The theory of dynamic equations has many interesting applications in control theory, mathematical economics, mathematical biology, engineering and technology. In some cases, there exists uncertainty, ambiguity or vague factors in such problems, and fuzzy theory and interval analysis are powerful tools for modeling these equations on time scales.

In this paper, we introduce the Adomian decomposition method for the following class of first order fuzzy dynamic equations where

Here $F\left(\mathbb{R}\right)$ denotes the set of all real fuzzy numbers, $\tilde{0}$ denotes the zero fuzzy number and ${\delta }_H$ denotes the first type fuzzy delta derivative on $\mathbb{T}$.

To the best of our knowledge, there is a gap in the references for investigations of the Adomian decomposition method for fuzzy dynamic equations on time scales. Here, in this paper we try to fill out this gap.

This paper is organized as follows. In the next section, we give some basic definitions and facts of fuzzy dynamic calculus on time scales. In Section 3, we give an exposition of the Adomian decomposition method on time scales. In Section 4 we introduce the Adomian decomposition method for the problem (1), (2). In Section 5, we give a numerical example.

In this section, we will give some basic definitions and fact of fuzzy dynamic calculus on time scales. For detailed study of fuzzy dynamic calculus on time scales we refer the reader to the book [2].

Suppose that $\mathbb{T}$ is a time scale with forward jump operator and delta differentiation operator $\sigma$ and $\Delta$, respectively. With $F\left(\mathbb{R}\right)$ we will denote the space of the real fuzzy numbers and with $D(·,·)$ we will denote the Hausdorff distance between the real fuzzy numbers. For more details for fuzzy numbers and Hausdorff distance between the real fuzzy numbers we refer the reader to the appendix of the book [2].

The defined ${\delta }_H$-derivative has the following properties.

Now, we introduce the conception for the first type fuzzy delta integration on time scales. Let $I\subset \mathbb{T}$.

In this section, we will describe the Adomian decomposition method on arbitrary time scale $\mathbb{T}$ with forward jump operator and delta differentiation operator $\sigma$ and $\Delta$, respectively. The exposition in this section, follows the exposition in [3].

Denote the set consisting of all possible strings of length n, containing exactly k times $\sigma$ and $n-k$ times $\Delta$ operators by ${\displaystyle S_k^{\left(n\right)}}$. Then we have for every ${\displaystyle t,s\in \mathbb{T}}$. For $s\in \mathbb{T}$, $l,m,n\in {\mathbb{N}}_0$, set and for any $m,n\in {\mathbb{N}}_0$, we have For $n\in {\mathbb{N}}_0$, $t,s\in \mathbb{T}$, define the polynomials Note that Note also that and whereupon and so on. Below we denote by $B_i^j$, $i,j\in \mathbb{N}$, the constants for which Suppose that $u:\mathbb{T}\to \mathbb{R}$ is a given function which has a convergent series expansion of the form Suppose also that $f:\mathbb{R}\to \mathbb{R}$ is a given analytic function such that where $A_n$, $n\in {\mathbb{N}}_0$, are given by Here the functions $c(\nu ,n)$ denote the sum of products of $\nu$ components $u_j$ of u given in (10), whose subscripts sum up to n, divided by the factorial of the number of repeated subscripts, i.e., and so on. Suppose now that u is given by the convergent series We wish to find the respected transformed series for $f\left(u\right)$. From (10), we have and hence, Thus, Hence, For $n=0$, we have Thus, For $n=1$, we find or whereupon For $n=2$, we have or Then or whereupon For $n=3$, we find or whereupon and so on. We get the following result.

In this section, we will introduce the Adomian decomposition method for the problem (1), (2). Firstly, note that the problem (1), (2) can be rewritten in the form Consider the problem where ${\underline{f}}^{\alpha }:\mathbb{R}\to \mathbb{R}$ is an analytic function. We propose a solution of the IVP (13), in the form Like in the general case, we suppose that We have and Let Then we have Taking the Laplace transform of both sides of the dynamic equation (13) we obtain Then we obtain Now, by taking inverse Laplace transform of both sides, we get Employing (14), we have In order to equate the coefficients of the time scale monomials $h_k(t,t_0)$ on both sides, we reorder the sums as follows. This results in the following nonlinear system for determining the constants ${\underline{c}}_j$, $j=0,1,\dots$. Notice that the system is infinite and nonlinear in its unknowns. However, the nonlinearity is of polynomial type. This is a results of the nonlinear structure of the function ${\underline{f}}^{\alpha }$.

Now, consider the problem where ${\overline{f}}^{\alpha }:\mathbb{R}\to \mathbb{R}$ is an analytic function. We will search a solution of the IVP (17), in the form Assume that We have and As above, we get the following system for the constants ${\overline{c}}_j$, $j=0,1,\dots$.

Consider the initial value problem associated with the first order nonlinear fuzzy dynamic equation of the form $\alpha \in [0,1]$. Consider the IVP Assume that the solution has the series representation where $c_j$, $j\in {\mathbb{N}}_0$ are the coefficients to be determined. where The infinite nonlinear system for this example has the form Solving this nonlinear system one can approximately obtain $c_i$, $i\in \mathbb{N}$, and hence, the approximate solution of the initial value problem which is As above, where Let $\mathbb{T}=2^{{\mathbb{N}}_0}$ and $t_0=1$. Then $\sigma \left(t\right)=2t$, $t\in \mathbb{T}$, and Next, Really, Moreover, Indeed, Note that Then For $n=1$, we get whereupon Therefore $B_1^1=1$. For $n=2$, we find or or whereupon we get the system whose solutions are Next, or $t\in \mathbb{T}$, or whereupon we get the system whose solutions are Now, we consider the first four equations of (23) with the following approximations or whereupon we get Replacing $\alpha$ with $2\alpha$, we find Therefore approximative solutions are and