Skip to main content
Springer Nature Link
Log in

Fuzzy bivariate triangular functions with application to nonlinear fuzzy Fredholm–Volterra integral equations in two dimensions

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper, an iterative numerical method based on two-dimensional triangular basis functions has been presented to obtain the numerical solution of fuzzy Fredholm–Volterra integral equations in two dimensions. Error estimation of the proposed method has been gained in terms of uniform and partial modulus of continuity. Also, numerical stability with respect to choice of the starting point of iteration has been also proved under the any theorem. Finally, some numerical examples have been included to demonstrate the convergence and accuracy of the proposed method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Abbasbandy S, Allahviranloo T (2006) The Adomian decomposition method applied to the fuzzy system of Fredholm integral equations of the second kind. Int J Uncertain Fuzziness Knowl-Based Syst 14:101–110

    Article  MathSciNet  MATH  Google Scholar 

  • Anastassiou GA (2010) Fuzzy mathematics: approximation theory. Springer, Berlin

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  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

    MathSciNet  MATH  Google Scholar 

  • Babolian E, Masouri Z, Hatamzadeh-Varmazyar S (2009) Numerical solution of nonlinear Volterra–Fredholm integro–differential equations via direct method using triangular functions. Comput Math Appl 58:239–247

    Article  MathSciNet  MATH  Google Scholar 

  • Babolian E et al (2010) Two-dimensional triangular functions and their applications to nonlinear 2D Volterra–Fredholm integral equations. Comput Math Appl 60:1711–1722

    Article  MathSciNet  MATH  Google Scholar 

  • Baghmisheh M, Ezzati R (2015) Numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using hybrid of block-pulse functions and Taylor series. Adv Differ Equ. https://doi.org/10.1186/s13662-015-0389-7

    Article  MathSciNet  MATH  Google Scholar 

  • Baghmisheh M, Ezzati R (2016) Error estimation and numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using triangular functions. J Intell Fuzzy Syst 30(2):639–649

    Article  MATH  Google Scholar 

  • Balachandran K, Parkash P, Kanagarajan K (2002) Existence of solutions of nonlinear fuzzy Volterra–Fredholm integral equations. Indian J Pure Appl Math 3:329–343

    MathSciNet  Google Scholar 

  • Balachandran K, Kanagarajan K (2005) Existence of solutions of general nonlinear fuzzy Volterra–Fredholm integral equations. J Appl Math Stoch Anal 3:333–343

    Article  MathSciNet  MATH  Google Scholar 

  • Barkhordari Ahmadi M, Khezerloo M (2011) Fuzzy bivariate Chebyshev method for solving fuzzy Volterra–Fredholm integral equations. Int J Ind Math 3(2):67–77

    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 

  • Behzadi SS, Allahviranloo T, Abbasbandy S (2014) The use of fuzzy expansion method for solving fuzzy linear Volterra–Fredholm integral equations. J Intell Fuzzy Syst 26(4):1817–1822

    Article  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 

  • 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 

  • Bica AM, Popescu C (2017) Fuzzy trapezoidal cubature rule and application to two-dimensional fuzzy Fredholm integral equations. Soft Comput 21(5):1229–1243

    Article  MATH  Google Scholar 

  • Bica AM, Ziari S (2017) Iterative numerical method for solving fuzzy Volterra linear integral equations in two dimensions. Soft Comput 21(5):1097–1108

    Article  MATH  Google Scholar 

  • Bica AM, Ziari S (2019) Open fuzzy cubature rule with application to nonlinear fuzzy Volterra integral equations in two dimensions. Fuzzy Sets Syst 358:108–131

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Das PC (2017a) Fuzzy normed linear space valued sequence space. Proyecciones 36(2):245–255

    Article  MathSciNet  MATH  Google Scholar 

  • Das PC (2017b) On fuzzy normed linear space valued statistically convergent sequences. Proyecciones 36(3):511–527

    Article  MathSciNet  MATH  Google Scholar 

  • Deb A, Dasgupta A, Sarkar G (2006) A new set of orthogonal functions and its application to the analysis of dynamic systems. J Frankl Inst 343:1–26

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (1987) Fuzzy numbers: an overview. In: Bezdek JC (ed) Analysis of fuzzy information, Mathematics and Logic, vol I. CRC Press, Boca Raton, pp 3–39

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

    MathSciNet  MATH  Google Scholar 

  • Ezzati R, Ziari S (2013b) Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind using fuzzy bivariate Bernstein polynomials. Int J Fuzzy Syst 15(1):84–89

