Existence and Uniqueness of Solution of an Uncertain Characteristic Cauchy Reaction-diffusion Equation by Adomian Decomposition Method

In this paper, the existence and uniqueness theorems for a solution to an uncertain characteristic Cauchy reaction-diffusion problem is studied by Adomian decomposition method. Sufficient conditions are presented for uniform convergence of the proposed method. Also, some illustrative examples are given for solving the problem.


INTRODUCTION
Mathematical uncertain or fuzzy models have attracted much attention in various fields of applied science [1,2,3].The general ideas and essential features of these models are their most applications, for instance, in the fuzzy integral equations [4,5,6,7,8] and the fuzzy differential equations [9,10,11].There is an interesting growth in fuzzy partial differential equations particularly in the past decade.In general, several systems are mostly related to uncertainty and inexactness.the problem of inexactness is discussed in general exact science, and that of uncertainty is discussed as vagueness or fuzzy.For fuzzy concepts, recently, the fuzzy partial differential equations have been thoroughly studied [12,13,14].

PRELIMINRIES
Let E be the set of all upper semi-continuous convex normal fuzzy numbers with bounded α -cut intervals [15].This implies that if is a complete metric space [17,18] and the following properties are wellknown: , also as a result we will have , and there exists Then we say u to be right (resp.left) H-differentiable in the direction e at ν , and call ).And if , then we say u to be H-differentiable in the direction e at ν , denote The fuzzy integral is also defined with respect to x as above. .More details of elementary definitions have been studied in [16,17,19].

UNCERTAIN CHARACTERISTIC CAUCHY PROBLEM (UCCP)
Reaction-diffusion equations describe a wide variety of nonlinear systems in physics, chemistry, ecology, biology and engineering [21,22,23] is the concentration parameter, and • φ is the reaction parameter.In this paper, we will consider three cases as follows: is the fuzzy extension of real-valued differentiable function, i.e. fuzzy function.Therefore we will have a fuzzy problem by the Eqs.( 1) and ( 2) called an UCCP.We consider φ as a known crisp function and, without losing generality, assume 0 < φ .Now an UCCP, using the parametric representation of fuzzy numbers is translated as system of equations in the crisp case.
is a fuzzy function with the parametric representation of then in the Eq. ( 1) ) , ( : Thus by assumptions 0 > η and 0 < φ , we can obtain ) , ( y x u by solving the following system in the crisp case The analytical approximate solution of this system is found by the Adomian Decomposition Method (ADM) [24,25,26,27,28,29] and an approximation is obtained for fuzzy solution of UCCP.

EXISTENCE AND UNIQUENESS OF FUZZY SOLUTION
In this section, the existence and uniqueness of fuzzy solution is obtained using successive iterations of ADM.Also we will show uniform convergence of ADM.
Consider the UCCP ), , ( ) where y L y ∂ ∂ = and where Then there exists a unique fuzzy solution ) , ( : of ( 4) and the successive iterations on Ω .First we prove the following Lemma.
As (11), we can obtain , )   (25) with the initial conditions (24).We obtain approximate solution of ( 23 as a approximation of the exact solution.Table 3 and Fig. 4 show the numerical results and the error function, respectively.

CONCLUSION
In this paper, we denoted existence and uniqueness theorems for an uncertain characteristic Cauchy problem with the fuzzy functions by successive iterations of ADM.We also introduced sufficient conditions for uniform convergence from successive iterations of ADM in an UCCP.Finally, using ADM and parametric representation of fuzzy numbers we proposed a strategy for solving an UCCP through several numerical examples.As future research we will apply this method and UCCP under generalized differentiability (see [30]).

Definition 2 . 1 .
can be embedded in E .For any R ∈ c, we define a fuzzy number c ~ by[16] A parametric representation of fuzzy number w is any pair ) − w is a nondecreasing function on [0,1], (ii) + w is a non-increasing function on [0,1],(iii) − w and + w are bounded and left continuous on (0,1] , and right continuous at 0

Lemma 4 . 1 .
If the conditions of Theorem 4.1 are satisfied and 1), ( Without losing generality, we assume )

ϕ
continuous, is similar to proving Theorem 4.1 and is omitted.Now, by the aforementioned hypotheses the uniform convergence of the sequence )is proved as follows.By relation(13) and their analogue components corresponding to 1 +

Fig. 2 .,
Fig. 2. Plot of D in Example 1.In the following examples can be seen that exist a monotonic convergence of the ADM without have the exact solution.The sequence of partial sums − k u and + k u are showed for various values of .Form figure and table in examples it can be seen that convergence is achieved rapidly within 20 15 − terms taken the decomposition series solution.Example 5.2.Consider 6 1 = η

Table 3
Comparison of the approximate values by the ADM in Example 3