FUZZY INTEGRO-DIFFERENTIAL EQUATIONS: DISCRETE SOLUTION AND ERROR ESTIMATION

Document Type: Research Paper

Authors

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

2 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.

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 Fredholmfuzzy 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,

 

 

A note on fuzzy integral equations

, Fuzzy Sets and Systems,

 

 

108

 

 

(1999), 193-200.

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

 

 

Existence of solutions of fuzzy

neutral differential equations in Banach spaces

 

 

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

(2005), 407-417.

[25] M. Puri and D. Ralescu,

Differentials of fuzzy functions, J. Math. Anal. Appl., 91

(1983),

552-558.

 

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

 

 

 

On solution of a class of fuzzy

BVPs

 

 

, Iranian journal of fuzzy systems, 9

(2012), 49-60.

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

 

 

A note on fuzzy Volterra integral equations

,

Fuzzy Sets and Systems,

 

 

81

(1996), 237-240.

[28] C. Wu and Z. Gong,

 

 

On Henstock integral of fuzzy-number-valued functions I

, Fuzzy Sets

and Systems,

 

 

120

(2001), 523-532.

[29] C. Wu and M. Ma,

 

 

 

Existence theorem to the Cauchy problem of fuzzy differential equations

under compactness-type conditions

 

 

, Information Sciences, 108

(1998), 123-134.