## 1. Introduction

The differential transform method (DTM) was firstly introduced by Pukhov [

1,

2,

3]. However, his work passed unnoticed. Zhou [

4] rediscovered it to solve the linear and nonlinear equations in electrical circuit problems. Recently, a detailed comparison between the DTM and the Taylor series method (TSM) was carried out by Bervillier [

5]. The author pointed out that the DTM exactly coincides with the TSM when DTM is applied to solve the ODEs. Whereas, the DTM is a semi-numerical-analytic method, through which one can generates a Taylor series solution in a different manner. Firstly, the given problem is converted to a recurrence relation by using this approach. Furthermore, we can easily obtain the coefficients of a Taylor series solution. With this technique, we can apply directly to nonlinear problems without linearization, discretization or perturbation, and obtain an explicit and numerical solution with minimal calculations. Many researchers have successfully applied the DTM and its modifications have to solve various functional equations [

6,

7,

8,

9,

10,

11,

12,

13,

14,

15] and systems of non-singular equations [

16,

17,

18,

19,

20,

21].

Although being powerful, there still exist some difficulties in solving various of equations by the DTM. One of obstacles is to find a simple and effective way to obtain the differential transforms of nonlinear components. The traditional method is to expand those nonlinearities in an infinite series, and obtain its transformed function by imposing the differential transform upon the series. Subsequently, the transformed functions of nonlinear terms would readily be achieved by the one of its equivalent series. By using this approach, the computational difficulties will inevitably arise in determining the transformed function of this infinity series. In [

8], Chang and Chang proposed a new algorithm for calculating the differential transform

$F(k),k=0,1,2,\cdots $ of nonlinear function

$f(u(x))$ through a straightforward manner: obtaining the first term

$F(0)$ according to the definition of transform function, and taking the differentiation operation upon

$f(u(x))$ to get an identity; the recursive relation would be deduced by combing this identity and

$F(0)$; finally, one may obtain the other terms

$F(k),k=1,2,\cdots $ through the recursive relation. By using the same manner, authors [

22] also considered the differential transform of nonlinear function

$f(u(x,y))$. If we apply this method to those differential equations which have two or more nonlinearities, the computational budget will also inevitably be increased. In [

23,

24], an alternative algorithm was presented to calculate the differential transform of nonlinearities by using the Adomian polynomials. Meanwhile, due to the analytic operations of addition and multiplication without the differentiation operator, Duan’s Corollary 3 algorithm [

25] is eminently convenient for symbolic implementation to compute the Adomian’s polynomials with the help of Maple or Mathematica. However, it should be pointed out that this new and effective technique to handle differential transform of nonlinearities by the Adomian polynomials was merely subject to the nonlinear function with one variable.

In this study, we shall apply the DTM to solve the following systems of equations of Lane-Emden type:

subject to the initial conditions

where

${\lambda}_{1},{\lambda}_{2},\alpha $ and

$\beta $ are real constants,

$u=u(x)$ and

$v=v(x)$ are the solutions of the given system to be determined. If we set

${\varphi}_{i}(x)=0,i=1,2$, system (

1) becomes

Systems (

1) and (

3), with the initial conditions (

2), are called nonhomogeneous and homogeneous systems, respectively.

It is worth mentioning that

${f}_{i}(u,v),i=1,2$ given in systems (

1) and (

3) are the analytical functions of two independent variables. We inevitably encounter the complicated differential transforms of those nonlinearities with multi-variables, if the traditional DTM is employed to obtain the solution of them. As far as we know, there is not any new work which engaged in calculating the differential transforms of nonlinear functions with multi-variables. In this regard, We firstly disclose the relation between the differential transform and the Adomian polynomials of those nonlinear functions with multi-variables, and then employe the Adomian polynomials to evaluate the differential transform of nonlinear functions

${f}_{i}(u,v),i=1,2$. Furthermore, as we mentioned above, some researchers [

16,

17,

18,

19,

20,

21] discussed the systems of differential equations by using the DTM. However, all of these mentioned systems are non-singular. Whereas, systems (

1) and (

3) discussed in this study have a singular point

$x=0$ represented as

x with shape factors

${\lambda}_{1}$ and

${\lambda}_{2}$.

