Semi Analytical Solution for Fuzzy Autonomous Differential Equations

A BSTRACT . In this work, we have used fuzzy Adomian decomposition method to find the fuzzy semi analytical solution of the fuzzy autonomous differential equations with fuzzy initial conditions. This method allows for the solution of the fuzzy initial value problems to be calculated in the form of an infinite fuzzy series in which the fuzzy components can be easily calculated. The fuzzy series solutions that we have obtained are accurate solutions and very close to the fuzzy exact analytical solutions. Some numerical results have been given to illustrate the efficiency of the used method.


Introduction
The topic of fuzzy semi analytical methods (fuzzy series method) for solving fuzzy initial value problems (FIVPs) has been rapidly growing in recent years. Several fuzzy semi analytical methods have been proposed to obtain the fuzzy series solution of the linear and non-linear FIVB.
Fuzzy Adomian decomposition method is one of the fuzzy semi analytical methods used to obtain the fuzzy series solution of the FIVBs.
Researchers and scientists are continuing to develop this method for solving various types of the FIVBs because it represents an efficient and effective technique (For more details, see [1,2,3,4,6,7,8,10]). In this work, we will need many basic concepts in the fuzzy theory. These concepts can be found in detail in [5,8,11].

Fuzzy Autonomous Differential Equations
A fuzzy ordinary differential equation is called autonomous if it is independent of its independent crisp variable x. This implies that the nth order fuzzy autonomous differential equation is of the form [11]: The general idea of solving the fuzzy differential equation is based on transforming this equation into a system of non-fuzzy (crisp) differential equations.
In order to illustrate the above, we give the following example: If we consider the second order fuzzy autonomous differential equation To convert problem (2.8) into a system of the second order crisp ordinary differential equations, we apply the following steps: With the fuzzy initial conditions: Then we get the following system of second order crisp ordinary differential equations: (2.12) With the initial conditions: (2.13) With the initial conditions: Then the unique fuzzy solution of problem (8) is (2.16)

Fuzzy Adomian Decomposition Method
To understand the fuzzy Adomian decomposition method, we consider the nth order fuzzy differential equation [2,3,7]: Where F represents a general nonlinear fuzzy ordinary (or fuzzy partial) differential operator including both linear and nonlinear terms, x denotes the independent crisp variable, ( ) and ( ) are unknown fuzzy functions.
From section (2), We can conclude that: The fuzzy linear terms of (3.
and [ ] is the remainder of the fuzzy linear operator.
Thus the equation (3.1) may be written as: Such that: Note that: and it can be calculated as follows: •) If = , then we have: , then we have: Also, Note that:

Applied Examples
In this section, we will solve three fuzzy problems to illustrate the accuracy of the fuzzy Adomian decomposition method.
Example 1: Consider the first order fuzzy autonomous differential equation . .
Also, we find:   In this work, we have studied the fuzzy semi analytical solutions of the fuzzy autonomous differential equations. We have used the fuzzy Adomian decomposition method to find these solutions. Based on the numerical results we obtained, the fuzzy Adomian decomposition method is a highly efficient method in solving and gives accurate results, and in some cases, this method gives us the exact analytical solution. The accuracy of this method varies from one fuzzy differential equation to another, and this depends on the type of fuzzy differential equation, whether it is of the first order or the highest order, and also depends on the nature of the fuzzy differential equation, whether it is linear or non-linear.
Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.