Soft Computing

, Volume 21, Issue 5, pp 1097–1108

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

Foundations

## 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.

### Keywords

Fuzzy-number-valued functions Numerical method Two-dimensional fuzzy Volterra integral equations Picard’s type iterations Fuzzy trapezoidal cubature rule

### References

1. 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–S141Google Scholar
2. Attari H, Yazdani A (2011) A computational method for fuzzy Volterra-Fredholm integral equations. Fuzzy Inf. Eng. 2:147–156
3. Balachandran K, Kanagarajan K (2005) Existence of solutions of general nonlinear fuzzy Volterra-Fredholm integral equations. J. Applied Math. Stochastic Analysis 3:333–343
4. Balachandran K, Prakash P (2004) On fuzzy Volterra integral equations with deviating arguments. J. Applied Math. Stochastic Analysis 2:169–176
5. Barkhordari Ahmadi M, Khezerloo M (2011) Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholm integral equations. Int J. Industrial Math. 3(2):67–77Google Scholar
6. Bede B (2013) Mathematics of Fuzzy Sets and Fuzzy Logic. Springer-Verlag, Berlin Heidelberg
7. Bede B, Gal SG (2004) Quadrature rules for integrals of fuzzy-number-valued functions. Fuzzy Sets Syst. 145:359–380
8. 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): S337–346Google Scholar
9. 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-00067Google Scholar
10. 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
11. Bica AM (2008) Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations. Information Sciences 178:1279–1292
12. Bica AM, Popescu C (2013) Numerical solutions of the nonlinear fuzzy Hammerstein-Volterra delay integral equations. Information Sciences 233:236–255
13. 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
14. 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
15. Diamond P (2002) Theory and applications of fuzzy Volterra integral equations. IEEE Transactions on Fuzzy Systems 10:97–102
16. Dubois D, Prade H (1982) Towards fuzzy differential calculus. Part 2: Integration of fuzzy intervals. Fuzzy Sets Syst. 8:105–116
17. 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
18. Friedman M, Ma M, Kandel A (1999) Numerical solutions of fuzzy differential and integral equations. Fuzzy Sets Syst. 106:35–48
19. Friedman M, Ma M, Kandel A (1999) Solutions to fuzzy integral equations with arbitrary kernels. Int. J. Approx. Reasoning 20:249–262
20. 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)Google Scholar
21. 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
22. Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets Syst 18:31–43
23. Gong Z, Wu C (2002) Bounded variation, absolute continuity and absolute integrability for fuzzy-number-valued functions. Fuzzy Sets Syst. 129:83–94
24. Haji Ghasemi S (2011) Fuzzy Fredholm-Volterra integral equations and existence and uniqueness of solution of them. Australian J. Basic Applied Sciences 5:1–8Google Scholar
25. Jafarian A, Measoomy Nia S, Tavan S (2012) A numerical scheme to solve fuzzy linear Volterra integral equations system. J. Appl. Math.: article ID 216923Google Scholar
26. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst. 24:301–317
27. Mordeson J, Newman W (1995) Fuzzy integral equations. Information Sciences 87:215–229
28. 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
29. Nieto JJ, Rodriguez-López R (2006) Bounded solutions of fuzzy differential and integral equations. Chaos Solitons & Fractals 27:1376–1386
30. Park JY, Lee SY, Jeong JU (2000) The approximate solutions of fuzzy functional integral equations. Fuzzy Sets Syst. 110:79–90
31. Park JY, Han HK (1999) Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets Syst. 105:481–488
32. Park JY, Jeong JU (1999) A note on fuzzy integral equations. Fuzzy Sets Syst. 108:193–200
33. Park JY, Jeong JU (2000) On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations. Fuzzy Sets Syst. 115:425–431
34. 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 00146Google Scholar
35. 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
36. 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):153–161Google Scholar
37. Sadatrasoul SM, Ezzati R (2014a) Quadrature rules and iterative method for numerical solution of two-dimensional fuzzy integral equations. Abstract Appl. Analysis, article ID 413570Google Scholar
38. 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
39. Salahshour S, Allahviranloo T (2013) Applicationm of fuzzy differential transform method for solving fuzzy Volterra integral equations. Appl. Math. Modelling 37:1016–1027
40. Salehi P, Nejatiyan M (2011) Numerical method for nonlinear fuzzy Volterra integral equations of the second kind. Int. J. Industrial Math. 3(3):169–179Google Scholar
41. Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst. 24:319–330
42. Shafiee M, Abbasbandy S, Allahviranloo T (2011) Predictor-corrector method for nonlinear fuzzy Volterra integral equations. Australian J. Basic Applied Sciences 5(12):2865–2874Google Scholar
43. 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
44. Subrahmanyam PV, Sudarsanam SK (1996) A note on fuzzy Volterra integral equations. Fuzzy Sets Syst. 81:237–240
45. Vali MA, Agheli MJ, Gohari Nezhad G (2014) Homotopy analysis method to solve two-dimensional fuzzy Fredholm integral equation. Gen. Math. Notes 22(1):31–43Google Scholar
46. Wu C, Gong Z (2001) On Henstock integral of fuzzy-number-valued functions (I). Fuzzy Sets Syst. 120:523–532