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

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 [1–6]. 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 [7–13] 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 [16–18] 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 𝑝∈[0,1], which is considered as a “small 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., [22–26]) 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, 28–31]. 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, 24–26, 35–37].

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 𝜆, 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 [3–5] 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] 𝑑𝑁𝑑𝑡=𝑟𝑁(1−𝑁/𝐾),(1.1) where 𝑟 and 𝐾 are positive constants. Here 𝑁=𝑁(𝑡) represents the population of the species at time 𝑡, 𝑟(1−𝑁/𝐾) is the per capita growth rate, and 𝐾 is the carrying capacity of the environment. We nondimensionalize (1.1) by setting 𝑢(𝜏)=𝑁(𝑡)𝐾,𝜏=𝑟𝑡,(1.2) and it becomes 𝑑𝑢𝑑𝜏=𝑢(1−𝑢).(1.3) If 𝑁(0)=𝑁₀, then 𝑢(0)=𝑁₀/𝐾. Therefore, the analytical solution of (1.3) is easily obtained: 𝑢(𝜏)=11+𝐾/𝑁₀−1𝑒^−𝜏.(1.4)

Second, we consider the Predator-Prey Models: Lotka-Volterra systems as an interacting species model to be governed by [3, 43, 44] 𝑑𝑁𝑑𝑑𝑃𝑡=𝑁(𝑎−𝑏𝑃),𝑑𝑡=𝑃(𝑐𝑁−𝑑),(1.5) where 𝑎,𝑏,𝑐, and 𝑑 are constants. Here 𝑁=𝑁(𝑡) is the prey population and 𝑃=𝑃(𝑡) that of the predator at time 𝑡. We nondimensionalize the system (1.5) [43] by setting 𝑢(𝜏)=𝑐𝑁(𝑡)𝑑,𝑣(𝜏)=𝑏𝑃𝑎(𝑡),𝜏=𝑎𝑡,𝛼=𝑑/𝑎,(1.6) and it becomes 𝑑𝑢𝑑𝑑𝜏𝑣=𝑢(1−𝑣),𝑑𝜏=𝛼𝑣(𝑢−1).(1.7)

Third, we present the heat equation [4]: 𝑢_𝑡=𝑢_𝑥𝑥+𝜖𝑢^𝑚,(1.8) where 𝑚=1,2,3,…, and 𝜖 is a parameter. Here, the indices 𝑡 and 𝑥 denote derivatives with respect to these variables. Unless 𝑚=1, (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 𝑢_𝑡=(𝐴(𝑢)𝑢_𝑥)_𝑥+𝐵(𝑢)𝑢_𝑥+𝐶(𝑢),(1.9) where 𝐴(𝑢), 𝐵(𝑢), and 𝐶(𝑢) 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]: 𝜕𝑢𝜕𝑡=𝜕𝜕𝑥𝑢^𝑚𝜕𝑢𝜕𝑥,(1.10) where 𝑚 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 “population pressure" in biological systems. This equation is called a degenerate parabolic differential equation because the diffusion coefficient 𝐷(𝑢)=𝑢^𝑚 does not satisfy the condition for classical diffusion equations, 𝐷(𝑢)>0 [49]. For the motion of thin viscous films, (1.10) with 𝑚=3 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 𝑚 can be determined by comparing known solutions with empirical data [49].

In the following section, we apply the ADM [7–13] to (1.3), (1.7)–(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] 𝑑𝑢𝑑𝜏=𝑢−𝑓(𝑢),𝑢(0)=𝛾,(2.1) where 𝑓 is a nonlinear function of 𝑢. We are looking for the solution 𝑢 satisfying (2.1.1). The decomposition method consists of approximating the solution of (2.1.1) as an infinite series: 𝑢=^∞_𝑛=0𝑢_𝑛,(2.2) and decomposing 𝑓 as 𝑓(𝑢)=^∞_𝑛=0𝐴_𝑛,(2.3) where 𝐴_𝑛's are the Adomian polynomials given by 𝐴_𝑛=1𝑛!𝑑𝑑𝜆^𝑛𝑛𝑓^∞_𝑛=0𝜆^𝑛𝑢_𝑛_𝜆=0,𝑛=0,1,2,….(2.4) 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 𝐿𝑢=𝑢−𝑓(𝑢),(2.5) where the notation 𝐿=𝜕/𝜕𝜏 symbolizes the linear differential operator. We assume the integration inverse operators 𝐿⁻¹exist, and it is defined as 𝐿⁻¹=∫^𝜏0(⋅)𝑑𝜏. Therefore, applying on both sides of (2.1.5) with 𝐿⁻¹ yields 𝑢(𝜏)=𝑢(0)+𝐿⁻¹𝑢(𝜏)−𝐿⁻¹𝑓(𝑢(𝜏)).(2.6) Using (2.1.2) and (2.1.3), it follows that ^∞_𝑛=0𝑢_𝑛=𝑢(0)+𝐿^∞−1_𝑛=0𝑢_𝑛−𝐿^∞−1_𝑛=0𝐴_𝑛.(2.7) Therefore, one determines the iterates in the following recursive way: 𝑢𝑢₀=𝑢(0)=𝛾,_𝑛+1=𝐿⁻¹𝑢_𝑛−𝐿⁻¹𝐴_𝑛,𝑛=0,1,2,….(2.8) We then define the solution 𝑢 as 𝑢=lim_{𝑛𝑛→∞}_𝑘=0𝑢_𝑘.(2.9)

