Adomian polynomials method for dynamic equations on time scales

A recent study on solving nonlinear di erential equations by a Laplace transform method combined with the Adomian polynomial representation, is extended to the more general class of dynamic equations on arbitrary time scales. The derivation of the method on time scales is presented and applied to particular examples of initial value problems associated with nonlinear dynamic equations of rst order.


Introduction
In a recent paper, a series solution method based on combining the Laplace transform and Adomian polynomial expansion was proposed to nd an approximate solution of nonlinear dierential equations [8]. It uses the expansion in Adomian polynomials dened in [1,2]. An important drawback of the Laplace transform method is the fact that it cannot be applied in the case of nonlinear dierential equation in general. In order to cope with this problem, the authors of [8] suggested the use of Adomian polynomial expansion of the nonlinear function of the dependent variable involved in the dierential equation.
In this work, we propose a counterpart of this method on an arbitrary time scale and derive its general formulation for a dynamic equation of any order. We conrm that when the time scale is the set of real numbers, our method reduces to that in [8].
Our presentation is organized as follows. First, we recollect some preliminary information on time scales in Secton 2. In Section 3, we derive the method for an n-th order nonlinear dynamic equation. The next section contains the application of the method to specic examples of rst order nonlinear dynamic equations. The last section is devoted to conclusion and some further directions for study.

Preliminaries
We start this section with a review of some basic concepts on time scales which are used throughout the paper. A detailed information on basic calculus on time scales can be found in [3,4,5]. Denition 2.1. A time scale, usually denoted by T, is an arbitrary nonempty closed subset of the real numbers. On a time scale T, 2. A function f : T −→ R which is regulated and continuous at right dense points of T is called rdcontinuous. The set of rd-continuous functions is denoted by C rd (T).

A continuous function
T κ \D is countable and contains no right-scattered elements of T, (c) f is dierentiable at each t ∈ D. Theorem 2.1 ([3], [4], [5]). Let t 0 ∈ T, x 0 ∈ R, f : T κ −→ R be a given regulated function. Then there exists exactly one pre-dierentiable function F satisfying Denition 2.5. If f : T −→ R is a regulated function, any function F dened in Theorem 2.1 is a preantiderivative of f . For a regulated function f the indenite integral is given as More details on delta integral can be found in [3], [4] and [5]. In the following discussion we need the denition of the generalized exponential function on time scales. Its denition is based on the regressive functions, that is, functions f : T → R satisfying The set of all regressive and rd-continuous functions f : T → R is usually denoted by R(T) or R. The set R endowed with the operation ⊕ dened as is a group called regressive group (R, ⊕). For any f ∈ R, we dene for all t ∈ T κ , and the operation in R as Clearly, for f, g ∈ R, we have We also need the Hilger complex numbers which are dened by for h > 0 and C 0 = C. We also dene for h > 0, and Z 0 = C. Finally, the cylindrical transformation ξ h : C h → Z h is dened as where Log is the principal logarithm function. If h = 0, we take ξ 0 (z) = z for all z ∈ C.
Denition 2.6. For f ∈ R, the generalized exponential function is dened as More infomation on the generalized exponential function can be found in [3,5]. Below we give the denition of Laplace transform on time scales.
Denition 2.7. [5,6] Denote by T 0 , a time scale such that 0 ∈ T 0 and sup T 0 = ∞. For a function f : T 0 → C, dene the set For all z ∈ D(f ), the Laplace transform of the function f is dened as Denition 2.8. The monomials h k (t, s), k ∈ N 0 on a time scale T are dened as follows [5].
The Taylor formula on a general time scale is given as follows.

Adomian polynomials method on time scales
In this section we derive the method and present its application to a dynamic equation of arbitray order with a nonlinear term.
Let T be a time scale with forward jump operator σ, delta dierentiation operator ∆ and graininess function µ. In the rest of the paper we assume that µ is delta dierentiable on T. Denote the set consisting of all possible strings Λ n,k of length n, containing exactly k times σ and n − k times ∆ operators by S The following theorem is needed in the derivation of the method. Theorem 3.1. [5] For every m, n ∈ N 0 we have By Theorem 3.1, for any m, n ∈ N 0 , we have For n ∈ N 0 , t, s ∈ T, dene the polynomials Note that on any time scale h 1 (t, s) = t − s and we have Note also that and by (3), we get and so on. Below we denote by B j i , i, j ∈ N, the constants for which Example 3.1. Let α ∈ R. Then using the Taylor formula and the fact that the Taylor series of e α (t, s) yields Now, suppose that u : T → R is a given function which has a convergent series expansion of the form Suppose also that f : R → R is a given analytic function such that where A n , n ∈ N 0 , are given by Here the functions c(ν, n) denote the sum of products of ν components u j of u given in (6), whose subscripts sum up to n, divided by the factorial of the number of repeated subscripts, i.e., A 0 = f (u 0 ), and so on. Suppose now that u is also given by the convergent series We wish to nd the respected transformed series for f (u). From (6), we have and hence, u n = c n H 1 n (t, t 0 ) n ∈ N 0 . Thus, we obtain a series representation for f of the form which compared with he expansion (7) gives the coecients A n (c 0 , c 1 , . . . , c n ) as A n (c 0 , c 1 , . . . , c n )H 1 n (t, t 0 ) = A n (u 0 , u 1 , . . . , u n ), n = 0, 1, . . . .
For n = 0, we have Thus, For n = 1, we nd For n = 3, we nd and continuing in this way we get the following result.
Theorem 3.2. Let u : T → R be a function with a convergent expansion given in (8). Let f : R → R be an analytic function having the form (7). Then where c 0 = 1, . . .
On the other hand, by Theorem 3.2, we obtain = 2c 0 c 1 and so on, i.e., we get (9).

Examples of IVPs for rst order nonlinear dynamic equations
As a particular case, we consider an IVP associated with a rst order dynamic equation of the form where f : R → R is an analytic function. We propose a solution of the IVP (11), in the form Like in the general case, we suppose that On the other hand, by (5) we have and Let L (y(t)) (z) = Y (z).
Taking the Laplace transform of both sides of the dynamic equation (11) we obtain Then we arrive at Now, by taking the inverse Laplace transform of both sides, we get Employing (12), we have In order to equate the coecients 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 c j , j = 0, 1, . . ..
Notice that the system is innite and nonlinear in its unknowns. However, the nonlinearity is of polynomial type. This is a result of the nonlinear structure of the function f .
for k ∈ N and hence, B j k = k!δ k,j for k ∈ N and j = 1, . . . k. In this case, the system (14) becomes which is consistent with the study given in [8].

· · ·
In the next two examples we take f to be a nonlinear function.
where e y(t) is the exponential function on the set of real numbers. Assume that the solution has the series representation where c j , j ∈ N 0 , are the coecients to be determined. By Theorem 3.