Review Article | Open Access

# A Review of Some Recent Results for the Approximate Analytical Solutions of Nonlinear Differential Equations

**Academic Editor:**Ji Huan He

#### Abstract

This paper features a survey of some recent developments in techniques for obtaining approximate analytical solutions of some nonlinear differential equations arising in various fields of science and engineering. Adomian's decomposition method is applied to some nonlinear problems, and some mathematical tools such as He's homotopy perturbation method and variational iteration method are introduced to overcome the shortcomings of Adomian's method. The results of some comparisons of these three methods appearing in the research literature are given.

#### 1. Introduction

Nonlinear phenomena play a crucial role in applied mathematics and engineering. Therefore, over the last ten years, so many mathematical methods that are aimed at obtaining analytical solutions of nonlinear differential equations arising in various fields of science and engineering have appeared in the research literature [1โ6]. However, most of them require a tedious analysis or a large computer memory to handle these problems.

In this paper we present and compare three methods which are recently studied by the scientists to obtain approximate analytical solutions of some nonlinear differential equations arising in various fields of science and engineering.

The first method is so-called Adomian decomposition method (ADM) which was introduced by Adomian [7โ13] in the beginning of the 1980s. This is an iterative method which provides approximate analytical solutions in the form of an infinite power series for nonlinear equations. It is well known that this method avoids linearization, discretization and scientifically unrealistic assumptions. It also provides an efficient numerical solution with high accuracy [6, 7, 14]. This method is modified and used by Jin and Liu [15] to improve the convergence of series solution. They apply the modified ADM to solve a kind of evolution equations. Also, the authors of [16โ18] apply the ADM to obtain the approximate analytical solutions for heat-like and wave-like equations with variable coefficients, for the wave equation in an infinite one-dimensional medium and for Bratu-type equations, respectively.

The second method is the homotopy perturbation method (HPM) which was proposed by He [19] in 1999. In this method, the solution is obtained as the summation of an infinite series, which converges to analytical solution. Using the homotopy technique from topology, a homotopy is constructed with an embedding parameter , which is considered as a โsmall parameter". The approximations obtained by the HPM are uniformly valid not only for small parameters but also for very large parameters. Also, this method is modified and used by some scientists to obtain a fast convergent rate (see, e.g., [20]).

The last method is the variational iteration method (VIM) which is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. This method has been proposed by Shou and He [21] and is thoroughly used by many researchers (see, e.g., [22โ26]) to handle linear and nonlinear problems. The VIM uses only the prescribed initial conditions and does not require a specific treatment.

Although it is revealed that modified form of HPM corresponds to ADM for certain nonlinear problems [27], many researchers find ADM very difficult to calculate the Adomian polynomials [23, 28โ31]. Also, ADM could not always satisfy all the boundary conditions of the nonlinear problems, leading to an error at the boundary of the domain in which the problem is solved [32].

On the other hand, the authors of [33, 34] overcome the shortcomings of the Adomian method using HPM and He polynomials, and they state that HPM and He polynomials can completely replace the Adomian method and Adomian polynomials.

Compared with Adomian method, HPM and He polynomials do have some obvious merits: (1) the method needs not to calculate Adomian polynomials; (2) the method is very straightforward, and the solution procedure is very simple [20, 24โ26, 35โ37].

In their calculations of the analytical solutions of various kinds of heat-like and wave-like equations, the authors of [21] pointed out that contrary to Adomian method, VIM needs no calculation of Adomian polynomial, only simple operation is needed. Another nice comparison between ADM and VIM is given by Wazwaz [38]. In his study he concludes the following: VIM gives several successive approximations through using the iteration of the correction functional. However, ADM provides the components of the exact solution, where these components should follow the summation of an infinite power series. Moreover, the VIM requires the evaluation of the Lagrangian multiplier , whereas ADM requires the evaluation of the Adomian polynomials that mostly require tedious algebraic calculations. More importantly, the VIM reduces the volume of calculations by not requiring the Adomian polynomials, hence the iteration is direct and straightforward. However, ADM requires the use of Adomian polynomials for nonlinear terms, and this needs more work. For nonlinear equations that arise frequently to express nonlinear phenomenon, He's VIM facilitates the computational work and gives the solution rapidly if compared with ADM.

Hojjati and Jafari [39] have made a comparison among these three methods, and they have concluded that although the numerical results are almost the same, HPM is much easier, more convenient and efficient than ADM and VIM.