2.2. Analysis of the Method for Interacting Species

In this section, we consider the system of the form [3] 𝑑𝑢𝑑𝑑𝜏𝑣=𝑢−𝑓(𝑢,𝑣),𝑑𝜏=𝛼[𝑔(𝑢,𝑣)−𝑣],(2.10) with initial data 𝑢(0)=𝛿,𝑣(0)=𝛽.(2.11) Here, 𝑓 and 𝑔 are nonlinear functions of 𝑢 and 𝑣. We are looking for the solutions (𝑢,𝑣) satisfying (2.2.1)-(2.2.2). The decomposition method consists of approximating the solutions of the above system as an infinite series: 𝑢=^∞_𝑛=0𝑢_𝑛,𝑣=^∞_𝑛=0𝑣_𝑛,(2.12) and decomposing 𝑓 and 𝑔 as [6] 𝑓(𝑢,𝑣)=^∞_𝑛=0𝐵_𝑛,𝑔(𝑢,𝑣)=^∞_𝑛=0𝐶_𝑛,(2.13) where 𝐵_𝑛 and 𝐶_𝑛 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 𝐿𝐿𝑣𝑢==𝑢𝛼−[𝑔𝑓(𝑢(,𝑢𝑣,)𝑣−),𝑣],(2.14) where the notation 𝐿=𝜕/𝜕𝜏 again symbolizes the linear differential operator. Therefore, applying on both sides of the equations of the system (2.2.5) with 𝐿⁻¹ yields [6] 𝑣𝑢((𝜏𝜏))==𝑢𝑣((00))++𝐿𝛼𝐿⁻¹𝑢(₋₁𝜏)−𝐿⁻¹𝑓(𝑢(𝜏),𝑣(𝜏)),𝑔(𝑢(𝜏),𝑣(𝜏))−𝐿⁻¹𝑣(𝜏).(2.15) Using (2.2.3) and (2.2.4), it follows that ^∞_𝑛=0𝑢_𝑛=𝑢(0)+𝐿^∞−1_𝑛=0𝑢_𝑛−𝐿^∞−1_𝑛=0𝐵_𝑛,_∞_𝑛=0𝑣_𝑛=𝑣(0)+𝛼𝐿^∞−1_𝑛=0𝐶_𝑛−𝐿^∞−1_𝑛=0𝑣_𝑛.(2.16) Therefore, one determines the iterates in the following recursive way: 𝑢𝑢₀=𝑢(0)=𝛿,_𝑛+1𝑣=𝐿⁻¹𝑢_𝑛−𝐿⁻¹𝐵_𝑛,𝑛=0,1,2,…,₀𝑣=𝑣(0)=𝛽,_𝑛+1=𝛼𝐿⁻¹𝐶_𝑛−𝐿⁻¹𝑣_𝑛,𝑛=0,1,2….(2.17)

We then define the solutions of the initial value problem (2.2.1)-(2.2.2) as (𝑢,𝑣)=lim_{𝑛𝑛→∞}_𝑘=0𝑢_𝑘,lim_{𝑛𝑛→∞}_𝑘=0𝑣_𝑘.(2.18)

2.3. Analysis of the Method for the Heat Equation 𝑢_𝑡=𝑢_𝑥𝑥+𝜖𝑢^𝑚

In this section, we consider (1.8) in an operator form [4] 𝐿_𝑡(𝑢(𝑥,𝑡))−𝐿_𝑥(𝑢(𝑥,𝑡))−𝜖𝑢^𝑚=0,(2.19) with the initial and boundary conditions, where the notations 𝐿_𝑡=𝜕/𝜕𝑡 and 𝐿_𝑥=𝜕²/𝜕𝑥² symbolize the linear differential operators. We assume the integration inverse operators 𝐿_𝑡−1 and 𝐿_𝑥−1 exist, and they are defined as 𝐿_𝑡−1=∫^𝑡0(⋅)𝑑𝑡 and 𝐿_𝑥−1=∬^𝑥0(⋅)𝑑𝑥𝑑𝑥, respectively. Therefore, we can write the solutions in 𝑡 and 𝑥 directions as [1, 2, 7] 𝑢𝑢((𝑥𝑥,,𝑡𝑡))==𝑢𝑢((𝑥0,,𝑡0))++𝑥𝐿𝑢_𝑡−1𝑥𝐿_𝑥(𝑢(𝑥,𝑡))+Φ(𝑢),(0,𝑡)+𝐿_𝑥−1𝐿_𝑡(𝑢(𝑥,𝑡))−Φ(𝑢),(2.20) respectively, where Φ(𝑢)=𝜖𝑢^𝑚. By ADM [7], one can write the solution in series form as 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑢_𝑛(𝑥,𝑡).(2.21)

