Document Type : Research Paper

**Authors**

Faculty of mathematical sciences, University of Tabriz, Tabriz, Iran

**Abstract**

This paper investigates existence and uniqueness results for the

first order fuzzy integro-differential equations. Then numerical results and

error bound based on the left rectangular quadrature rule, trapezoidal rule

and a hybrid of them are obtained. Finally an example is given to illustrate

the performance of the methods.

first order fuzzy integro-differential equations. Then numerical results and

error bound based on the left rectangular quadrature rule, trapezoidal rule

and a hybrid of them are obtained. Finally an example is given to illustrate

the performance of the methods.

**Keywords**

[1] S. Abbasbandy and T. Allahviranloo,*Numerical solution of fuzzy differential equation by *Runge-Kutta method, Nonlinear Stud., **11 **(2004), 117-129.

[2] S. Abbasbandy and T. Allahviranloo,*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(1) **2006), 101-110.

[3] T. Allahviranloo and M. Afshar Kermani, *Numerical methods for fuzzy linear partial differential *equations under new definition for derivative, Iranian journal of fuzzy systems, **7**(2010), 33-50.

[4] E. Babolian, H. Sadeghi and S. Javadi, *Numerically solution of fuzzy differential equations *by Adomian method, Appl. Math. Comput., **149 **(2004), 547-557.

[5] K. Balachandran and K. Kanagarajan, *Existence of solutions of general nonlinear fuzzy *Volterra-Fredholm integral equations, J. Appl. Math. Stochast. Anal., **3 **(2005), 333-343.

[6] K. Balachandran and P. Prakash,*Existence of solutions of nonlinear fuzzy integral equations *in Banach spaces, Libertas Math., **21 **(2001), 91-97.

[7] K. Balachandran and P. Prakash, *Existence of solutions of nonlinear fuzzy Volterra-Fredholm *integral quations , Indian J. Pure Appl. Math., **33(3) **(2002), 329-343.

[8] B. Bede and S. G. Gal,*Generalizations of the differentiability of fuzzy-number-valued functions *with applications to fuzzy differential equations, Fuzzy Sets and Systems, **151 **(2005),581-599.

[9] B. Bede and S. G. Gal,*Qudrature rules for integrals of fuzzy-number-valued functions*, Fuzzy Sets and systems,**145 **(2004), 359-380.

[10] A. M. Bica,*Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm *integral equations, Information Sciences, **178 **(2008), 1279-1292.

[11] M. Friedman, M. Ma and A. Kandel, *Numerical solutions of fuzzy differential and integral *equations, Fuzzy Sets and Systems, **106**(1999), 35-48.

[12] M. Friedmann, M Ming and A. Kandel, *Solution to fuzzy integral equations with arbitrary *kernels, Int. J. Approx. Reason., **20 **(1999), 249-262.

[13] R. Goetschel and W. Voxman,*Elementary fuzzy calculus*, Fuzzy Sets and Systems, **18 **(1986),31-43.

[14] W. Hackbusch, *Integral equations: theory and numerical treatment*, Birkhuser Verlag, Basel,1995.

[15] O. Kaleva, *Fuzzy differential equations*, Fuzzy Sets and Systems, **24 **(1987), 301-317.

[16] O. Kaleva,*The Cauchy problem for fuzzy differential equations*, Fuzzy Sets and Systems, **35**(1990), 389-96.

[17] V. Lakshmkanihtan and R. N. Mohapatra, *Theory of fuzzy differential equations and inclusions,*Taylor and Francis, London, 2003.

[18] V. Lakshmkanihtan, K. N. Murty and J. Turner,*Two point boundary value problems associated *with nonlinear fuzzy differential equations, Math. Inequal. Appl., **4 **(2003), 527-533.

[19] M. Ma , M. Friedman and Abraham Kandel, *Numerical solutions of fuzzy differential equations,*Fuzzy Sets and Systems,**105 **(1999), 133-138.

[20] A. Molabahrami, A. Shidfar and A. Ghyasi, *An analytical method for solving linear Fredholm*fuzzy integral equations of the second kind, Computers and Mathematics with applications,61(2011), 2754-2761.

[21] J. J. Nieto,*The Cauchy problem for continuous fuzzy differential equations*, Fuzzy Sets and Systems,**102 **999), 259-262.

[22] D. O’Regan, V. Lakshmikantham and J. J. Nieto,*Initial and boundary value problems for *fuzzy differential equations, Nonlinear Anal., **54 **(2003), 405-415.

[23] J. Y. Parka and J. U. Jeong,

, Fuzzy Sets and Systems,

108

(1999), 193-200.

[24] P. Prakash, J. J. Nieto, J. H. Kim and R. Rodriguez-Lopez,

neutral differential equations in Banach spaces

, Dyn. Syst. Appl., **14(3-4) **

(2005), 407-417.

[25] M. Puri and D. Ralescu,

(1983),

552-558.

[26] O. Solaymani Fard, A. Esfahani and A. Vahidian Kamyad,

BVPs

, Iranian journal of fuzzy systems, **9 **

(2012), 49-60.

[27] P. V. Subrahmanyam and S. K. Sudarsanam,

,

Fuzzy Sets and Systems,

(1996), 237-240.

[28] C. Wu and Z. Gong,

, Fuzzy Sets

and Systems,

(2001), 523-532.

[29] C. Wu and M. Ma,

under compactness-type conditions

, Information Sciences, **108 **

(1998), 123-134.

January and February 2013

Pages 107-122