Solving Intuitionistic Fuzzy Transport Equations by Intuitionistic Fuzzy Laplace Transforms

In this paper, we use an intuitionistic fuzzy Laplace transforms for solving intuitionistic fuzzy hyperbolic equations precisely the transport equation with intuitionistic fuzzy data under strongly generalized H-differentiability concept. For this purpose, the intuitionistic fuzzy transport equation is converted to the intuitionistic fuzzy boundary value problem (IFBVP) based on the intuitionistic fuzzy laplace transform. The re-lated theorems and properties are proved in detail. Finally, we solve an example to illustrate this method.


Introduction
A simple approach to model propagation phenomena that emerge naturally under uncertainty is to use intuitionistic fuzzy partial differential equations (IFPDE).Problems implying time t as an independent variable usually involve parabolic or hyperbolic equations.An archetype of the intuitionistic fuzzy hyperbolic equations is the transport equation, wich can appear in many applications such as fluid mechanics, the dynamics of particuler interacting with matter (neutrons in a fissile material, photons in a planetary or stellar atmosphere, electrons and holes in a semiconductor, etc).
The notion of intuitionistic fuzzy set was first presented by Atanassov [1,2] as a generalization of the notion of fuzzy set, that is introduced by Zadeh (1965) [13], the authors in [3,4] are discussed Fuzzy laplace transform and Solving fuzzy Duffing's equation by the laplace transform decomposition, the idea of intuitionistic fuzzy metric space and Fuzzy differential systems under generalized metric spaces approach are presented in [7,9], while in [] the theorem of the existence of the soultion for intuitionistic fuzzy transport equations are proved.
This work is motivated by the solution of a intuitionistic fuzzy transport equation using the intuitionistic fuzzy Laplace transform.This appears to be one of the first attempts to solve one of the first attempts to solve these well known intuitionistic fuzzy partial differential equations under a strongly generalized Hdifferentiability.This paper is organized as follows: We provide some preliminaries that we will use all along this work, a result for intuitionistic fuzzy laplace transform is discussed in section 3, forthemore, the transport equation with intuitionistic fuzzy data is presented and solved with intuitionistic fuzzy laplace transform method in section 4, and the conclusion is made in section 5.

Preliminaries
An intuitionistic fuzzy set A ∈ X is given by define respectively the degree of membership and degree of non-membership of the element x ∈ X to the set A, which is a subset of X, and for every Obviously, every fuzzy set has the form the collection of all intuitionistic fuzzy number.An element u, v of IF 1 is said an intuitionistic fuzzy number if it satisfies the following conditions • u, v is normal i.e there exists x 0 , x 1 ∈ R such that u(x 0 ) = 1 and v(x 1 ) = 1.
• The non-membership function v is fuzzy concave i.e v(λ • u is upper semi-continuous and v is lower semicontinuous wich satisfy the following requirements: • u, v + (α) is a bounded monotonic increasing continuous function, • u, v + (α) is a bounded monotonic decreasing continuous function, • u, v − (α) is a bounded monotonic increasing continuous function, For α ∈ [0, 1] and u, v ∈ F 1 , the upper and lower αcuts of u, v are defined by We define 0 (1,0) ∈ IF 1 as we define the following operations by : For u, v , z, w ∈ IF 1 and λ ∈ R, the addition and scaler-multiplication are defined as follows we define the following sets: On the space IF 1 we will consider the following metric, Where .denotes the usual Euclidean norm in R n .
and the limits (in the metric D) ) and the limits ) and the limitis • for all h > 0 sufficiently small, ∃ , where g : R × IF 1 → IF 1 is supposed to be continuous, if equivalent to one of the integral equations : 3 The Intuitionistic Fuzzy Laplace Transform Method Theorem 3.1 Let u(x,t) be an intuitionistic fuzzy-valued function on [a, ∞) represented by assume that u(x,t, α), u(x,t, α), u(x,t, α), u(x,t, α) are Riemannintegrabl on [a,b] for every b ≥ a, and assume that for positive constants M(α), M(α), M(α) and M(α) such that

Then u(x,t) is an improper intuitionistic fuzzy Riemann-integrabl on [a, ∞) and the improper intuitionistic fuzzy Riemann-integral is an intuitionistic fuzzy number.
Furthermore, we have: Theorem 3.3 Let u (x,t) be an integrable fuzzyvalued function, and u(x,t) is the primitive of u (x,t) on [0, ∞).Then Where u is (i)-differentiable. Or Where u is (ii)-differentiable.

Remark 3 Let u(x,t) be continuous intuitionistic fuzzy-valued function on
dx and also we have

Application
Let us consider the following intuitionistic fuzzy nonhomogeneous transport equation By applying intuitionistic fuzzy Laplace transforms method, we get Therefore, Hence, So, aftewards, we find the values of the constants C 1 , C 2 , C 3 and C 4 , by replacing the value of the parametric form of u(0,t) in the system 8, we have Now, after using the inverse classical Laplace transform, u(x, s, u(x, s), u(x, s) and u(x, s) are calculated as follows where

Conclusion
We remark that u(x, s) u(x, s) and u(x, s) u(x, s) also the functions u(x, s), u(x, s) are increasing with respect to α and the functions u(x, s), u(x, s) are decreasing with respect to α.
So, this we shown that (u(x, s), u(x, s), u(x, s), u(x, s)) is the parametric form of the solution of the problem (1).