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

𝑢𝑢₀=𝑢(𝑥,0),𝑢_𝑛+1=𝐿_𝑡−1𝐿_𝑥𝑢_𝑛+𝐴_𝑛,𝑛≥0,(2.22)₀=𝑢(0,𝑡)+𝑥𝑢_𝑥(0,𝑡),𝑢_𝑛+1=𝐿_𝑥−1𝐿_𝑡𝑢_𝑛−𝐴_𝑛,𝑛≥0,(2.23) respectively, where the Adomian polynomials are [1, 2, 7] 𝐴_𝑛=1𝑛!𝑑𝑑𝜆^𝑛𝑛Φ^∞_𝑛=0𝜆^𝑛𝑢_𝑛_𝜆=0,𝑛≥0.(2.24) We obtain the first few Adomian polynomials for Φ(𝑢)=𝜖𝑢^𝑚 as 𝐴₀=𝜖𝑢^𝑚0, 𝐴₁=𝑚𝜖𝑢^0𝑚−1𝑢₁, 𝐴₂=(𝑚𝜖/2)[(𝑚−1)𝑢^0𝑚−2𝑢²¹+2𝑢₂𝑢^0𝑚−1], 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 |𝑢(𝑥,𝑡)−𝜙_𝑛(𝑥,𝑡)| in Tables 1–4, where 𝑢(𝑥,𝑡) is the particular analytical solution and 𝜙_𝑛(𝑥,𝑡) is the partial sum: 𝜙_𝑛(𝑥,𝑡)=^𝑛_𝑘=0𝑢_𝑘(𝑥,𝑡),𝑛≥0.(2.25) As it is clear from (2.3.4) and (2.3.8), we have 𝑢(𝑥,𝑡)=lim_𝑛→∞𝜙_𝑛(𝑥,𝑡).(2.26)

Table 1 

Numerical results (in t direction) for |𝑢(𝑥,𝑡)−𝜙₃₀(𝑥,𝑡)| where 𝑢(𝑥,𝑡)=𝑒^{(1−𝜋2)𝑡}cos(𝜋𝑥) for (3.12)–(3.14).

Table 2 

Numerical results (in x direction) for |𝑢(𝑥,𝑡)−𝜙₃₀(𝑥,𝑡)| where 𝑢(𝑥,𝑡)=𝑒^{(1−𝜋2)𝑡}cos(𝜋𝑥) for (3.12)–(3.14).

Table 3 

Numerical results (in t direction) for |𝑢(𝑥,𝑡)−𝜙₂₀(𝑥,𝑡)| where 𝑢(𝑥,𝑡)=(1+2𝑥)/(𝑥²+𝑥+6𝑡+1) for (3.19), (3.24)-(3.25).

Table 4 

Numerical results (in x direction) for |𝑢(𝑥,𝑡)−𝜙₂₀(𝑥,𝑡)| where 𝑢(𝑥,𝑡)=(1+2𝑥)/(𝑥²+𝑥+6𝑡+1) for (3.19), (3.24)-(3.25).

2.4. Analysis of the Method for the Porous Media Equation

Equation (1.10) can be written in an operator form [5] 𝐿_𝑡(𝑢)=𝐿_𝑥𝑢^𝑚𝐿_𝑥𝑢,(2.27) with the initial and boundary conditions, where the notations 𝐿_𝑡=𝜕/𝜕𝑡 and 𝐿_𝑥=𝜕/𝜕𝑥 symbolize the linear differential operators. We assume the integration inverse operators 𝐿_𝑡−1 and 𝐿_𝑥−1 exist, and they are defined as 𝐿_𝑡−1=∫^𝑡0(⋅)𝑑𝑡 and 𝐿_𝑥−1=∫^𝑥0(⋅)𝑑𝑥, respectively. Therefore, one can write the solution in 𝑡 direction as [7] 𝑢(𝑥,𝑡)=𝑢(𝑥,0)+𝐿_𝑡−1𝐿_𝑥(Φ(𝑢)),(2.28) where Φ(𝑢)=𝑢^𝑚𝑢_𝑥. By ADM [7] one can write the solution in series form as 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑢_𝑛(𝑥,𝑡).(2.29)

To find the solutions in 𝑡 direction, one solves the recursive relations: 𝑢₀=𝑢(𝑥,0),𝑢_𝑛+1=𝐿_𝑡−1𝐿_𝑥𝐴_𝑛,𝑛≥0,(2.30) respectively, where the Adomian polynomials are [1, 2, 7] 𝐴_𝑛=1𝑛!𝑑𝑑𝜆^𝑛𝑛Φ^∞_𝑛=0𝜆^𝑛𝑢_𝑛_𝜆=0,𝑛≥0.(2.31) We obtain the first few Adomian polynomials for Φ(𝑢)=𝑢^𝑚𝑢_𝑥 as 𝐴𝐴₀₁=𝑢^𝑚0(𝑢₀)_𝑥,𝐴₂=𝑚𝑢^0𝑚−1𝑢₁(𝑢₀)_𝑥+𝑢^𝑚0(𝑢₁)_𝑥,=𝑚𝑢^0𝑚−1𝑢₂(𝑢₀)_𝑥+𝑚𝑢^0𝑚−1𝑢₁(𝑢₁)_𝑥+𝑢^𝑚0𝑢₂_𝑥+𝑚2(𝑚−1)𝑢^0𝑚−2𝐴₃𝑢²¹(𝑢₀)_𝑥,=𝑚𝑢^0𝑚−1𝑢₃(𝑢₀)_𝑥+𝑚(𝑚−1)𝑢^0𝑚−2𝑢₁𝑢₂𝑢₀_𝑥+𝑚2(𝑚−1)𝑢^0𝑚−2𝑢³¹(𝑢₀)_𝑥⋮+𝑚𝑢^0𝑚−1𝑢₂(𝑢₁)_𝑥+𝑚𝑢^0𝑚−1𝑢₁(𝑢₂)_𝑥+𝑢^𝑚0(𝑢₃)_𝑥,(2.32)

