Intuitionistic fuzzy Dirichlet problem

In the present paper, a new definition of intuitionistic fuzzy derivative is presented, which a generalization of fuzzy fractional derivative and is compatible with the “crisp” definition of fractional derivative. We prove some properties concerning this definition. Finally, the existence of Dirichlet problem is proven.


Introduction
The present paper gives a methode of solving the following intuitionistic fuzzy problem by means of the intuitionistic fuzzy Laplace transform.
where 0 < β ≤ 1, the operator c gH D β denotes the Caputo fractional generalized derivative of order β, f : Fractional calculus is a mathematical branch investigating the properties of derivatives and integrals of non-integer orders. It applies in modelling of many physical and chemical processes and in engineering [3]. Podlubny [12] and Kilbas et al [9] gave the idea of fractional calculus and consider Riemann-Liouville differentiability to solve FFDEs. Agarwal et al. [1] proposed the concept of solutions for fractional differential equations with uncertainty.
Intuitionistic fuzzy sets (IFS) were first formulated by Atanassov (see [4,5]). An IFS is a generalization of fuzzy sets introduced by Zadeh [14]. In fuzzy sets, the membership value, (µ(x)), of x ∈ X (called the universe) is just a single real number, usually in [0, 1] and the nonmembership of x is taken as (1 − µ(x)). But for intuitionistic fuzzy sets the membership value (µ(x)) as well as the non-membership value (ν(x)) should be taken into account for describing any x in X such that the sum of membership and non-membership is less than or equal to 1. Thus, an IFS is expressed by an ordered pair of real numbers (µ(x), ν(x)) and π(x) = 1 − µ(x) − ν(x) is called the hesitancy. There have been several attempts to quantify the uncertainty associated with fuzzy sets as well as with intuitionistic fuzzy sets [7]. The authors in [8] studied the solution concept of fractional differential equations with intuitionistic fuzzy initial data under generalized fuzzy Caputo derivative. In [11] Melliani et al. introduced the extension of Hukuhara difference in the intuitionistic fuzzy case. Allahviranloo, Armand and Gouyandeh in [2] solve the fuzzy fractional differential equations under generalized fuzzy Caputo derivative. From this last paper, we introduce in this paper the concept of generalized intuitionistic fuzzy Caputo derivative and we will build our ideas and properties in order to solve the fractional equation by means of the Laplace transform, which is the one of the interesting transforms used for solving intuitionistic fuzzy differential equations. Using this transform allows reduction of the problem. The advantage of intuitionistic fuzzy Laplace transform is to solve the problem directly without determining a general solution.
The present paper investigate the analytic solution of the following problem where FT is the set of all intuitionistic fuzzy triangular numbers, and f is a function defined from ]a, b[ into FT . This paper is organized as follows. After this introduction, we recall in Section 2 some concept concerning the fuzzy metric space and generalized Hukuhara's difference, which inspired the method of build a fuzzy Banach space. The generalized differentiability takes place in the later section. We will presented the main result in Section 5 and we end our work by an example in order to illustrate our results.

Preliminaries
In this section we will present some definitions and properties, which we will build our work upon.

Definition 1. [10]
The set of all intuitionistic fuzzy numbers is given by with the following conditions: 1. For each u, v ∈ IF 1 is normal, i.e, ∃x 0 , x 1 ∈ R, such that u(x 0 ) = 1 and v(x 1 ) = 1.
2. For each u, v ∈ IF 1 is a convex intuitionistic set, i.e., u is fuzzy convex and v is fuzzy concave.
3. For each u, v ∈ IF 1 , u is lower continuous and v is upper continuous.
[10] For α ∈ [0, 1], we define the upper and lower t-cut by Definition 3. The intuitionistic fuzzy zero is an intuitionistic fuzzy set defined by We define two operations on IF 1 by According to Zadeh extension, we have } be a family of subsets in R satisfying the following conditions 2. M α and M s are nonempty compact convex sets in R, for each α ∈ [0, 1].

