Numerical Investigations on Hybrid Fuzzy Fractional Differential Equations by Improved Fractional Euler Method

In this paper, the improved Euler method is used for solving hybrid fuzzy fractional differential equations (HFFDE) of order q ∈ (0,1) under Caputo-type fuzzy fractional derivatives. This method is based on the fractional Euler method and generalized Taylor’s formula. The accuracy and efficiency of the proposed method is demonstrated by solving numerical examples.


Introduction
The study of fractional differential equations (FDE) forms a suitable setting for mathematical model of real world in various fields viz.Physical and chemical processes.Several forms of FDE have been proposed in more accurate models, and there has been considerable interest in developing numerical methods.M. Mazandarani et.al.proposed the modified fractional Euler method (fuzzy-context) to solve fuzzy fractional differential equations [16,11].Hybrid fuzzy differential equations (HFDE) have been focus of many studies due to natural way to model dynamic system with embedded uncertainty [12,13,14].So far, numerical methods have been used to solve these equations such as, Euler method in [12], Runge-Kutta method in [13].For instance, Pederson et.al [14] investigated the hybrid fractional differential equations.The aim of this paper is to solve the HFFDE by the improved Euler method under Caputo-type fuzzy fractional derivatives.The paper is prepared as follows.After a preliminary section, we will study the Caputo-type fuzzy fractional derivatives.The next section, we discuss HFFDE.Consecutively, we briefly describe the improved fractional Euler method.In the penult section, we present numerical examples to illustrate the theory.Finally in the last section, we give concluding remarks.

Preliminaries
We denote by R F the class of fuzzy subsets u : R → [0, 1] satisfying the following properties: (a) u is normal, that is, there exist x 0 ∈ R with u(x 0 ) = 1.
(c) u is upper semi-continuous on R.
(d) cl {x ∈ R|u(x) > 0} is compact where cl denotes the closure of a subset.
Then the α-level set [u] α is a non-empty compact interval for all 0 ≤ α ≤ 1 and any u ∈ R. The notation [u] α = [u α , u α ] denotes explicitly the α-level set of u.We refer to u and u as the lower and upper branches on u respectively.For u ∈ R, we define the length of u by: len(u) = u − u.For u, v ∈ R F and λ ∈ R, the sum u + v and the product λ u are defined by α means the usual addition of two intervals (subsets) of R and λ [u] α means the usual products between a scalar and a subset of R. The metric structure is given by the Hausdorff distance d : Then it is easy to see that d is a metric in R F and has the following properties: Definition 2.2.Let F : I → R F and fix t 0 ∈ (a, b).We say that F is (1)-differentiable at t 0 , if there exists an element F ′ (t 0 ) ∈ R F such that for all h > 0 sufficiently near to 0 such that F(t 0 + h) ⊖ F(t 0 ), F(t 0 ) ⊖ F(t 0 − h) and the limits (in the metric D) We say that F is (2)-differentiable if for all h > 0 sufficently near to 0 such that and the limits (in the metric D) If t 0 is the end point of I, then we consider the corresponding one-sided derivative.
International Scientific Publications and Consulting Services Theorem 2.2.Let F, G : I → R F be integrable and λ ∈ R. Then (1)

Fuzzy fractional integral and derivative
The space of all continuous fuzzy number valued functions on I, the space of all absolutely continuous fuzzy number valued functions on I and the space of all Lebesgue integrable fuzzy number valued functions on I are respectively denoted by C F (I), (AC) F (I) and L F (I). Throughout this paper, let β ∈ (0, 1).Definition 3.1.[5,16] Let f ∈ L F (I).The Riemann-lioville fractional integral of order β of the fuzzy number valued function f is defined as follows: where Γ(β ) is the well-known Gamma function.
Theorem 3.1.[11] Let f ∈ L F (I).The Riemann-Liouville fractional integral of order β of the fuzzy number valued function f , based on its α-cut representation, can be expressed as where Definition 3.2.[16] If f ∈ AC(I), then Riemann-Lioville fractional derivative of order β of the crisp function f exists almost everwhere on I and can be represented by Note that Riemann-Lioville fractional derivative of order β of f is the first order derivative of the fractional integral 1 − β of f .Definition 3.3.[16] If f ∈ AC(I), then Caputo fractional derivative of order β of the crisp function f exists almost everywhere on I and can be represented by Note that Caputo fractional derivative of order β of f is the fractional integral 1 − β of the first order derivative of f .Definition 3.4.[16] Let f ∈ (AC) F (I) and If the fuzzy number valued function f is (1)-differentiable, then f is said to be Caputo differentiable in the first form and denoted by C a D ] .

Theorem 3.3. (Characterization theorem) Let us consider the fuzzy fractional initial value problem
where f : (A2) x α and x α are continuous and uniformly bounded on any bounded set.that is for any then the FFIVP (3.1) and the system of fractional differential equations (FDEs)

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Proof.Assume the hypothesis (A1)-(A3) are satisfied.First fix ε > 0. Choose δ = ε H and suppose ∥(t, x, y) − (t, x 1 , y 1 )∥ < δ .Then, Next we must show that f α , f α are uniformly bounded on any bounded set.Let S be any bounded Hence, f α is uniformly bounded on S. Similarly, f α is uniformly bounded on any bounded set.Therefore, eqn.
Generalized Taylor's formula under the Caputo-type fuzzy fractional derivative was introdued in [11].
4 The hybrid fuzzy fractional differential system Consider the hybrid fuzzy differential system International Scientific Publications and Consulting Services where Here we assume the existence and uniqueness of solutions of the hybrid system hold on each [t m ,t m+1 ].To be specific the system would look like: Pederson and Sambandham [14] introduced hybrid terms in the fractional differential equations.We note that β ∈ (0, 1) and C a D β x m represents some type of fractional differentiation(fixed for all m ′ s).By the solution of Equation ( 4.3) we mean the following function: ] , (A5) f α m and f α m are equicontinuous and bounded on any bounded set.That is, for any ε > 0 there is a δ m (ε Then, (3.1) and the hybrid system of FDEs are equivalent.