In Section 3, we provide some examples and demonstrate the absolute errors |𝑢(𝑥,𝑡)−𝜙_𝑛(𝑥,𝑡)| in Tables 5-6, where 𝑢(𝑥,𝑡) is the particular analytical solution and 𝜙_𝑛(𝑥,𝑡) is the partial sum: 𝜙_𝑛(𝑥,𝑡)=^𝑛_𝑘=0𝑢_𝑘(𝑥,𝑡),𝑛≥0.(2.33) Equations of the form (1.10) admit traveling-wave solutions 𝑢=𝑢(𝑘𝑥+𝜆𝑡) where 𝑘 and 𝜆 are constants [48].

Table 5 

Numerical results for |𝑢(𝑥,𝑡)−𝜙₅₀(𝑥,𝑡)| where 𝑢(𝑥,𝑡)=1/(𝑥−𝑡) for (3.33).

Table 6 

Numerical results for |𝑢(𝑥,𝑡)−𝜙₆(𝑥,𝑡)| where 𝑢(𝑥,𝑡)=(2𝑥−3𝑡)^−3/4 for (3.38).

3. Applications of ADM

Example 3.1. We first consider (1.3) with initial data 𝑢(0)=𝑁₀/𝐾. We proceed as in Section 2.1. We take 𝑓(𝑢)=𝑢² and 𝛾=𝑁₀/𝐾. Adomian polynomials can be derived as follows: 𝑓(𝑢)=𝑢²=^∞_𝑛=0𝐴_𝑛=(𝑢₀+𝑢₁+𝑢₂+⋯)²=+𝑢²⁰2𝑢+₀2𝑢₀𝑢₁+2𝑢₀𝑢₂+𝑢²¹+2𝑢₀𝑢₃+2𝑢₁𝑢₂𝑢₄+2𝑢₁𝑢₃+𝑢²²+2𝑢₀𝑢₅+2𝑢₁𝑢₄+2𝑢₂𝑢₃+⋯.(3.1) Therefore, we get the following Adomian polynomials [14]: 𝐴𝐴₀₁=𝑢²⁰,𝐴₂=2𝑢₀𝑢₁,𝐴₃=2𝑢₀𝑢₂+𝑢²¹,𝐴₄=2𝑢₀𝑢₃+2𝑢₁𝑢₂,𝐴₅=2𝑢₀𝑢₄+2𝑢₁𝑢₃+𝑢²²,⋮=2𝑢₀𝑢₅+2𝑢₁𝑢₄+2𝑢₂𝑢₃,(3.2) For numerical purposes we take 𝑁₀=2 and 𝐾=1. Therefore, 𝑢𝑢₀=𝑢(0)=𝑁₀/𝐾=2,₁𝑢=𝐿⁻¹𝑢₀−𝐿⁻¹𝐴₀=−2𝜏,₂𝑢₃=𝐿⁻¹𝑢₁−𝐿⁻¹𝐴₁=3𝜏²,𝑢₄=𝐿⁻¹𝑢₂−𝐿⁻¹𝐴₂=−13/3𝜏³,𝑢₅=𝐿⁻¹𝑢₃−𝐿⁻¹𝐴₃=25/4𝜏⁴,⋮=𝐿⁻¹𝑢₄−𝐿⁻¹𝐴₄=−541/60𝜏⁵,(3.3) 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 𝑢(𝜏)==𝑢2₀−(𝜏2)𝜏++𝑢3₁𝜏(𝜏₂)+𝑢₂(𝜏)+𝑢₃(𝜏)+𝑢₄(𝜏)+𝑢₅(𝜏)+⋯−13/3𝜏³+25/4𝜏⁴−541/60𝜏⁵+⋯,(3.4) which gives the analytical solution obtained in (1.4) in the closed form, with 𝑁₀=2,𝐾=1. We let 𝜙_𝑛 be the 𝑛th partial sums of the series in (2.1.2), that is, 𝜙_𝑛=^𝑛_𝑘=0𝑢_𝑘,𝑛≥0,(3.5) and compare the analytical solution with (3.4) in Figure 1.

Figure 1 

Solution to the logistic growth model (1.3) with 𝑢(0)=2) in a population.

