Numerical solution of first-order fully fuzzy differential equations by Runge-Kutta Fehlberg method under strongly generalized H-differentiability

In this paper we propose Runge-Kutta Fehlberg method for solving fully fuzzy differential equations (FFDEs) of the form y ′ (t) = a⊗ y(t), y(0) = y0, t ∈ [0,T ] under strongly generalized H-differentiability. The algorithm used here is based on cross product of two fuzzy numbers. Using cross product we investigate the problem of finding a numerical approximation of solutions. The convergence of this method is discussed and numerical example is included to verify the reliability of proposed method.


Introduction
The research of fuzzy differential equations (FDE) form an appropriate setting for mathematical modelling of real world problems in which uncertainties or imprecision pervades.The solutions of a FDE with fuzzy initial conditions are used in science and engineering fields, thus fuzzy initial value problem (FIVP) should be solved [4].The term "fuzzy differential equation" was first coined in 1978 [17].The idea of a fuzzy derivative was defined by Chang and Zadeh [2].It was followed by Dubois and Prade [11], who used the extension principle.The brief sketch of FIVP was proposed by Seikkala and Kalava [16] and other researchers began to improve the fuzzy theory.

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There have been many ideas for the definition of fuzzy derivative to study FDE.The first and the most popular approach are using the Hukuhara differentiability for fuzzy-value functions.The strongly generalized differentiability was introduced Bede et.al (for e.g.[8], [7]).The strongly generalized derivative is defined for a longer class of fuzzy number valued functions than the Hukuhara derivative.The numerical methods for solving FDEs y ′ (x) = f (x, y) where x 0 is real number and y(x 0 ) = y 0 are introduced in Allahviranloo [2], Abbasbandy et al. (see e.g.[3], [4], [7]) applied the concept of strongly generalized H-differentiability to solve linear first-order FDEs.It should be noted that in all mentioned numerical methods, finding a numerical solution for FDE is only possible with real coefficients.
In this paper, we find the numerical solution for FFDE in the form y ′ (t) = a ⊗ y(t), y(0) = y 0 , t ∈ [0, T ] where a is a fuzzy number.First, by choosing different types of derivatives and sign of a and y(t), FFDE is divided into four differential equations.Since each of the divided differential equations satisfies the Lipschitz condition, they should have a unique solution and Runge-Kutta Fehlberg method is used to find the numerical solutions.
The paper is organised as follows: In section 2, we give some basic definitions and cross product is defined.In section 3, first-order fully fuzzy differential equation is introduced.In section 4, Runge-Kutta Fehlberg method is presented in detailed and its convergence result is discussed.In section 5, a numerical example is included to illustrate the theory.