In [40], the author features a survey of some recent developments in asymptotic technics, which are valid not only for weakly nonlinear equations but also for strongly ones. The limitations of the traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In [41], the author pays particular attention throughout the paper to give an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, VIM, parameter-expansion method, exp-function method, HPM, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-infinity theory in high-energy physics, Kleiber's 3/4 law in biology, possible mechanism in spider-spinning process, and fractal approach to carbon nanotube are briefly introduced. In [42], the same author presents a coupling method of a homotopy technique and a perturbation technique to solve nonlinear problems. In contrast to traditional perturbation methods, HPM does not require a small parameter in the equation.

We now present some of the equations from our last work [3โ5] related to nonlinear problems of various fields of science and engineering.

First, we consider the * logistic growth * in a population as a single species model to be governed by [43]
where and are positive constants. Here represents the population of the species at time , is the per capita growth rate, and is the carrying capacity of the environment. We nondimensionalize (1.1) by setting
and it becomes
If , then . Therefore, the analytical solution of (1.3) is easily obtained:

Second, we consider the Predator-Prey Models: Lotka-Volterra systems as an interacting species model to be governed by [3, 43, 44] where , and are constants. Here is the prey population and that of the predator at time . We nondimensionalize the system (1.5) [43] by setting and it becomes

Third, we present the heat equation [4]: where , and is a parameter. Here, the indices and denote derivatives with respect to these variables. Unless , (1.8) is a nonlinear heat equation. Construction of particular analytical solutions for nonlinear equations of the form (1.8) is an important problem. Especially, finding an analytical solution that has a biological interpretation is of fundamental importance. Recently, some new methods such as Lie symmetry reduction method [45], and antireduction method [46] which transforms the nonlinear PDEs to a system of ODEs have been introduced in the research literature to find particular analytical solutions to PDE. Finding analytical solutions of most nonlinear PDE generally requires new methods.

The particular analytical solutions of the nonlinear reaction diffusion equations of the form where , , and are specially chosen smooth functions, are obtained in [47]. This equation usually arises in mathematical biology [43, 44]. In fact, (1.8) is a particular case of the last equation.

We last consider the nonlinear heat equation called the porous media equation [5]: where is a rational number.

Finding the particular analytical solutions that have a physical or biological interpretation for the nonlinear equations of the form (1.10) is of fundamental importance. This equation often occurs in nonlinear problems of heat and mass transfer, combustion theory, and flows in porous media. For example, it describes unsteady heat transfer in a quiescent medium with the heat diffusivity being a power-law function of temperature [48].

Equation (1.10) has also applications to many physical systems including the fluid dynamics of thin films [49]. Murray [43] describes how this model has been used to represent โpopulation pressure" in biological systems. This equation is called a degenerate parabolic differential equation because the diffusion coefficient does not satisfy the condition for classical diffusion equations, [49]. For the motion of thin viscous films, (1.10) with can be derived from the Navier-Stokes equations. Lacking a physical law to describe the complex behavior in a system, an appropriate value for the parameter can be determined by comparing known solutions with empirical data [49].

In the following section, we apply the ADM [7โ13] to (1.3), (1.7)โ(1.10), respectively.

#### 2. Adomian's Decomposition Method

##### 2.1. Analysis of the Method for Single Species

In this section we consider the model equation of the form [3] where is a nonlinear function of . We are looking for the solution satisfying (2.1.1). The decomposition method consists of approximating the solution of (2.1.1) as an infinite series: and decomposing as where 's are the Adomian polynomials given by The convergence of the decomposition series (2.1.3) is studied in [50]. Applying the decomposition method [7, 14], (2.1.1) can be written as where the notation symbolizes the linear differential operator. We assume the integration inverse operators exist, and it is defined as . Therefore, applying on both sides of (2.1.5) with yields Using (2.1.2) and (2.1.3), it follows that Therefore, one determines the iterates in the following recursive way: We then define the solution as

##### 2.2. Analysis of the Method for Interacting Species

In this section, we consider the system of the form [3] with initial data Here, and are nonlinear functions of and . We are looking for the solutions satisfying (2.2.1)-(2.2.2). The decomposition method consists of approximating the solutions of the above system as an infinite series: and decomposing and as [6] where and are the Adomian polynomials that can be generated for any form of nonlinearity. Applying the decomposition method, the system (2.2.1) can be written as where the notation again symbolizes the linear differential operator. Therefore, applying on both sides of the equations of the system (2.2.5) with yields [6] Using (2.2.3) and (2.2.4), it follows that Therefore, one determines the iterates in the following recursive way:

We then define the solutions of the initial value problem (2.2.1)-(2.2.2) as

##### 2.3. Analysis of the Method for the Heat Equation

In this section, we consider (1.8) in an operator form [4] with the initial and boundary conditions, where the notations and symbolize the linear differential operators. We assume the integration inverse operators and exist, and they are defined as and , respectively. Therefore, we can write the solutions in and directions as [1, 2, 7] respectively, where . By ADM [7], one can write the solution in series form as

To find the solutions in and directions, one solves the recursive relations:

respectively, where the Adomian polynomials are [1, 2, 7] We obtain the first few Adomian polynomials for as , , , and so on. The convergence of the decomposition series given by (2.3.4) is studied in [50].