Example 3.2. We now consider the initial value problem given by (1.7) with initial data 𝑢(0)=𝛿=1.3, 𝑣(0)=𝛽=0.6. We proceed as in Section 2.2. We take 𝛼=1, 𝑓(𝑢,𝑣)=𝑔(𝑢,𝑣)=𝑢𝑣. Therefore, from (2.2.4) we obtain 𝐵_𝑛=𝐶_𝑛, 𝑛=0,1,2,…, and Adomian polynomials can be derived as follows: 𝑓(𝑢,𝑣)===𝑔𝑢𝑣_∞(𝑢,𝑣)_𝑛=0𝐵_𝑛=^∞_𝑛=0𝑢_𝑛^∞_𝑛=0𝑣_𝑛=_∞_𝑛=0^𝑛_𝑘=0𝑢_𝑘𝑣_𝑛−𝑘.(3.6) Hence, we get the following Adomian polynomials: 𝐵_𝑛=𝐶_𝑛=^𝑛_𝑘=0𝑢_𝑘𝑣_𝑛−𝑘,𝑛=0,1,2,….(3.7) From this equality, we have 𝐵𝐵₀₁=𝐶₀=𝑢₀𝑣₀,𝐵₂=𝐶₁=𝑢₀𝑣₁+𝑢₁𝑣₀,𝐵₃=𝐶₂=𝑢₀𝑣₂+𝑢₁𝑣₁+𝑢₂𝑣₀,⋮=𝐶₃=𝑢₀𝑣₃+𝑢₁𝑣₂+𝑢₂𝑣₁+𝑢₃𝑣₀,(3.8) Let us now compute the 𝑢_𝑘 and 𝑣_𝑘 from (2.2.8): 𝑣𝑢₀=𝑢(0)=𝛿=1.3,₀𝑢=𝑣(0)=𝛽=0.6,₁𝑣=𝐿⁻¹𝑢₀−𝐿⁻¹𝐵₀=0.52𝜏,₁𝑢=𝐿⁻¹𝐶₀−𝐿⁻¹𝑣₀=0.18𝜏,₂𝑣₂=𝐿⁻¹𝑢₁−𝐿⁻¹𝐵₁=−0.013𝜏²,𝑢₃=𝐿⁻¹𝐶₁−𝐿⁻¹𝑣₁=0.1830𝜏²,𝑣₃=𝐿⁻¹𝑢₂−𝐿⁻¹𝐵₂=−0.1122𝜏³,𝑢₄=𝐿⁻¹𝐶₂−𝐿⁻¹𝑣₂=0.0469𝜏³,𝑣₄=𝐿⁻¹𝑢₃−𝐿⁻¹𝐵₃=−0.0497𝜏⁴,⋮=𝐿⁻¹𝐶₃−𝐿⁻¹𝑣₃=0.0099𝜏⁴,(3.9) and so on, in this manner the rest of the terms of the decomposition series have been calculated using 𝑀𝑎𝑡ℎ𝑐𝑎𝑑7. Substituting these terms into (2.2.3), we obtain the following approximate solutions to the initial value problem given by (1.7) with initial data 𝑢(0)=𝛿=1.3, 𝑣(0)=𝛽=0.6: 𝑢(𝜏)==1𝑢₀.3(𝜏+)0+.5𝑢2₁𝜏(𝜏−)+0.0𝑢1₂3(𝜏𝜏₂)+𝑢₃(𝜏)+𝑢₄(𝜏)+⋯𝑣(𝜏)=𝑣₀−0.1122𝜏³−0.0497𝜏⁴−⋯,(3.10)=0.6(𝜏+)0+.1𝑣8₁𝜏(𝜏+)0+.1𝑣8₂3(0𝜏𝜏)₂+𝑣₃(𝜏)+𝑣₄(𝜏)+⋯+0.0469𝜏³+0.0099𝜏⁴+⋯.(3.11)

