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Paper Information

Journal:   JOURNAL OF SCIENCES (ISLAMIC AZAD UNIVERSITY)   WINTER 2010 , Volume - , Number 74/2 (MATHEMATICS ISSUE); Page(s) 43 To 52.
 
Paper: 

NUMERICAL SOLUTION OF FUZZY FUNCTIONAL INTEGRAL EQUATIONS BY USING LAGRANGE INTERPOLATION

 
 
Author(s):  PARANDIN N.*, FARIBORZI ARAGHI M.A.
 
* MATHEMATICS DEPARTMENT, SCIENCE AND RESEARCH BRANCH, ISLAMIC AZAD, UNIVERSITY, TEHRAN, IRAN
 
Abstract: 
introduction: In this paper, we propose a numerical method for approximating the solution of the fuzzy functional integral equations of Volterra and Fredholm type by using Lagrange interpolation. For this purpose, we convert the fuzzy Fredholm and Volterra integral equations to the crisp systems of integral equations. The proposed method is illustrated by various fuzzy numerical examples.
Aim: In almost, the integral equations cannot be solved analytically or simply.
Therefore, we propose the numerical methods for solving the integral equations.
Material and Method: For solving fuzzy functional integral equations, at first, we introduce the parametric form of them and then we obtained (2n+2) x (2n+2) systems via Lagrange interpolation. In the following, this system convert to two (n+1) x (n+I) systems.
By solving them, we have the support points which can be approximate the exact solution.
Results: In this work, for solving Fredholm or Volterra fuzzy functional integral equation, we replaced each of this fuzzy integral equation by two crisp integral equations. For numerical solution of these equations, we applied the Lagrange interpolation with different r_cuts that is between zero and one. At last, by substituting x in the crisp equations by Xj for j = 0,1,...,n we obtained a linear system of equations with 2n + 2 equations and 2n + 2 unknown. By solving two (n+l) x (n+l) systems, y(xj.;r) and ȳ (xj;r) for j=0,1,...,n and 0
£r£1 are calculated. Consequently, the approximation for exact solution is given by putting y(xj .;r) and . ȳ (xj;r) for j = 0, 1,...,n in Lagrange interpolation function. The advantage of this method in comparison with the other methods is that the solution of the integral equation is approximated by having the supported points. Also, the proposed method in comparison with Freidman et al.'s method converges to the exact solution with less number of the iterations and node points.
Conclusion: This By the proposed method, we can numerically solve the fuzzy functional integral equations of Volterra and Fredholm of the second kind, Also, the proposed method in comparison with Freidman et al.' s method converges to the exact solution with less number of the iterations and node points.
 
Keyword(s): FUZZY LAGRANGE INTERPOLATION, PARAMETRIC FORM OF A FUZZY NUMBER, FUZZ) FUNCTIONAL INTEGRAL EQUATIONS
 
References: 
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