Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth Order Runge-Kutta Method

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.


Introduction
Integral equations discover specific pertinence in numerous logical and numerical mathematical models. In reality, it isn't continuously basic, but too outlandish to demonstrate the deficient physical models utilizing Integral equations to determine certain parameters in real life models so in that case fuzzy set theory be a powerful tool to treat these models mathematically. One of the significant components of a fuzzy analytical theory that plays a key role in numerical analysis is fuzzy integral equations. Zadeh introduced the concept of the fuzzy sets [1]. The theory has developed since, and is now a separate branch of applied mathematics and used to solve many mathematical problems in the filed on differential equations [2,3]. The elementary fuzzy calculation based on the theory of extension and integration was studied first by Dubois and Prade [4]. An integrated Rieman-type method was advocated by Goetschal and Voxman [5], and Kaleva [6] defined the integration of fuzzy functions using the Lebesgue-type integration theory. Fuzzy integral equations with arbitrary kernels were studied by Lakshmikantham and Mohapatra [7]. A number of researchers have been explored by the theoretical properties of fuzzy integral equations [7][8][9][10][11][12][13]. Friedman et.al investigated that core of numerical approaches to solving fuzzy integral equations with arbitrary kernels [14].
Although some numerical approaches in last decade were used to solve FVIE2 such as trapezoidal quadrature rule method [15] solved several types of FVIE2, in [16] a numerical method based on quadrature rules is suggested to solve nonlinear FVIE2, Iterative numerical method [17] is introduced to solve some examples involved nonlinear FVIE2 and so on. From the previous analysis, it would be important to analyze the RKM6 which was not used to solve FVIE2.
The general numerical solution of Runge-Kutta methods for solving crisp FVIE2 were discussed in [18,19]. In this work, we present the use fuzzy set theory properties to formulate RKM6 for solving and analyze numerical solution FVIE2 involving linear test example in the form of table and figures.
The paper is structured accordingly: Some significant meanings and basic concepts to be used in this paper are given in Section 2. In section3, we introduce the fuzzy analysis of FVIE2. We are addressing RKM6 in section 4 to find a numerical solution for FVIE2In section 5, a numerical illustration illustrates the proposed procedure. Finally, in section 6, a short description of this study is provided.

Preliminaries
In this section, some basic concepts for fuzzy calculus are presented as follows: Definition 2.1 [20]. "Fuzzy numbers constitute a subset of the real numbers, reflecting unknown values. Fuzzy numbers are correlated with membership degrees that state how valid it is to tell if anything belongs to a given set or not. If the fuzzy number is described by three numbers  <  <  then µ is called a triangular fuzzy number as in Figure 1, where μ(x) graph is a triangle with an interval base and vertex [,  ] of =  with membership function as follows:  which satisfies the following requirements: We will let denote the set of fuzzy numbers on .
. The α -level represent of a fuzzy number̃ , denoted by [̃] α , is defined as { ỹ(x; α)} It is clear that the α -Level representation of a fuzzy number ̃ is a compact convex subset of R. Thus, if ̃ is a fuzzy number, then Sometimes, we will write y α and y α replace of ( ) and ( ), respectively, ∀ α ∈ [0, 1] .
The limit is in metric D given if the fuzzy function ( ) in metric D is continuous [3], then the following integral exists:

Analysis of FVIE2
The general form of FVIE2 is defined in the following form

Analysis of RKM6 in Fuzzy Domain
The exact and approximate solution at  To specify a particular method, we need to provide the integer p (the number of stages), and the coefficients (for =1, 2,.., p-1), (for 1 ≤ ≤ p).these data are usually arranged in a co-called Butcher tableau [18]. Now, the method defined in Eq. where and satisfy Eq. (4.1) and Eq. (4.3) respectively.

Conclusions
The RKM6 has been formulated into fuzzy domain and analyzed in order to solve fuzzy Volterra equation integral equation numerically. Comparison of the conclusion obtained from this method with the exact solution of a given test problem involving a linear FVIE2 displayed in and figures. The numerical results of the proposed Runge -Kutta method for FVIE2 indicate that this method is appropriate one for solving such problem. Furthermore, the obtain result was accurate, even with small step size and satisfy the fuzzy number properties.