Example 3.3. If we take 𝜖=1 and 𝑚=1 in the (1.8), we obtain the linear heat equation, namely, 𝑢_𝑡=𝑢_𝑥𝑥+𝑢.(3.12) We impose the initial condition 𝑢(𝑥,0)=cos(𝜋𝑥),(3.13) and boundary conditions 𝑢(0,𝑡)=𝑒^{(1−𝜋2)𝑡},𝑢_𝑥(0,𝑡)=0.(3.14)
To obtain the solution in 𝑡 direction, we use the recursive relation (2.3.5) by simply taking 𝑢₀=cos(𝜋𝑥). In this case the Adomian Polynomials are 𝐴₀=𝑢₀,𝐴₁=𝑢₁,𝐴₂=𝑢₂, and so on. Therefore, we have 𝑢₁=1−𝜋²𝑡cos(𝜋𝑥),𝑢₂=12!1−𝜋²²𝑡²cos(𝜋𝑥),𝑢₃=13!(1−𝜋²)³𝑡³cos(𝜋𝑥),(3.15) and so on, in this manner the rest of the components of the series (2.3.4) have been calculated using 𝑀𝑎𝑡ℎ𝑐𝑎𝑑7. Putting these individual terms in (2.3.4) one gets the analytical solution: 𝑢(𝑥,𝑡)=cos(𝜋𝑥)+1−𝜋²𝑡cos(𝜋𝑥)+12!1−𝜋²𝑡²cos(𝜋𝑥)+13!(1−𝜋²=𝑒_{(1−𝜋2)𝑡})³𝑡³cos(𝜋𝑥)+⋯cos(𝜋𝑥),(3.16) which can be verified through substitution.
Similarly, to obtain the solution in 𝑥 direction, we use the recursive relation (2.3.6) by taking 𝑢₀=𝑒^{(1−𝜋2)𝑡}, where the 𝐴_𝑛 are the same as above. We therefore have 𝑢₁=−(𝜋𝑥)²2!𝑒^{(1−𝜋2)𝑡},𝑢₂=(𝜋𝑥)⁴4!𝑒^{(1−𝜋2)𝑡},𝑢₃=−(𝜋𝑥)⁶6!𝑒^{(1−𝜋2)𝑡},(3.17) 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: 𝑢(𝑥,𝑡)=𝑒^{(1−𝜋2)𝑡}−(𝜋𝑥)²2!𝑒^{(1−𝜋2)𝑡}+(𝜋𝑥)⁴4!𝑒^{(1−𝜋2)𝑡}−(𝜋𝑥)⁶6!𝑒^{(1−𝜋2)𝑡}=𝑒_{(1−𝜋2)𝑡}+⋯cos(𝜋𝑥).(3.18)

Example 3.4. In the second example, we consider the nonlinear heat equation (1.8) with 𝜖=−2 and 𝑚=3, that is, 𝑢_𝑡=𝑢_𝑥𝑥−2𝑢³.(3.19) In [46] the authors solve (3.19) using antireduction method, and give the solution by means of ansatz (𝜑_𝑖,𝑖=1,2) as follows: 𝑢(𝑥,𝑡)=𝜑₁(𝑡)+2𝜑₂(𝑡)𝑥(1+𝜑₁(𝑡)𝑥+𝜑₂(𝑡)𝑥²)⁻¹,(3.20) where 𝜑₁ and 𝜑₂ satisfy the ordinary differential equations: 𝜑^1=−6𝜑₁𝜑₂,𝜑^2=−6𝜑²².(3.21) We impose 𝜑₁(0)=𝜑₂(0)=1, and solve the above ordinary differential equations, and obtain 𝜑₁(𝑡)=𝜑₂(𝑡)=16𝑡+1.(3.22) Therefore, we have the partial analytical solution of (3.19) as 𝑢(𝑥,𝑡)=1+2𝑥𝑥²+𝑥+6𝑡+1.(3.23) We now solve (3.19) using ADM with the initial condition: 𝑢(𝑥,0)=1+2𝑥𝑥²+𝑥+1,(3.24) and the boundary conditions: 𝑢(0,𝑡)=16𝑡+1,𝑢^𝑥(0,𝑡)=12𝑡+1(6𝑡+1)².(3.25) For the solution of this equation in the 𝑡 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 𝐴₀=−2𝑢³⁰,𝐴₁=−6𝑢²⁰𝑢₁,𝐴₂=−6(𝑢₀𝑢²¹+𝑢²⁰𝑢₂), and so on. Therefore, we obtain 𝑢₀=1+2𝑥𝑢𝑥²+𝑥+1₁,=𝐿_𝑡−1𝐿_𝑥𝑢₀−2𝐿_𝑡−1𝑢³⁰=−6(1+2𝑥)(𝑥²+𝑥+1)²𝑢𝑡,₂=𝐿_𝑡−1𝐿_𝑥𝑢₁−6𝐿_𝑡−1𝑢²⁰𝑢₁=36(1+2𝑥)(𝑥²+𝑥+1)³𝑢₃𝑡²,=𝐿_𝑡−1𝐿_𝑥𝑢₂−6𝐿_𝑡−1𝑢²⁰𝑢₂+𝑢²¹𝑢₀=−216(1+2𝑥)(𝑥²+𝑥+1)⁴𝑡³,(3.26) and so on, in this manner the rest of the terms of the decomposition series have been calculated using 𝑀𝑎𝑡ℎ𝑐𝑎𝑑7. Substituting (3.26) into the decomposition series (2.3.8), we obtain 𝑢(𝑥,𝑡)==𝑢₀1(𝑥+,𝑡2)𝑥+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃(𝑥,𝑡)⋯𝑥²+𝑥+1−(𝑥6²(+1+𝑥+2𝑥1))²+36(1+2𝑥)𝑡(𝑥²+𝑥+1)³𝑡²−216(1+2𝑥)(𝑥²+𝑥+1)⁴𝑡³⋯,(3.27) 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 𝑥 direction, we use the recursive relation given by (2.3.6) to determine the individual terms of the series (2.3.8): 𝑢₀=1𝑢₁6𝑡+1+𝑥(162𝑡𝑡++11)²,==𝑥𝐿_𝑥−12𝐿_𝑡𝑢₀+2𝐿_𝑥−1𝑢³⁰−(61𝑡8+𝑡−1)2³+𝑥³−72𝑡²+1+𝑥₄(6𝑡+1)⁴180𝑡²+30𝑡+1(6𝑡+1)⁵+𝑥⁵432𝑡³−108𝑡²−36𝑡−2(6𝑡+1)⁶,(3.28) 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 𝑢(𝑥,𝑡)==𝑢₀(1𝑥,𝑡)+𝑢₁(𝑥,𝑡)+⋯+6𝑡𝑥+₃1+𝑥(162𝑡𝑡++11)²+𝑥²−(61𝑡8+𝑡−1)2³−72𝑡²+1(6𝑡+1)⁴+𝑥⁴180𝑡²+30𝑡+1+𝑥₅(6𝑡+1)⁵432𝑡³−108𝑡²−36𝑡−2(6𝑡+1)⁶⋯,(3.29) which again gives the analytical solution given by (3.23) in the closed form.