For any non-decreasing sequence
We define u and v by Remark 2. [10] We have that IF 1 , d p is a complete metric space.

The generalized Hukuhara derivative of an intuitionistic fuzzy-valued function
The concept of intuitionistic fuzzy Hukuhara difference was introduced by the authors in [11]. In this paper we will give the definition of generalized Hukuhara difference between two intuitionistic fuzzy numbers.
Definition 4. The generalized Hukuhara difference of two fuzzy numbers u, v and u , v ∈ IF 1 is defined as follows Note that the α-level representation of a fuzzy-valued function f : Definition 5. The generalized Hukuhara derivative of an intuitionistic fuzzy-valued function f : We can define the generalized derivative of higher order by (2) gh exists and continues, with respect to metric d ∞ .
We set As in the previuos definition, we will give the definition of intuitionistic fuzzy fractional Riemann-Liouville integral. If the α-levels represents an intuitionistic fuzzy element and the integral preserves the monotony, then we have by Theorem 1 the family define an intuitionistic fuzzy element, which is the Riemann-Liouville fractional integral of f on

Intuitionistic fuzzy generalized Hukuhara partial differentiation
Throught this paper we denote T F the sets of all triangular intuitionistic fuzzy numbers. We use the same proof as Bede in [6], we can show that if u, v , u , v ∈ T F , then the difference u, v − g u , v always exists in T F and u, v − g u , v = (−1)( u, v − g u , v ). Thus we get FT, . p is a Banach space. In this section f : D ⊂ R × R + → FT is called the two variable fuzzy-valued function. The parametric representation of the fuzzy-valued function fis expressed by f (x, t, α − ) = f (x, t, α − ), f (x, t, α − ) and f (x, t, α − ) = f (x, t, α + ), f (x, t, α + ) .
f (x, t; α − ) both partial differentiable w.r.t. t at (x 0 , t 0 ). We say that Inspired from [13], we present the following definition.
Definition 12. f : R × R + → T F . We say that the function t = h(x), is switching boundary for the differentiability of f (x, t) with respect to t, if for all x belonging to the domain of h and for all t ∈ R + , there exist points t 0 < t 1 < t 2 such that 1. at (x, t 1 ) (1) holds while (2) does not hold and at (x, t 2 ) (2) holds and (1) does not hold, or 2. at (x, t 1 ) (2) holds while (1) does not hold and at (x, t 2 ) (1) holds and (2) does not hold.
and f (x, t) = p(x, t)u(x). Then ∂ t gH f (x, t) exists and In fact, the function h 2 (t) is switching boundary type 1 for differentiability of f (x, t) with respect to t.
Proof. Since p is valued in R + then we can set f (x, t; α) by report at t. In the same manner, if h 2 (t) < x < h 3 (t) we get with respect to the second variable t.

Intuitionistic fuzzy Dirichlet problem in one dimension
In this section, we will investigade the following problem where f is a suitable function.
We introduce the auxiliary function v verifying u = v and v = f .
And A mapping u : [a, b] → T F is a solution of (3) if only if v is a solution of the following problem: Proof. Since u, v and f are at values in IF 1 , we put u = u 1 , We will discuss four cases.
In this case we have Using Theorem 2, we get v = f .
2. If u is (ii)-diff and v is (ii)-diff.
In this case we have Using Theorem 2 we get v = f .
In this case we have Also by Theorem 2 we get v = − gH f .
In this case we have Also by Theorem 2 we get v = − gH f .
Before we turn to the solution, we will give a property of the map v. Lemma 1. v is a continuous function.
a t a f (τ )dτ dt . Since The function is monotone increasing on I with respect to α, In the same manner, the we get The function B 2 (α + ) (b−a)(t−a) − gH t a f (τ, α + )dτ is monotone decreasing on I with respect to α.
By Theorem 1 we get the following property.
Thus v is a continuous derivable map on [a, b].  which implies that φ is a solution of (3). If x is (i)-diff and x is (ii)-diff.
On the other hand, the problem