Skip to main content
SpringerLink
Log in

Fuzzy Bezier splines with application to fuzzy functional integral equations

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper, we present the description of fuzzy Bezier splines and as an application we propose an iterative numerical method for approximating the solution of fuzzy functional integral equations of Fredholm type. The convergence of the method is proved by providing an error estimate and it is tested on some numerical examples. The numerical stability regarding the choice of the first iteration is investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  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(1):107–115

    Article  MathSciNet  MATH  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 

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

    MathSciNet  MATH  Google Scholar 

  • Bede B (2013) Mathematics of fuzzy sets and fuzzy logic. Springer, Berlin

    Book  MATH  Google Scholar 

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

    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 Sci 60(4):383–392

    MathSciNet  MATH  Google Scholar 

  • Bica AM (2007) Algebraic structures for fuzzy numbers from categorial point of view. Soft Comput 11:1099–1105

    Article  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 

  • Bica AM (2011) The numerical method of successive interpolations for Fredholm functional integral equations. Numer Algorithms 58:351–377

    Article  MathSciNet  MATH  Google Scholar 

  • Bica AM, Popescu C (2013) Numerical solutions of the nonlinear fuzzy Hammerstein–Volterra delay integral equations. Inf Sci 223:236–255

    Article  MathSciNet  MATH  Google Scholar 

  • Bica AM, Popescu C (2014) Approximating the solution of nonlinear Hammerstein fuzzy integral equations. Fuzzy Sets Syst 245:1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Blaga P, Bede B (2006) Approximation by fuzzy B-spline series. J Appl Math Comput 20:157–169

    Article  MathSciNet  MATH  Google Scholar 

  • Buckley JJ, Eslami E, Feuring T (2002) Fuzzy integral equations. In: Fuzzy mathematics in economics and engineering, studies in fuzziness and soft computing, vol 91. Springer, Physica-Verlag, Heidelberg, pp 229–241

  • Congxin W, Cong W (1997) The supremum and infimum of the set of fuzzy numbers and its applications. J Math Anal Appl 210:499–511

    Article  MathSciNet  MATH  Google Scholar 

  • Diamond P, Kloeden P (2000) Metric topology of fuzzy numbers and fuzzy analysis. In: Dubois D, Prade H (eds) Fundamentals of fuzzy sets. The handbook of fuzzy sets series, vol 7. Kluwer Academic Publishers, Dordrecht, pp 583–641

    Chapter  MATH  Google Scholar 

  • Enkov S, Georgieva A, Nikolla R (2016) Numerical solution of nonlinear Hammerstein fuzzy functional integral equations. In: AIP conference proceedings, vol 1789, no 1. https://doi.org/10.1063/1.4968452

  • Ezzati R, Ziari S (2011) Numerical solution and error estimation of fuzzy Fredholm integral equation using fuzzy Bernstein polynomials. Aust J Basic Appl Sci 5(9):2072–2082

    Google Scholar 

  • Ezzati R, Ziari S (2012) Approximation of fuzzy integrals using fuzzy Bernstein polynomials. Fuzzy Inf Eng 4:415–423

    Article  MathSciNet  MATH  Google Scholar 

  • Ezzati R, Ziari S (2013) Numerical solution of nonlinear fuzzy Fredholm integral equations using iterative method. Appl Math Comput 225:33–42

    MathSciNet  MATH  Google Scholar 

  • Feng W, Zhang D (2012) The local existence and uniqueness of solutions for fuzzy functional Volterra integral equations. In: Proceedings of the 9th international conference on fuzzy systems and knowledge discovery, FSKD, Article ID 6233993, pp 184–187

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

    Article  MathSciNet  MATH  Google Scholar 

  • Gal SG (2000) Approximation theory in fuzzy setting. In: Anastassiou GA (ed) Handbook of analytic-computational methods in applied mathematics, cap. 13. Chapman & Hall, Boca Raton, pp 617–666

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

    Article  MathSciNet  MATH  Google Scholar 

  • Goghary HS, Goghary MS (2005) Two computational methods for solving linear Fredholm fuzzy integral equations of the second kind by Adomian method. Appl Math Comput 161:733–744

    MathSciNet  MATH  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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Kiasari SMK, Khezerloo M, Dogani AMH (2010) Numerical solution of linear Fredholm fuzzy integral equations by modified homotopy perturbation method. Aust J Basic Appl Sci 4:6416–6423