Example 3.5. Let us take 𝑚=1 in (1.10). We obtain 𝜕𝑢𝜕𝑡=𝜕𝜕𝑥𝑢𝜕𝜕𝑥𝑢.(3.30) We impose the initial condition 𝑢(𝑥,0)=𝑥.(3.31)
To obtain the solution, we use the recursive relation (2.4.4) by taking 𝑢₀=𝑥. In this case the first Adomian Polynomial is 𝐴₀=𝑥. Therefore, we have 𝑢₁=𝑡 and 𝐴₁=𝑡. Finally, 𝑢₂=0 which follows that 𝑢_𝑛(𝑥,𝑡)=0 for 𝑛≥2. Putting these individual terms in (2.4.3), one gets the analytical solution 𝑢(𝑥,𝑡)=𝑥+𝑡,(3.32) which can be verified through substitution.

Example 3.6. When 𝑚=−1, (1.10) becomes 𝜕𝑢𝜕𝑡=𝜕𝜕𝑥1𝑢𝜕𝜕𝑥𝑢.(3.33) In [48] the authors give a particular analytical solution to (3.33) as follows: 𝑢(𝑥,𝑡)=(𝑐₁𝑥−𝑐²¹𝑡+𝑐₂)⁻¹,(3.34) where 𝑐₁ and 𝑐₂ are arbitrary constants. We take 𝑐₁=1 and 𝑐₂=0 for simplicity. Therefore, with these choices of 𝑐₁ and 𝑐₂ their solution becomes 𝑢(𝑥,𝑡)=1𝑥−𝑡.(3.35) We now solve (3.33) using ADM with the initial condition: 𝑢(𝑥,0)=1𝑥.(3.36) 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 𝑢₀=1/𝑥, 𝐴₀=−1/𝑥, 𝑢₁=𝑡/𝑥², 𝐴₁=−𝑡/𝑥², 𝑢₂=𝑡²/𝑥³, 𝐴₂=−𝑡²/𝑥³, 𝑢₃=𝑡³/𝑥⁴, and so on. In this manner the rest of the terms of the decomposition series have been calculated using 𝑀𝑎𝑡ℎ𝑐𝑎𝑑7. Substituting these individual terms in (2.4.3), we obtain 𝑢(𝑥,𝑡)==𝑢1₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃(𝑥,𝑡)+⋯𝑥+𝑥𝑡²+𝑡²𝑥³+𝑡³𝑥⁴+⋯,(3.37) which gives the analytical solution obtained in (3.35) in the closed form. This result can be verified through substitution.

Example 3.7. If 𝑚=−4/3, (1.10) reads 𝜕𝑢𝜕𝑡=𝜕𝜕𝑥𝑢^−4/3𝜕𝑢𝜕𝑥.(3.38) In [48], a particular analytical solution to (3.38) is given as follows: 𝑢(𝑥,𝑡)=(2𝑐₁𝑥−3𝑐²¹𝑡+𝑐₂)^−3/4,(3.39) where 𝑐₁ and 𝑐₂ are arbitrary constants, and we take 𝑐₁=1 and 𝑐₂=0 for simplicity. Therefore, one has 𝑢(𝑥,𝑡)=(2𝑥−3𝑡)^−3/4.(3.40) We now solve (3.38) using ADM with the initial condition: 𝑢(𝑥,0)=(2𝑥)^−3/4.(3.41) 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 𝑢₀=(2𝑥)^−3/4, 𝐴₀=−3×2^−7/4𝑥^−3/4, 𝑢₁=9×2^−15/4𝑥^−7/4𝑡, 𝐴₁=−27×2^−19/4𝑥^−7/4𝑡, 𝑢₂=189×2^−31/4𝑥^−11/4𝑡², 𝐴₂=−567×2^−35/4𝑥^−11/4𝑡², 𝑢₃=2079×2^−43/4𝑥^−15/4𝑡³, and so on. In this manner the rest of the terms of the decomposition series have been calculated using 𝑀𝑎𝑡ℎ𝑐𝑎𝑑7. Substituting these individual terms in (2.4.3), one obtains 𝑢(𝑥,𝑡)==𝑢(2₀𝑥(𝑥),𝑡)_−3/4+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃(𝑥,𝑡)+⋯+9×2^−15/4𝑥^−7/4𝑡+189×2^−31/4𝑥^−11/4𝑡²+2079×2^−43/4𝑥^−15/4𝑡³+⋯,(3.42) 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: 𝐴(𝑢)−𝑓(𝐫)=0,𝐫∈Ω,(4.1) with boundary conditions: 𝐵(𝑢,𝜕𝑢/𝜕𝑛)=0,𝐫∈Γ,(4.2) where 𝐴 is a general differential operator, 𝐵 is a boundary operator, 𝑓(𝐫) is a known analytic function, and Γ is the boundary of the domain Ω.