    MathSciNet  MATH  Google Scholar 

  • Ezzati R, Sadatrasoul SM (2017) Application of bivariate fuzzy Bernstein polynomials to solve two-dimensional fuzzy integral equations. Soft Comput 21(14):3879–3889

    Article  MATH  Google Scholar 

  • Fard OS, Sanchooli M (2010) Two successive schemes for numerical solution of linear fuzzy Fredholm integral equations of the second kind. Aust J Bsic Appl Sci 4:817–825

    Google Scholar 

  • Fariborzi Araghi MA, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Khajehnasiri A (2016) Numerical solution of nonlinear 2D Volterra–Fredholm integro–differential equations by two-dimensional triangular function. Int J Appl Comput Math 2(4):575–591

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Maleknejad K, Jafari Behbahani Z (2012) Applications of two-dimensional triangular functions for solving nonlinear class of mixed Volterra–Fredholm integral equations. Math Comput Model 55:833–844

    MathSciNet  Google Scholar 

  • Mokhtarnejad F, Ezzati R (2015) The numerical solution of nonlinear Hammerstein fuzzy integral equations by using fuzzy wavelet like operator. J Intell Fuzzy Syst 28(4):1617–1626

    Article  MathSciNet  MATH  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 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Nieto JJ, Rodriguez-Lopez R (2006) Bounded solutions of fuzzy differential and integral equations. Chaos Solitons Fractals 27:1376–1386

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Park JY, Jeong JU (1999) A note on fuzzy integral equations. Fuzzy Sets Syst 108:193–200

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  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 

  • Rivaz A, Yousefi F, Salehinejad H (2012) Using block pulse functions for solving two-dimensional fuzzy Fredholm integral equations of the second kind. Int J Appl Math 25(4):571–582

    MathSciNet  MATH  Google Scholar 

  • Sadatrasouland SM, Ezzati R (2014) Quadrature rules and iterative method for numerical solution of two-dimensional fuzzy integral equations. Abstr Appl Anal, Article ID 413570

  • Sadatrasoul SM, Ezzati R (2015) Iterative method for numerical solution of two-dimensional nonlinear fuzzy integral equations. Fuzzy Sets Syst 280:91–106

    Article  MathSciNet  MATH  Google Scholar 

  • Sadatrasouland SM, Ezzati R (2016) Numerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula. J Comput Appl Math 292(15):430–446

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319–330

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Baruah A (2010) Norlund and Riesz mean of sequences of fuzzy real numbers. Appl Math Lett 23:651–655

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Das PC (2012) On convergence of series of fuzzy real numbers. Kuwait J Sci Eng 39(1 A):57–70

    MathSciNet  Google Scholar 

  • Vali MA, Agheli MJ, Gohari Nezhad G (2014) Homotopy analysis method to solve two-dimensional fuzzy Fredholm integral equation. Gen Math Notes 22(1):31–43

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Wu C, Song S, Wang H (1998) On the basic solutions to the generalized fuzzy integral equation. Fuzzy Sets Syst 95:255–260

    Article  MathSciNet  MATH  Google Scholar 

  • Ziari S (2018) Iterative method for solving two-dimensional nonlinear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimation. Iran J Fuzzy Syst 15(1):55–76

    MathSciNet  MATH  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, Ezzati R, Abbasbandy S (2012) Numerical solution of linear fuzzy fredholm integral equations of the second kind using fuzzy haar wavelet. In: Greco S, Bouchon-Meunier B, Coletti G, Fedrizzi M, Matarazzo B, Yager RR (eds) Advances in computational intelligence. IPMU 2012. Communications in Computer and Information Science, vol 299. Springer, Berlin, Heidelberg

  • 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

Acknowledgements

The authors declare that the research was not funded by any grant.

Author information

Authors and Affiliations

  1. Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

    S. Karamseraji & R. Ezzati

  2. Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

    S. Ziari

Authors
  1. S. Karamseraji
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. R. Ezzati
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. S. Ziari
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to S. Ziari.

Ethics declarations

Conflict of interest

The authors declare that there is 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 V. Loia.

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

Karamseraji, S., Ezzati, R. & Ziari, S. Fuzzy bivariate triangular functions with application to nonlinear fuzzy Fredholm–Volterra integral equations in two dimensions. Soft Comput 24, 9091–9103 (2020). https://doi.org/10.1007/s00500-019-04439-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04439-9

Keywords