A new method to find fuzzy n th order derivation and applications to fuzzy n th order arithmetic based on generalized h-derivation

In this paper, fuzzy nth-order derivative for n ∈ N is introduced. To do this, nth-order derivation under generalized Hukuhara derivative here in discussed. Calculations on the fuzzy nthorder derivative on fuzzy functions and their relationships, in general, are introduced. Then, the fuzzy nth-order differential equations is solved, for n ∈ N .


Introduction
The H-derivative of fuzzy number valued function was introduced by Siekalla in [22].This derivation amplifies the fuzziness when time goes by [8], thus strongly general differentiability was introduced in this paper and have been studied by many researchers, this concept allows us to solve the problems of H-derivative.The fuzzy derivations are very important for solving fuzzy equations for instance, fuzzy differential equations and fuzzy integro-differential equations.
The first order equations under H-derivation studied by Bede initially at [3,8].He explained four cases of derivatives for fuzzy first order derivative.Two cases of them are always very important and the two others are important to acquire switching point.He used these four cases of derivatives for solving fuzzy differential equations.Chalco, used first two cases of derivations, because the two others cases are constant, [11].
General H-derivative has been used to study the second order derivation by Allahviranloo, [5] and Zhang, [26].Their studies were used to get the existence of the fuzzy second order equations under general H-derivative.Allahviranloo et.al obtained the solutions of nth-order fuzzy linear differential equations by approximating method in [1,2].Allahviranloo and hooshangian introduced fuzzy generalized H-differential and used it to solve fuzzy differential equations of secondorder, [4].
In this paper we use general H-derivative to find high order derivation.We acquire cases of derivations and we use them to invent relations Corresponding author: L.hooshangian@gmail.com(LalehHooshangian), Tell, Fax:+982144865030.
105 between derivatives and their cases, then we apply them to investigate summation and minus of fuzzy derivatives and relationships between them.Indeed, with general differentiability, we can find more relationships for a larger classes of them rather than using H-derivative.
In section 2, we review briefly some needed concepts.In section 3 we introduce nth-order derivation for all n ∈ N , the minus and summation of two the fuzzy functions under H-derivative which are approved for nth-order derivation.Indeed, an algorithm is introduced to find fuzzy nth-order derivation and its cases for all, in general.In section 4, the fuzzy H-difference between two nth-order derivative of fuzzy functions is demonstrated and the examples to illustrate more are presented.In the final section, fuzzy differential equations in general form are solved.Our solution have been based on the generalized H-derivation.Finally, conclusion will be drawn in Section 5.

Basic Concepts
The basic definitions of a fuzzy number are given as follows: Definition 1. [14] A fuzzy number is a fuzzy set like u : R→ [0, 1] which satisfies: Definition 2. [14] The metric structure is given by Hausdorff distance satisfying the following properties, that R F is denoted the class of fuzzy subsets of real axis: ) is a complete metric space and following properties are well known: y and it is denoted x ⊖ y.Definition 4. [7] Let F : I → R F and t 0 ∈ I.We say that F is differentiable at t 0 if there is F ′ (t 0 ) ∈ R F such that either (I) For h > 0 sufficiently close to 0, the Hdifferences F (t 0 +h)⊖F (t 0 ) and F (t 0 )⊖F (t 0 −h) exist and the following limits For h > 0 sufficiently close to 0, the Hdifferences F (t 0 )⊖F (t 0 +h) and F (t 0 −h)⊖F (t 0 ) exist and the following limits For h > 0 sufficiently close to 0, the Hdifferences F (t 0 +h)⊖F (t 0 ) and F (t 0 −h)⊖F (t 0 ) exist and the following limits For h > 0 sufficiently close to 0, the Hdifferences F (t 0 )⊖F (t 0 +h) and F (t 0 )⊖F (t 0 −h) exist and the following limits in the first form (I), then f α and g α are differentiable functions and

Generalized Fuzzy N th-order Derivative
In this article is necessary to introduce the E and E j items in the following terms: , for all j that j = 2, 3, ..., n.
Theorem 4. For all F, G ∈ R F and c ∈ R, for all j = 1, 2, ..., n, is approved the following items:

Proof. The proof is clear.
At first we approve a theorem on the Hakuhara difference that are needed here under: a) The proof is trivial.b) ⊖x = ⊖y ⇒ 0 ⊖ x = 0 ⊖ y, thus by Definition 3, there exists u ∈ R F that 0⊖x = 0⊖y = u, thus 0 = x + u and also 0 = y + u, thus x + u = y + u and it is mean x = y c) If ⊖(⊖x) exists, then there is u ∈ R F that 0 ⊖ (⊖x) = u.In following we prove that u = x: 0⊖(⊖x) = u, thus 0 = u+(⊖x), then 0⊖u = ⊖x and by using (a) we have 0⊖u = ⊖u = ⊖x, using (b) we can result x = u.d) If x ⊖ y = z, thus we have x = z + y, then we can write x = y + z ⇒ x ⊖ z = y e) If x ⊖ (y + z) exists, then there exists u ∈ R F that x ⊖ (y + z) = u now by Definition 3 x = u + y + z, thus x ⊖ y = u + z, now we can gain x ⊖ y ⊖ z = u f) If there exists x ⊖ (y ⊖ z), then u ∈ R F which x⊖(y⊖z) = u, by using Definition 3 we can write x = u + (y ⊖ z) and x = u + y ⊖ z, now we can write x + z = u + y and therefore x + z ⊖ y = u.g) By using (a) we have ⊖ax = 0 ⊖ ax, thus there exists u ∈ R F which ⊖ax = 0 ⊖ ax = u now by Definition 3 we have 0 = u + ax, thus 0 = u a + x and by using (a) we write 0 ⊖ x = u a , thus 0 + (⊖x) = u a and 0 + a(⊖x) = u therefore a(⊖x) = u.h) If there exists ⊖(x ⊖ y) then there is a u ∈ R F which ⊖(x ⊖ y) = u, thus 0 ⊖ (x ⊖ y) = u and by Definition 3 we have 0 = u + x ⊖ y and 0 + y = u + x, thus 0 + y ⊖ x = u therefore y ⊖ x = u.
In Definition 7, it is clear that the nth-ordered derivative is depend on the (n − 1)th-ordered derivative, (n − 1)th-ordered derivative depend on the (n − 2)th-ordered derivative and so on.Using this dependance and by using Theorem 5, for F : I −→ R F , we have four cases of derivatives that can be proved as follows: Theorem 6.For all integer n -even and oddwe have four cases for H-derivative: (A): If n = 2k, k = 1, 2, ... we have four cases: (1): If even quantity of F (i) (t 0 ), i = 1, 2, ..., n are differentiable in case (I) and the rest in case (II) of Definition 7: (2): If odd quantity of F (i) (t 0 ), i = 1, 2, ..., n−1 are in case (I) and the rest in case (II) of Definition 7: (3): If even quantity of F (i) (t 0 ), i = 1, 2, ..., n are differentiable in case (III) and the rest in case (IV) of Definition 7: (4): If odd quantity of F (i) (t 0 ), i = 1, 2, ..., n−1 be in case (III) and the rest in case (IV) of Definition 7: .., we have four cases.
(1): If odd quantity of F (i) (t), i = 1, 2, ..., n are differentiable in case (I) and the rest in case (II) of Definition 7: (2): If even quantity of F (i) (t 0 ), i = 1, 2, ..., n− 1 are in case (I) and the rest be in case (II) of Definition 7: (3): If odd quantity of F (i) (t 0 ), i = 1, 2, ..., n are differentiable in case (III) and the rest in case (IV) of Definition 7: (4): If even quantity of F (i)(t 0 ) , i = 1, 2, ..., n − 1 are in case (III) and the rest be in case (IV) of Definition 7: )(E j (F (t 0 + jh))) Proof.By induction, we consider the method for nth-order fuzzy derivative as accurate, the method should be approved for (n + 1)th-order fuzzy derivation.The theorem is proved for case (1) of (A), in the other cases are proved similarly.In the case (I) of Definition 7, the nth-order derivative is in following: and (n + 1)th-order derivation in case (I) is: in the other hand by Theorem 6 for even n, we have: By replacing elements of Eq. ( 11) by (10) we have By expanding limits and by the following formulate: we can reach the followings: thus Remark 3. Now by replacing n = 1 in Eqs. ( 5), ( 6), ( 7) and ( 8), the Definition 4 and the other definitions in Ref [3] can be got.
Theorem 7. Let F : I → R f is nth-ordered differentiable on each t ∈ I in the case (III) or (IV) in Definition 7. Then F (n) ∈ R for all t ∈ I.
Proof.Suppose that, (I) and (III) are coincided simultaneously.Then there are A, B, C ∈ R F , which for only two first limits in cases ( 1) and (3) in Theorem 6 we have here: and and Thus we get

