On new solutions of linear system of first-order fuzzy differential equations with fuzzy coefficient

In this paper, we firstly introduce system of first order fuzzy differential equations. Then, we convert the problem to two crisp systems of first order differential equations. For numerical aspects, we apply exponentially fitted Runge Kutta method to solve the fuzzy problems. We solve some well-known examples in order to demonstrate the applicability and accuracy of results.


Exponentially-fitted Runge Kutta method
There has been many different approaches in the literature for numerically solving the initial value differential system y(x 0 ) = y 0 .
In this chapter we apply the exponentially-fitted Runge Kutta method, proposed by [13], on this problem.A brief introduction on exponentially-fitted Runge Kutta method is first presented along with some explanations on the ideas of the method.Let us consider the following explicit four stage Runge-Kutta method: where f , y k ,Y i ∈ R m k = n, n + 1, i = 1, . . . 4. Alternatively, in the tableau form we have We preserve some values for the components of c as presented in the classical fourth-order Runge Kutta scheme, i.e. c 1 = 0; c 2 = c 3 = 1 2 and c 4 = 1.By integrating all stages and the final step of method (2.6), i.e. y(x) = exp(x), the above conditions give rise to the following system of equations: We choose a 31 = a 42 = 0; this choice is inspired by the classical method.We also consider γ 4 = 1 and solve all ten equations to get

Problem description
In [1], the authors have investigated the solutions of the following fuzzy system of linear first-order differential equations Now by using Seikkala differentiability, Eq. (1.1) is converted to the following system of ODEs: International Scientific Publications and Consulting Services where AX r (t) and AX r (t) are defined as follows: Next the authors interpreted system (1.2) in two different ways: In the next section, we solve Equation (2.3) by the exponentially fitted Runge Kutta method which is introduced in the previous section. ) 2 , e t − r(e t −e −t ) 2 ] ) )

Table 2 :
Results for Example 3.1, lower approximations International Scientific Publications and Consulting Services

Table 3 :
Results for Example 3.1, upper approximations

Table 4 :
Results for Example 3.2, lower approximations International Scientific Publications and Consulting Services