Basic concepts
In this section, we recall the basic notation of fuzzy numbers, strongly generalized H-differentiability and the cross product.A non-empty subset A of R is called convex if and only if (1 − k)x + ky ∈ A for every x, y ∈ A and k ∈ [0, 1].By P k (R), we denote the family of all non-empty compact convex subsets of R.There are various definitions for the concepts of fuzzy numbers [11].
The set of all fuzzy real numbers is denoted by R F .Obviously,R ⊂ R F .For 0 < r ≤ 1, the r-level set is denoted by [u] r = {x ∈ R; u(x) ≥ r} and [u] 0 = {x ∈ R; u(x) ≥ 0}.Then, it is well-known that for any r ∈ [0, 1], [u] r is a bounded closed interval.For u, v ∈ R F and λ ∈ R, the sum u + v and the product λ .uare defined by means the conventional addition of two intervals(subsets) of R, and λ [u] r = {λ x : x ∈ [u] r } means the conventional product between a scaler and a subset of R in [11].
Definition 2.2.The parametric form of a fuzzy number u(r) is a pair [u(r), u(r)] of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfy the following conditions: and International Scientific Publications and Consulting Services respectively, for every r ∈ [0, 1].We denote by −u = (−1)u ∈ R F the symmetricc of u ∈ R F .The producct u.v of fuzzy numbers u and v, based on Zadeh's extension principle, is defined by where Definition 2.3.A fuzzy number u ∈ R F is said to be positive if u(1) ≥ 0, strict positive if u(1) > 0, negative if u(1) ≤ 0 and strict negative if u(1) < 0. We say that u and v have the same sign if they are both strict positive or both strict negative.If u is positive(negative) then −u is negative(positive).
In this paper, we denote an arbitrary fuzzy number with compact support by a pair of functions (u(r), u(r)) , 0 ≤ r ≤ 1.Also, we use the Hausdorff distance between fuzzy numbers.This fuzzy number space as has beeen shown in Bede and Gal [8] can be embedded into the Banach space with the usual metric defined as follows: Let R F be the set of all upper semi-continuous normal convex fuzzy numbers with bounded r-level sets.Since the r-cuts of fuzzy numbers are always closed and bounded, the intervals are written as [u] r = [u(r), u(r)], for all r.We denote by ω all of non empty convex compact sets.Recall that Note that the notation is consistent, since ρ(a, The following properties are well known: [13,21] Definition 2.4.[12] Let f : R → R F be a fuzzy-valued function.If for arbitrary fixed t 0 ∈ R and ε > 0, δ > 0 such that f is said to be continuous.We say that f is strongly generalized H-differentiable at t 0 , if there exists an element f ′ (t 0 ) ∈ R F , such that (i) for all h > 0 sufficiently near to 0, there exist the H-difference f International Scientific Publications and Consulting Services (ii) for all h < 0 sufficiently near to 0, there exists the H-difference f Notice that we say fuzzy-valued function f is (i)-differentiable if satisfy in the first form (i) in Definition (2.6), and we say f is (ii)-differentiable if satisfies in the second form (ii) in definition 2.6.
As a special case when f is a fuzzy-valued function, we have the following results.
then f (t; r) and f (t; r) have first-order and second-order derivatives and f Lemma 2.1.[18] For x 0 ∈ R, the fuzzy differential equation y supposed to be continuous, is equivalent to the one of the integral equations or y(0) = y(x) + (−1).
Here, the equivalence between two equations means that any solution of an equation is a solution for the other one.
Remark 2.1.[8] In the case of strongly generalized differentiability, to the fuzzy differential equation y we may attach two different integral equations, while in the case of H-differentiability, we may attach only one.The second integral equations in Lemma 2.1 can be written in the form y(0) = y(x) + (−1).

∫ x
x 0 f (t, y(t))dt.The following theorems concern the existence of solutions of a fuzzy initial-value problem under generalized differentiability in [8].
Theorem 2.3.Suppose that following conditions hold: (a) Let R 0 = [x 0 , x 0 + p] × B(y 0 , q), p, q > 0, y 0 ∈ R F where B(y 0 , q) = {y ∈ R F : D(y, y 0 ) ≤ q} denote a closed ball in R F and let f :  has two solutions (one (i)-differentiable and the other on (ii)-differentiable) y, y ′ : [x 0 , x 0 + d] → B(y 0 , q) where r = min { s, q m , q m 1 , d } and the successive iteration converge to these two solutions, respectively.We denote, the space of continuous functions y : where ρ ∈ R is fixed [14].

The cross product
In this section, we recall summary from the theoretical properties of the cross product of two fuzzy numbers.Let R * F = {u ∈ R F : u is positive or negative}.First, Ban and Bede begin with a theorem which has been obtained using the stacking theorem [20], for more details see [6,9,19].

Theorem 2.4. If u and v are positive fuzzy numbers then w
for every r ∈ [0, 1], is a positive fuzzy number.

Remark 2.2. Let u and v be two fuzzy numbers. 1.If u is positive and v is negative then u
) is a negative fuzzy number.

Remark 2.3. The cross product is difined for any fuzzy numbers in R ∧
F = {u ∈ R * F ; there exists an unique x 0 ∈ R such that u(x 0 ) = 1}, so implicitily for any triangular fuzzy number.Remark 2.4.The below formulas of calculus can be easily proved (r ∈ [0, 1]): 1), if u and v are negative, 1), if u and v are negative, 3 First-order fully fuzzy differential equation In this section, we are going to show that, FFDE satisfies in the Lipschitz condition and, therefore, has unique solution.An FFDE has the following equation: where a and y 0 are triangular fuzzy numbers in this paper.The Lipschitz condition for problem (3.7) is introduced in following lemma.
Lemma 3.1.[10] Let a be an triangular fuzzy number, for each t