In Section 3, we provide a couple of examples and demonstrate the absolute errors in Tables 1โ4, where is the particular analytical solution and is the partial sum: As it is clear from (2.3.4) and (2.3.8), we have

##### 2.4. Analysis of the Method for the Porous Media Equation

Equation (1.10) can be written in an operator form [5] with the initial and boundary conditions, where the notations and symbolize the linear differential operators. We assume the integration inverse operators and exist, and they are defined as and , respectively. Therefore, one can write the solution in direction as [7] where . By ADM [7] one can write the solution in series form as

To find the solutions in direction, one solves the recursive relations: respectively, where the Adomian polynomials are [1, 2, 7] We obtain the first few Adomian polynomials for as

In Section 3, we provide some examples and demonstrate the absolute errors in Tables 5-6, where is the particular analytical solution and is the partial sum: Equations of the form (1.10) admit traveling-wave solutions where and are constants [48].

#### 3. Applications of ADM

*Example 3.1. *We first consider (1.3) with initial data . We proceed as in Section 2.1. We take and . Adomian polynomials can be derived as follows:
Therefore, we get the following Adomian polynomials [14]:
For numerical purposes we take and . Therefore,
and so on, in this manner the rest of the terms of the decomposition series have been calculated using *Mathcad7 *. Substituting these terms into (2.1.2), we obtain
which gives the analytical solution obtained in (1.4) in the closed form, with . We let be the th partial sums of the series in (2.1.2), that is,
and compare the analytical solution with (3.4) in Figure 1.

*Example 3.2. *We now consider the initial value problem given by (1.7) with initial data , . We proceed as in Section 2.2. We take , . Therefore, from (2.2.4) we obtain , , and Adomian polynomials can be derived as follows:
Hence, we get the following Adomian polynomials:
From this equality, we have
Let us now compute the and from (2.2.8):
and so on, in this manner the rest of the terms of the decomposition series have been calculated using . Substituting these terms into (2.2.3), we obtain the following approximate solutions to the initial value problem given by (1.7) with initial data , :

*Example 3.3. *If we take and in the (1.8), we obtain the linear heat equation, namely,
We impose the initial condition
and boundary conditions

To obtain the solution in direction, we use the recursive relation (2.3.5) by simply taking . In this case the Adomian Polynomials are , and so on. Therefore, we have
and so on, in this manner the rest of the components of the series (2.3.4) have been calculated using . Putting these individual terms in (2.3.4) one gets the analytical solution:
which can be verified through substitution.

Similarly, to obtain the solution in direction, we use the recursive relation (2.3.6) by taking , where the are the same as above. We therefore have
and so on, in this manner the rest of the components of the series (2.3.4) have been calculated. From the decomposition series (2.3.4), we again obtain the analytical solution:

*Example 3.4. *In the second example, we consider the nonlinear heat equation (1.8) with and , that is,
In [46] the authors solve (3.19) using antireduction method, and give the solution by means of ansatz () as follows:
where and satisfy the ordinary differential equations:
We impose , and solve the above ordinary differential equations, and obtain
Therefore, we have the partial analytical solution of (3.19) as
We now solve (3.19) using ADM with the initial condition:
and the boundary conditions:
For the solution of this equation in the direction, we use the recursive relation given by (2.3.5) to obtain the terms of the decomposition series (2.3.8). In this case the Adomian Polynomials are , and so on. Therefore, we obtain
and so on, in this manner the rest of the terms of the decomposition series have been calculated using . Substituting (3.26) into the decomposition series (2.3.8), we obtain
which gives the analytical solution obtained in (3.23) in the closed form. This result can be verified through substitution.

On the other hand, to obtain the solution in the direction, we use the recursive relation given by (2.3.6) to determine the individual terms of the series (2.3.8):
and so on. In this manner the rest of the terms of the decomposition series (2.3.8) have been calculated. Substituting (3.28) into (2.3.8) gives
which again gives the analytical solution given by (3.23) in the closed form.

*Example 3.5. *Let us take in (1.10). We obtain
We impose the initial condition

To obtain the solution, we use the recursive relation (2.4.4) by taking . In this case the first Adomian Polynomial is . Therefore, we have and . Finally, which follows that for . Putting these individual terms in (2.4.3), one gets the analytical solution
which can be verified through substitution.

