandy S, Amirfakhrian M (2006) The nearest approximation of a fuzzy quantity in parametric form. Appl Math Comput 172:624–632

    MathSciNet  MATH  Google Scholar 

  • Abbasbandy S, Babolian E (1998) Interpolation of fuzzy data by natural splines. Appl Math Comput 5:457–463

    MathSciNet  MATH  Google Scholar 

  • Abbasbandy S, Ezzati R, Behforooz H (2008) Interpolation of fuzzy data by using fuzzy splines. Int J Uncertain Fuzziness Knowl Based Syst 16(7):107–115

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Allahviranloo T, Ghanbari M, Hosseinzadeh AA, Haghi E, Nuraei R (2011) A note on fuzzy linear systems. Fuzzy Sets Syst 177:87–92

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Amirfakhrian M (2012) Some approximation methods in fuzzy logic. Lambert Academic Publishing (LAP), Germany

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

    MathSciNet  MATH  Google Scholar 

  • Bayona V, Moscoso M, Kindelan M (2011) Optimal constant shape parameter for multiquadric based RBF-FD method. J Comput Phys 230:73847399

    Article  MathSciNet  MATH  Google Scholar 

  • Behforooz H, Ezzati R, Abbasbandy S (2010) Interpolation of fuzzy data by using E(3) cubic splines. Int J Pure Appl Math 60(4):383–392

    MathSciNet  MATH  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  MathSciNet  MATH  Google Scholar 

  • Congxin W, Ma M (1992) On embedding problem of fuzzy number spaces. Fuzzy Sets Syst 44:33–38

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (1982) Towards fuzzy differential calculus. Fuzzy Sets Syst 8:225–233

    Article  MathSciNet  MATH  Google Scholar 

  • Dehghan B, Hashemi M (2006) Iterative solution of fuzzy linear systems. Appl Math Comput 175:645–674

  • Fasshauer G, Zhang J (2007) On choosing optimal shape parameters for RBF approximation. Numer Alg 45:345–368

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Hardy L (1978) The application of multiquadric equations and point mass anomaly models to crustal movement studies. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Survey. Natl Geod Surv 55(76)

  • Hardy RL (1990) Multiquadric equation of topography and other irregular surfaces. J Geophys Res 76:1905–1915

    Article  Google Scholar 

  • Jafari H, Hosseinzadeh H, Mohamadzadeh S (2010) Numerical solution of system of linear integral equations by using Legendre wavelets. Int J Open Probl Comput Sci Math 5:63–71

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Kaleva O (1994) Interpolation of fuzzy data. Fuzzy Sets Syst 61:63–70

    Article  MathSciNet  MATH  Google Scholar 

  • Kansa EJ (1990) Multiquadricsa scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximation and partial dervative estimates. Comput Math Appl 19:127–145

    Article  MathSciNet  MATH  Google Scholar 

  • Lowen R (1990) A fuzzy Lagrange interpolation theorem. Fuzzy Sets Syst 34:33–38

    Article  MathSciNet  MATH  Google Scholar 

  • Luh Lin-Tian (2016) The mystery of the shape parameter III. Appl Comput Harmon Anal 40:186–199

    Article  MathSciNet  MATH  Google Scholar 

  • Luh Lin-Tian (2014) The mystery of the shape parameter IV. Eng Anal Bound Elem 48:2431

    Article  MathSciNet  Google Scholar 

  • Nanda S (1989) On integration of fuzzy mappings. Fuzzy Sets Syst 32:95–101

    Article  MathSciNet  MATH  Google Scholar 

  • Powell MJD (1990) Univariate multiquadric approximation: reproduction of linear polynomials. Multivar Approx Interpol 94:227–240

  • Sugeno M (1974) Theory of fuzzy integral and its application. Ph.D. dissertation. Tokyo Institute of Technology

  • Wright B (2003) Radial basis function interpolation: numerical and analytical developments. A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Applied Mathematics

  • Wu HC (2000) The fuzzy Riemann integral and it’s numerical integration. Fuzzy Sets Syst 110:1–25

    Article  MathSciNet  Google Scholar 

  • Wu H (2000) The fuzzy Riemann integral and its numerical integration. Fuzzy Sets Syst 110:1–25

    Article  MathSciNet  MATH  Google Scholar 

  • Wu ZM, Schaback R (1994) Shape preserving properties and convergence of univariate multiquadric quasi-interpolation. Comput Math Appl 10:441–446

    MathSciNet  MATH  Google Scholar 

  • Yue-Xing W, Lin X, Xiao-qian C (2009) The radial basis function shape parameter chosen and its application in engineering. Intell Comput Intell Syst 1:79–83

    Google Scholar 

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

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Correspondence to M. Amirfakhrian.

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Author A declares that she has no conflict of interest. Author B declares that she has no conflict of interest. Author C declares that he has no conflict of interest.

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Communicated by V. Loia.

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Amirfakhrian, M., Shakibi, K. & Rodríguez López, R. Fuzzy quasi-interpolation solution for Fredholm fuzzy integral equations of second kind. Soft Comput 21, 4323–4333 (2017). https://doi.org/10.1007/s00500-016-2065-6

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