7) if and only if y is continuous and satisfies one of the following conditions:
(a) The following lemma is needed to prove the initial-value problem (3.7) has a unique solution.
4 Runge-Kutta Fehlberg method and its convergence In this section, we describe our purpose approach for solving FFDE (3.7), then we analyze the convergence of this method.In the beginning suppose that the discrete equally spaced grid points Then, Runge-Kutta Fehlberg method to approximate the solution of (3.7) is as follows: where ∆w, Since in constituting the cross product a ⊗ w(t i ), the signs of a and w(t i ) are important, to determine w(t i+1 ) the sign of w(t i ) should be found in each step.Then, we consider four cases as follows: Case (1): In this case, we assume that y(t) is (i)-differentiable and a ≥ 0.Then, the Runge-Kutta Fehlberg method to approximate w(t i+1 ) is as follows: where,                Here, U 0 = V 0 = 0.If h → 0 we obtain U N → 0, V N → 0 and the proof is complete.

Example
In this section, an example is given to illustrate our method.Moreover, we plot the obtained solutions and approximate them based on the r-cut representation at each case.
are intervals that on them, y and y ′ are valid(are fuzzy numbers).Therefore, on I 1 ∩ I 2 = [0, +∞), y is (i)-differentiable solution.Approximatioe solutions w, w can be found by solving ODEs (4.9), (4.10)(see Figs. 1 and 2).The numerical values are given in tables 1, 2 and error ∆w is shown at t=0.1.Approximatioe solutions w, w can be found by solving ODEs (4.13), (4.14) (see Figs. 3 and 4)   In this paper, Runge-Kutta Fehlberg method is employed to solve FFDEs under strongly generalized H-differentiability.The convergence of this method is discussed and using an algorithm for Runge-Kutta Fehlberg method, the solution is approximated in each case.In the end, we considered a suitable example to illustrate the theory.Also, for future research work we will apply higher order Runge-Kutta methods.
(a) u(r) is a monotonically increasing left continuous function.(b) u(r) is a monotonically decreasing left continuous function.(c) u(r) ≤ u(r), 0 ≤ r ≤ 1.It should be noted that for a < b < c, a, b, c ∈ R, a triangular fuzzy number u = (a, b, c) is given such that u = b − (1 − r)(b − a) and u = b + (1 − r)(c − b) are the end point of the r-cut set for all 0 ≤ r ≤ 1.In this paper we use triangular fuzzy numbers.Here, u(r) = u(r) = b is denoted by [u] 1 .For arbitrary fuzzy number [u] r = [u(r), u(r)] and [v] r = [v(r), v(r)] and k ∈ R, we define addition and multiplication as

Definition 2 . 5 . [ 7 ]
Let x, y ∈ R F .If there exists z ∈ R F such that x = y + z, then z is called the H-difference of x and y, and it is denoted by x ⊖ y, Definition 2.6.[7] Let f : (a, b) → R F and t 0 ∈ (a, b).
(t, y 1 n )dt is defined for any n ∈ N. Then the fuzzy initial value problem { y ′ = f (x, y), y(x 0 ) = y 0 , (2.3) International Scientific Publications and Consulting Services

is a negative fuzzy number. 3 .
If u and v are negative then u ⊗ v = (−u) ⊗ (−v) is a positive fuzzy number.Now they (Ban and Bede) defined the cross product as follows: Definition 2.7.(cross product) The binary operation ⊗ on R * F introduced by Theorem 2.4 and Remark 2.2 is called cross product of two fuzzy numbers.

( 2 . 6 ) 2 . 5 .
Remark The cross product extends the scalar multiplication of fuzzy numbers.Indeed, if one of operands is the real number k identified with its characteristic function then for all r ∈ [0, 1], k r = k r = k and following the above formulas of calculus we get the results.

Figure 3 :
Figure 3: The approximate solution to the FFDE at t = 0.1 and h = 0.01

Table 1 .
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Table 2 .
r RK-Fehlberg RK-5 Exact Solution w(t i ; r) w(t i ; r) w(t i ; r) w(t i ; r) y(t i ; r) y(t i ; r)