*Example 3.6. *When , (1.10) becomes
In [48] the authors give a particular analytical solution to (3.33) as follows:
where and are arbitrary constants. We take and for simplicity. Therefore, with these choices of and their solution becomes
We now solve (3.33) using ADM with the initial condition:
For the solution of this equation, we use the recursive relation given by (2.4.4) to obtain the terms of the decomposition series (2.4.3). In this case , , , , , , , and so on. In this manner the rest of the terms of the decomposition series have been calculated using . Substituting these individual terms in (2.4.3), we obtain
which gives the analytical solution obtained in (3.35) in the closed form. This result can be verified through substitution.

*Example 3.7. *If , (1.10) reads
In [48], a particular analytical solution to (3.38) is given as follows:
where and are arbitrary constants, and we take and for simplicity. Therefore, one has
We now solve (3.38) using ADM with the initial condition:
We again use the recursive relation given by (2.4.4) to obtain the terms of the decomposition series (2.4.3). In this case , , , , , , , and so on. In this manner the rest of the terms of the decomposition series have been calculated using . Substituting these individual terms in (2.4.3), one obtains
which gives the analytical solution obtained in (3.40) in the closed form. This result can also be verified through substitution.

#### 4. The Idea of Homotopy Perturbation Method

The basic idea of the homotopy perturbation method (HPM) can be illustrated as follows [19]: we consider the nonlinear differential equation: with boundary conditions: where is a general differential operator, is a boundary operator, is a known analytic function, and is the boundary of the domain .

In general, one divides the operator into two parts and , where is linear, while is nonlinear. Therefore, (4.1) is written as follows:

By the homotopy technique [19, 51], one constructs a homotopy which satisfies or where is an embedding parameter, is an initial approximation of (4.1), which satisfies the boundary conditions. It is clear that the changing process of from zero to unity is just that of from to .

According to the HPM, we can first use the embedding parameter as a โsmall parameter", and assume that the solution of (4.4) and (4.5) can be written as a power series in : Setting results in the approximate solution of (4.1):

The combination of the perturbation method and the homotopy method is called the homotopy perturbation method, which has eliminated limitations of the traditional perturbation methods.

The series (4.8) is convergent for most cases, however, the convergent rate depends on the nonlinear operator (the following opinions are suggested by He [19]).

(1) The second derivative of with respect to must be small because the parameter may be relatively large, that is, .

(2) The norm of must be smaller than one so that the series converges.

#### 5. Applications of HPM

*Example 5.1. *We now solve (1.3) using HPM with the initial condition , as chosen in Example 3.1. We rewrite (1.3) in the form [52]
where is an embedding parameter. As in He's HPM, it is clear that when , (5.1) becomes a linear equation; when , it becomes the original nonlinear one. We consider the imbedding parameter as a โsmall parameter". We assume the solution of (5.1) is expressed as a power series given in (4.7). Substituting (4.7) into (5.1), and equating coefficients of like , we obtain the following differential equations:
where โprimes" denote differentiation with respect to . Thus, solving the equations above yields

Substituting these in (4.7) gives
Hence, by (4.8) one has
which is exactly the same solution obtained in (3.4). Also, the solution in (5.5) is equal to
in the closed form which is exactly the same as in (1.4) with (see Example 3.1).

*Example 5.2. *We now solve (1.7) using HPM with as taken in Example 3.2. We rewrite (1.7) in the form [52]
where is an embedding parameter. As in He's HPM, it is clear that when , (5.7) becomes a linear system; when , it becomes the original nonlinear one. We consider the imbedding parameter as a โsmall parameter". We assume the solutions of (5.7), are expressed as power series:
respectively. Substituting (5.8) and (5.9) into the system (5.7), and equating coefficients of like , we obtain the following systems of differential equations:
where โprimes" denote differentiation with respect to . Thus, solving the above systems of equations yields

Substituting these , into (5.8), and (5.9), respectively, we have
Letting one obtains
which are exactly the same solutions obtained in (3.10) and (3.11), respectively.

#### 6. The Modified HPM for the Porous Media Equation

In this section we outline the modified HPM studied by Chun et al. [35]. In this work, they have solved the porous media equation using modified HPM with initial condition: Here . Equation (6.1) is the same equation we have solved in Section 2.4 using ADM, and we have provided some numerical examples for it in Examples 3.5โ3.7. In their work they construct the following homotopy with and , To deal with the nonlinear term, they employ He polynomials considered in [33, 34] which is given by where Substituting (6.4) into (6.3), and equating coefficients of like , one obtains and so on, which forms the basis of a complete determination of the components