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Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions

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Abstract

In this paper, we propose an iterative numerical method for two-dimensional fuzzy Volterra linear integral equations. The method uses a recent fuzzy trapezoidal cubature formula for Lipschitzian fuzzy-number-valued functions applied at each iterative step. The convergence of the method is proved providing an error estimate and some numerical experiments illustrate the accuracy of this method confirming the theoretical results.

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References

  • Allahviranloo T, Khezerloo M, Sedaghatfar O, Salahshour S (2013) Toward the existence and uniqueness of solutions of second-order fuzzy Volterra integro-differential equations with fuzzy kernel. Neural Comput & Applic 22 (Supl 1): S133–S141

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

    Article  MathSciNet  MATH  Google Scholar 

  • Balachandran K, Kanagarajan K (2005) Existence of solutions of general nonlinear fuzzy Volterra-Fredholm integral equations. J. Applied Math. Stochastic Analysis 3:333–343

    Article  MathSciNet  MATH  Google Scholar 

  • Balachandran K, Prakash P (2004) On fuzzy Volterra integral equations with deviating arguments. J. Applied Math. Stochastic Analysis 2:169–176

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • Bede B (2013) Mathematics of Fuzzy Sets and Fuzzy Logic. Springer-Verlag, Berlin Heidelberg

    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 

  • Behzadi SS, Allahviranloo T, Abbasbandy S (2012) Solving fuzzy second-order nonlinear Volterra-Fredholm integro-differential equations by using Picard method. Neural Comput & Applic 21 (Supl 1): S337–346

  • Behzadi SS (2011) Solving fuzzy nonlinear Volterra-Fredholm integral equations by using homotopy analysis and Adomian decomposition methods, J. Fuzzy Set Valued Analysis: article ID jfsva-00067

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Bica AM, Popescu C (2013) Numerical solutions of the nonlinear fuzzy Hammerstein-Volterra delay integral equations. Information Sciences 233:236–255

    Article  MathSciNet  MATH  Google Scholar 

  • Bica AM, Popescu C (2015) Fuzzy trapezoidal cubature rule and application to two-dimensional fuzzy Fredholm integral equations. Soft Computing. doi:10.1007/s00500-015-1856-5

    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 

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

    Article  MATH  Google Scholar 

  • Dubois D, Prade H (1982) Towards fuzzy differential calculus. Part 2: Integration of fuzzy intervals. Fuzzy Sets Syst. 8:105–116

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Gal SG (2000) Approximation theory in fuzzy setting, in: G.A. Anastassiou (ed.), Handbook of analytic-computational methods in applied mathematics, pp. 617–666, Chapman & Hall/CRC Press, Boca Raton, (Chapter 13)

  • Ghanbari M (2010) Numerical solution of fuzzy linear Volterra integral equations of the second kind by homotopy analysis method. Int. J. Industrial Math. 2(2):73–87

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Gong Z, Wu C (2002) Bounded variation, absolute continuity and absolute integrability for fuzzy-number-valued functions. Fuzzy Sets Syst. 129:83–94

    Article  MathSciNet  MATH  Google Scholar 

  • Haji Ghasemi S (2011) Fuzzy Fredholm-Volterra integral equations and existence and uniqueness of solution of them. Australian J. Basic Applied Sciences 5:1–8

    Google Scholar 

  • Jafarian A, Measoomy Nia S, Tavan S (2012) A numerical scheme to solve fuzzy linear Volterra integral equations system. J. Appl. Math.: article ID 216923

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    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 

  • 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 

  • Rivaz A, Yousefi F (2013) Kernel iterative method for solving two-dimensional fuzzy Fredholm integral equations of the second kind. J. Fuzzy Set Valued Analysis, article ID 00146

  • Rivaz A, Yousefi F (2012) Modified homotopy perturbation method for solving two-dimensional fuzzy Fredholm integral equation. Int. J. Appl. Math. 25(4):591–602

    MathSciNet  MATH  Google Scholar 

  • Rouhparvar H, Allahviranloo T, Abbasbandy S (2009) Existence and uniqueness of fuzzy solution for linear Volterra fuzzy integral equations proved by Adomian decomposition method. ROMAI J. 5(2):153–161

  • Sadatrasoul SM, Ezzati R (2014a) Quadrature rules and iterative method for numerical solution of two-dimensional fuzzy integral equations. Abstract Appl. Analysis, article ID 413570

  • Sadatrasoul SM, Ezzati R (2014b) Iterative method for numerical solution of two-dimensional nonlinear fuzzy integral equations. Fuzzy Sets Syst. doi:10.1016/j.fss.2014.12.008

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

    Article  MathSciNet  MATH  Google Scholar 

  • Salehi P, Nejatiyan M (2011) Numerical method for nonlinear fuzzy Volterra integral equations of the second kind. Int. J. Industrial 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 

  • Shafiee M, Abbasbandy S, Allahviranloo T (2011) Predictor-corrector method for nonlinear fuzzy Volterra integral equations. Australian J. Basic Applied Sciences 5(12):2865–2874

    Google Scholar 

  • Song S, Liu Q-Y, Xu Q-C (1999) Existence and comparison theorems to Volterra fuzzy integral equation in (En, D). Fuzzy Sets Syst. 104:315–321

    Article  MATH  Google Scholar 

  • Subrahmanyam PV, Sudarsanam SK (1996) A note on fuzzy Volterra integral equations. Fuzzy Sets Syst. 81:237–240

    Article  MathSciNet  MATH  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 

Download references

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Oradea, Universitatii 1, 410087, Oradea, Romania

    Alexandru Mihai Bica

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

    Shokrollah Ziari

Authors
  1. Alexandru Mihai Bica
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  2. Shokrollah Ziari
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Correspondence to Shokrollah Ziari.

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The authors declare that they have no conflict of interest.

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Communicated by A. Di Nola.

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Bica, A.M., Ziari, S. Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions. Soft Comput 21, 1097–1108 (2017). https://doi.org/10.1007/s00500-016-2085-2

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