Arithmetics on the Fuzzy N th-ordered Derivations
In this section calculations of the fuzzy nthordered derivation and their relationships are researched.These calculations are concluded summation and minus of two fuzzy functions and scalar multipliers of one fuzzy function.
Proof.Without loosing generality for even n, if even quantity of f (i) , i = 1, 2, ..., n are differentiable in case (I) and f (t) = c ⊙ g(t), for all t ∈ I are considered.Using Theorem 6 we will have: h n by putting f (t) = c ⊙ g(t), the above equations will be written as below: h n By Theorem 6 the equations can be written in the following case: )(E j (g(t + (j + 1)h)) h n then we have )(E j (g(t + (j + 1)h)) proof for the other cases is similar and omitted.
We show that (a) is correct, the other results are provable similar.
Proof.Since f is continuous, it must be integrable.Is considered that ( 14) is solution of initial value problem (13).It is obvious that the solution for the following problem: x (n+1) (t) = f (t, x(t), x ′ (t), x ′′ (t), ..., x (n) (t)), x(t 0 ) = k 0 , x ′ (t 0 ) = k 1 , . . .By exercising integral over [t 0 , t], we can equivalently have: Then the solution of this fuzzy differential equation for all t ∈ [0, ∞] is Let x(t) and x ′ (t) are (I)-differentiable, the solution by Theorem 16 is obtained in the following: Now we can solve this interval-value integral equation, it means two crisp integral equation should be solve.The solution is gained by the Modified Adomian method in the following: Let x(t) and x ′ (t) be (II)-differentiable, the solution by Theorem 16 is gained in the following interval equation: means, the solution by solving two crisp integral equations by Modified Adomian method will be obtained in the following term: Let x(t) be (I)-differentiable and x ′ (t) be (II)differentiable, the solution by Theorem 16 is in the bottom interval equation: It means, the solution by solving a crisp integral equation system by Modified Adomian method will be obtained in the bottom term: Let x(t) be (II)-differentiable and x ′ (t) be (I)differentiable, the solution by Theorem 16 is obtained in the following: It means, the solution by solving a crisp integral equation system by Modified Adomian method will be gained in the sequence: 2 (α − 1)].

Conclusion
In this work, we introduced a new method for finding generalized fuzzy nth order derivative and we proved some theorems in the relationships between fuzzy derivatives of nth order and we presented the solution of fuzzy differential equations of nth order.For future research one can use generalized fuzzy nth order derivative for obtaining the switching point of fuzzy differential equations that is introduced by Bede [8].

1 .
u is an upper semi-continuous function, 2. u(x) = 0 outside some interval [a,d], 3.There are real numbers b, c such as a ≤ b ≤ c ≤ d and 3.1 u(x) is a monotonic increasing function on [a, b], 3.2 u(x) is a monotonic decreasing function on [c, d], 3.3 u(x) = 1 for all x ∈ [b, c].

1 , 2 ,Theorem 14 .
..., n.Then the initial value problem (13) has a unique solution on [a, b] in each sense of differentiability.See Theorem 3.3 in [21].For even number n, if f : [a, b] −→ R F and let a = b 0 < b 1 < ... < b n = b be a division of the interval [a, b] such that f is n-order differentiable of (I) or (II) differentiable in the sense of Definition 7 on each of the intervals [b i−1 , b i ], i = 1, 2, ..., n, with the same case of (n − 1)-order differentiable on each subinterval.Then: 1) j+1 ⊖ ( n j )f (t)(t + (j + 1)h) h n dt ...)dt)dt Theorem 15.For odd number n, if f : [a, b] −→ R F and let a = b 0 < b 1 < ... < b n = b be a division of the interval [a, b] such that f is norder differentiable of (I) or (II) differentiable in the sense of Definition 7 on each of the intervals [b i−1 , b i ], i = 1, 2, ..., n, with the same case of (n − 1)-order differentiable on each subinterval.Then:

k 2 2 !fExample 3 .x
(t − t 0 ) 2 + ... + c n−1 ( k n−1 (n−1)!(t − t 0 ) n−1 + (s, x(s), x ′ (s), ... , x (n) (s))ds...dsds)))).Let following fuzzy differential equation with initial values:′′ (t) = x(t) x(0) = [α − 1, 1 − α], x ′ (0) = [α, 2 − α] [24]nition 5.[24]Let F : I → R F be a setvalued function.A point t 0 ∈ I is said to be a switching point for the differentiability of F , if in any neighborhood T of t 0 there exist points t 1 < t 0 < t 2 such that: Type 1: F is differentiable at t 1 in the sense (I) of Definition 4 while it is not differentiable in the sense (II) of Definition 4 and F is differentiable at t 2 in the sense (II) of Definition 4 while it is not differentiable in the sense (I) of Definition 4. or Type 2: F is differentiable at t 1 in the sense (II) of Definition 4 while it is not differentiable in the sense (I) of Definition 4 and F is differentiable at t 2 in the sense (I) of Definition 4 while it is not differentiable in the sense (II) of Definition 4.
[20]rem 2.[20]Let F : I → R F be differentiable on each t ∈ I in the sense (III) or (IV) in Definition 4. Then F ′ (t) ∈ R for all t ∈ I.