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Proof.Assume the hypothesis of Theorem 3.3.Suppose x(t) is a solution of eqn.(3.1).Fix m = 0, 1, 2, ...,.For a hybrid fuzzy fractional differential Equation (4.3), we develop the improved fractional Euler's method via an application for fuzzy fractional differential equations in [11].The HFFIVP (4.3) is equivalent to the following systems of fractional ordinary differential equations Numerical method for (4.3) is the same for Caputo-differentiable of the two forms.We assume that x is Caputodifferentiable of form C a D β x(t).The initial value problem (4.3) is equivalent to the following integral equations: By substituting t = t 1 into eqn.(4.9) and approximation of the J β F(t, x α , x α ), J β G(t, x α , x α ) by the modified trapezoidalrule with h = t 1 − t 0 , we have x(t) about t 0 = 0 and neglect the second order term(involving h 2β ), the formula for the fractional Euler method is as follows: A system of points that approximates the solution of x(t) is produced by above recursive method.At each step, the fractional Euler method used as a prediction, and the modified trapezoidal rule is used to make a correction to get the finite value.The general formula for the improved fractional Euler algorithm is as follows: Theorem 4.2.[14] Consider the systems (4.8) and (4.12).For a fixed k ∈ Z + and r Proof.Fix k ∈ Z + and α ∈ [0, 1].Choose ε > 0. For each i = 0, 1, ...., k we will find a δ * i > 0 such that h i < δ * i implies where the h i values are allowable by regular partition of the [t i ,t i+1 ]'s.By convergence of numerical method [15] over We may assume δ * k < 1.Then h k < 1.By numerical stability there exists a δ k > 0 such that Therefore if h k < δ * k and (4.15) holds then ) By numerical stability there exists a Therefore if h k−1 < δ * k−1 and (4.18) holds then By numerical stability there exists a δ i > 0 such that Therefore if h i < δ * i and (4.21) holds then In particular,there exists a δ * 1 > 0 such that if h 1 < δ * 1 and (4.21) holds with i = 1 then By convergence of the numerical method over [t 0 ,t 1 ], we may choose δ * 0 > 0 such that h 0 < δ * 0 implies y α (t 1 ; 1) − x α 0,N 0 (α) < δ 1 and y α (t 1 ; 1) − x α 0,N 0 (α) < δ 1 Numerically, Pederson and Sambandam [12,13] solved some examples in fuzzy context with integer order.To give a clear overview of our study and to illustrate the above discussed method, here and in this section.
Example 5.1.[12] Consider the following HFFIVP where The HFFIVP (5.26) is equivalent to the following system of HFFIVP: ) is a continuous function of t, x, and λ k (x(t k )) and HFFIVP has a unique solution on [t k ,t k+1 ].To numerically solve the HFFIVP (5.26) we use the improved fractional Euler method for hybrid fuzzy fractional equations the system (5.26).The results are shown in Table 1 and 2. Furthermore, the approximate solutions in the interval [0,2] are illustrated in Fig. 1.
The numerical results are shown in Table 3 and 4. Furthermore, approximate values in the interval [0, 2] are illustrated in Fig. 2.

Conclusion
In this paper, we utilized the improved fractional Euler method to solve hybrid fuzzy fractional differential equations of order q ∈ (0, 1).The fractional derivative is considered under Caputo-type fuzzy fractional derivative based on strongly generalized fuzzy differentiability.Consistency, convergence of the numerical method is discussed.The solution obtained using the suggested method and show that this technique can be solved the problem effectively.All numerical results are obtained using Matlab.Higher order methods will be consider in our future work.
y and it is denoted x ⊖ y.Throughtout this paper, the sign ⊖ always stands for H-difference and we remark that x ⊖ y ≤ x + (−1)y in general.Usually we denote x + (−1)y by x − y.In the sequel, we fix I = [a, b], for a, b ∈ R.

Theorem 2 . 1 .
[11] Let F : [0, ∞) → R F .Assume that F α (x) and F α (x) are Riemann-integrable on [a, b] for every b ≤ a and assume that there are two positive functions M α , M α such that ∫ b a F α (x)dx ≤ M α and ∫ b a F α (x)dx ≤ M α for every b ≤ a. Then F(x) is improper fuzzy Riemann-integrable on [0, ∞) and the improper fuzzy Riemann-integrable is a fuzzy number.Furthermore [ ∫ ∞ 0 F(x)dx

(4. 9 )
Let g(t) be a crisp continuous function and (⌈β ⌉)-times differentiable in the independent variable t over the interval of differentiation(integration) [0, b].Let the interval [0, b] be subdivided into N subintervals [t j ,t j+1 ] of step size h = b N using the nodes t j = jh for j = 0, 1, ....N.Consider the following Riemann-Liouville integral

Figure 1 :
Figure 1: The approximate solution to the HFFIVP

Figure 2 :
Figure 2: The approximate solution to the HFFIVP