This system has arisen in the modelling of several physical problems such as pattern formation, population evolution, chemical reactions, and so on (see for example [

26] and references therein). Many researchers [

27,

28,

29,

30,

31,

32,

33] have focused their studies on the existence, uniqueness and classification of the systems by using the different methods. Compared with quite a number of the works on theoretical aspects, studies of the analytical approximate solutions of the systems have proceeded rather slowly. The numerical approaches for single Lane-Emden equation were presented in [

34,

35,

36] and references therein. In this work, we discuss the different systems (

1) and (

3), which have two equations. To our best knowledge, there were only two works devoted to this topic. In [

37], Wazwaz proposed the variation iteration method (VIM) to solve the systems, including the homogeneous and nonhomogeneous cases. In [

38], the authors obtained the analytical approximate solutions of the homogeneous systems by the Adomian decomposition method (ADM). The main difficulty of the systems of Lane-Emden type equations is the singular behavior at the origin. Both the VIM and the ADM overcome this obstacle by finding a corresponding Volterra integral form for the given system. However, there exists an inherent inaccuracy in identifying the Lagrange multiplier for the VIM, and the ADM is subject to a complicated

$n-$fold integration for solving the systems of differential equations. Here, we want to make fully use of those advantages of the DTM to reconsider the solutions of problems (

1) and (

3) under initial conditions (

2).

The rest of the paper is organized as follows.

Section 2 introduces the concept and fundamental operations of the DTM and the Adomian polynomials. In

Section 3, we shall present an easy and effective formula by using the Adomian polynomials to calculate the differential transform of any analytic nonlinearity. Some systems, including homogeneous and nonhomogeneous, are listed in

Section 4 to testify the validity and applicability of the proposed method. A brief conclusion is given in

Section 5 to end this paper.

## 3. Differential Transform of Nonlinearities

In this section, we shall present an easy and effective formula to evaluate the differential transform of any desired nonlinear function of multi-variables by using the Adomian polynomials.

**Theorem** **1.** If $u(x)={\sum}_{m=0}^{\infty}{a}_{m}{x}^{m}$, and $v(x)={\sum}_{m=0}^{\infty}{b}_{m}{x}^{m}$, thenwhere ${A}_{m}={A}_{m}({a}_{0},\cdots ,{a}_{m};{b}_{0},\cdots ,{b}_{m})$ are the Adomian polynomials of nonlinear function $f(u,v)$ with two variables. More generally, we have

**Theorem** **2.** If ${u}^{(i)}(x)={\sum}_{m=0}^{\infty}{a}_{m}^{i}{x}^{m}$, for $1\le i\le n$, thenwhere ${A}_{m}={A}_{m}({a}_{0}^{(1)},\cdots ,{a}_{m}^{(1)};\cdots ;{a}_{0}^{(n)},\cdots ,{a}_{m}^{(n)})$ are the Adomian polynomials of function $f({u}^{(1)},{u}^{(2)},\cdots ,{u}^{(n)})$ with multi-variables. **Theorem** **3.** Denote the differential transform of function $f(u,v)$ by $F(k)$, it holds thatwhere ${A}_{k},k=0,1,2,\cdots $ are the Adomian polynomials of nonlinear function $f(u,v)$, $U(k)$ and $V(k)$ are the transformed functions of $u(x)$ and $v(x)$, respectively. **Proof.** According to Equation (

5), we have

Using Theorem 1, we obtain

Furthermore, the following relation can be deduced by applying the differential transform to both sides of Equation (

8):

where

$DT\left\{\right\}$ denotes the differential transform for short. Noting that

${A}_{m}$ is independent of

x, and using the properties listed in

Table 1, Equation (

9) yields:

□

Theorem 3 enables us to derive the differential transform of any nonlinear term $f(u,v)$ by merely calculating the relevant Adomian polynomials. Once obtaining the Adomian polynomial of $f(u,v)$, the only thing we have to do is to replace ${u}_{k}$, ${v}_{k}$ by $U(k),V(k)$, respectively. This widens the applicability of the DTM as the Adomian polynomials can be generated quickly by a variety of algorithms with the help of computer algebraic systems, such as Maple. Furthermore, by using Theorem 2 and the similar manner to prove Theorem 3, we have the general result as follows:

**Theorem** **4.** Denote the differential transform of function $f({u}^{(1)},{u}^{(2)},\cdots ,{u}^{(n)})$ by $F(k)$, it holds thatwhere ${A}_{k},k=0,1,2,\cdots $ are the Adomian polynomials of nonlinear function $f({u}^{(1)},{u}^{(2)},\cdots ,{u}^{(n)})$.