    Google Scholar 

  • Mareš M (1997) Weak arithmetics of fuzzy numbers. Fuzzy Sets Syst 91:143–153

    Article  MathSciNet  MATH  Google Scholar 

  • Mirzaee F, Yari MK, Hoseini SF (2015) A computational method based on hybrid of Bernstein and block-pulse functions for solving linear fuzzy Fredholm integral equations system. J Taibah Univ Sci 9:252–263

    Article  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  MathSciNet  MATH  Google Scholar 

  • Mosleh M, Otadi M (2011) Numerical solution of fuzzy integral equations using Bernstein polynomials. Aust J Basic Appl Sci 5(7):724–728

    MATH  Google Scholar 

  • Mosleh M, Otadi M (2013) Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis. J Adv Inf Technol 4(3):148–155

    Google Scholar 

  • Nouriani H, Ezzati R (2017) Quadrature iterative method for numerical solution of two-dimensional linear fuzzy Fredholm integral equations. Math Sci 11:63–72

    Article  MathSciNet  MATH  Google Scholar 

  • Nouriani H, Ezzati R (2018) Numerical solution of two-dimensional linear fuzzy Fredholm integral equations by the fuzzy Lagrange interpolation. Adv Fuzzy Syst Article ID 5405124

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Sadatrasoul SM, Ezzati R (2014) Quadrature rules and iterative method for numerical solution of two-dimensional fuzzy integral equations. Abstr Appl Anal Article ID 413570. https://doi.org/10.1155/2014/413570

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

    Article  MathSciNet  MATH  Google Scholar 

  • Sadatrasoul SM, Ezzati R (2016b) Analysis of global solution of functional fuzzy integral equations using Krasnoselskii–Burton fixed point theorem. J Intell Fuzzy Syst 30(4):2203–2212

    Article  MATH  Google Scholar 

  • Sahu PK, Ray SS (2016) Comparison of accurate solutions of nonlinear Hammerstein fuzzy integral equations. Math Commun 21:283–299

    MathSciNet  MATH  Google Scholar 

  • Sahu PK, Ray SS (2017) A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein–Volterra delay integral equations. Fuzzy Sets Syst 309:131–144

    Article  MathSciNet  MATH  Google Scholar 

  • Wahab AF, Zulkifly MIE (2018) Cubic Bézier curve interpolation by using intuitionistic fuzzy control point relation. In: AIP conference proceedings, vol 1974, Article ID UNSP 020031

  • Wahab AF, Zulkifly MIE, Husain MS (2016) Bezier curve modeling for intuitionistic fuzzy data problem. In: AIP conference proceedings, vol 1750, Article ID 030047

  • Wahab AF, Zulkifly MIE (2017) A new fuzzy Bezier curve modeling by using fuzzy control point relation. Appl Math Sci 11:39–57

    Google Scholar 

  • Zeinali M (2017) Approximate solution of fuzzy Hammerstein integral equation by using fuzzy B-spline series. Sohag J Math 4(1):19–25

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Ziari S, Bica AM (2016) New error estimate in the iterative numerical method for nonlinear fuzzy Hammerstein–Fredholm integral equations. Fuzzy Sets Syst 295:136–152

    Article  MathSciNet  MATH  Google Scholar 

  • Ziari S, Bica AM (2019) An iterative numerical method to solve nonlinear fuzzy Volterra–Hammerstein integral equations. J Intell Fuzzy Syst 37(5):6717–6729

    Article  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

    MATH  Google Scholar 

  • Ziari S, Perfilieva I, Abbasbandy S (2019) Block-pulse functions in the method of successive approximations for nonlinear fuzzy Fredholm integral equations. Differ Equ Dyn Syst. https://doi.org/10.1007/s12591-019-00482-y

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Oradea, Universitatii Street No. 1, 410087, Oradea, Romania

    Alexandru Mihai Bica & Constantin Popescu

Authors
  1. Alexandru Mihai Bica
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Constantin Popescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandru Mihai Bica.

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

Communicated by A. Di Nola.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bica, A.M., Popescu, C. Fuzzy Bezier splines with application to fuzzy functional integral equations. Soft Comput 24, 6069–6084 (2020). https://doi.org/10.1007/s00500-020-04740-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-020-04740-y

Keywords

Search

Navigation