In general, one divides the operator 𝐴 into two parts 𝐿 and 𝑁, where 𝐿 is linear, while 𝑁 is nonlinear. Therefore, (4.1) is written as follows: 𝐿(𝑢)+𝑁(𝑢)−𝑓(𝐫)=0.(4.3)

By the homotopy technique [19, 51], one constructs a homotopy 𝑣(𝐫,𝑝)∶Ω×[0,1]→𝐑 which satisfies 𝐻(𝑣,𝑝)=(1−𝑝)𝐿(𝑣)−𝐿𝑢₀+𝑝[𝐴(𝑣)−𝑓(𝐫)]=0,𝑝∈[0,1],𝐫∈Ω,(4.4) or 𝐻(𝑣,𝑝)=𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝[𝑁(𝑣)−𝑓(𝐫)]=0,(4.5) where 𝑝∈[0,1] is an embedding parameter, 𝑢₀ is an initial approximation of (4.1), which satisfies the boundary conditions. It is clear that 𝐻𝐻((𝑣𝑣,,10))==𝐴𝐿((𝑣𝑣))−−𝐿𝑓(𝐫𝑢)₀==0,0,(4.6) the changing process of 𝑝 from zero to unity is just that of 𝑣(𝐫,𝑝) from 𝑢₀(𝐫) to 𝑢(𝐫).

According to the HPM, we can first use the embedding parameter 𝑝 as a “small parameter", and assume that the solution of (4.4) and (4.5) can be written as a power series in 𝑝: 𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+⋯.(4.7) Setting 𝑝=1 results in the approximate solution of (4.1): 𝑢=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂+⋯.(4.8)

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 𝐴(𝑣) (the following opinions are suggested by He [19]).

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

(2) The norm of 𝐿⁻¹𝜕𝑁/𝜕𝑣 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 𝑢(0)=2, as chosen in Example 3.1. We rewrite (1.3) in the form [52] 𝑑𝑢𝑑𝜏𝑢=(0𝑝𝑢)=(12−,𝑢),(5.1) where 𝑝∈[0,1] is an embedding parameter. As in He's HPM, it is clear that when 𝑝=0, (5.1) becomes a linear equation; when 𝑝=1, it becomes the original nonlinear one. We consider the imbedding parameter 𝑝 as a “small 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 𝑝, we obtain the following differential equations: 𝑝𝑝⁰∶𝑣^0=0,𝑣₀(0)=2,₁𝑝∶𝑣^1=𝑣₀−𝑣²⁰,𝑣₁(0)=0,₂𝑝∶𝑣^2=𝑣₁−2𝑣₀𝑣₁,𝑣₂(0)=0,₃𝑝∶𝑣^3=𝑣₂−𝑣²¹−2𝑣₀𝑣₂,𝑣₃(0)=0,₄𝑝∶𝑣^4=𝑣₃−2𝑣₀𝑣₃−2𝑣₁𝑣₂,𝑣₄(0)=0,₅⋮∶𝑣^5=𝑣₄−𝑣²²−2𝑣₁𝑣₃−2𝑣₀𝑣₄,𝑣₅(0)=0,(5.2) where “primes" denote differentiation with respect to 𝜏. Thus, solving the equations above yields 𝑣𝑣₀=2,₁𝑣=−2𝜏,₂𝑣₃=3𝜏²,𝑣₄=−13/3𝜏³,𝑣₅=25/4𝜏⁴,⋮=−541/60𝜏⁵,(5.3)
Substituting these in (4.7) gives 𝑣=2−2𝑝𝜏+3𝑝²𝜏²−13/3𝑝³𝜏³+25/4𝑝⁴𝜏⁴−541/60𝑝⁵𝜏⁵+⋯.(5.4) Hence, by (4.8) one has 𝑢=2−2𝜏+3𝜏²−13/3𝜏³+25/4𝜏⁴−541/60𝜏⁵+⋯,(5.5) which is exactly the same solution obtained in (3.4). Also, the solution in (5.5) is equal to 𝑢=22−𝑒^−𝜏,(5.6) in the closed form which is exactly the same as in (1.4) with 𝐾=1,𝑁₀=2 (see Example 3.1).

Example 5.2. We now solve (1.7) using HPM with 𝛼=1,𝑢(0)=1.3,𝑣(0)=0.6 as taken in Example 3.2. We rewrite (1.7) in the form [52] 𝑑𝑢𝑑𝜏=𝑝𝑢(1−𝑣),𝑑𝑑𝜏𝑣=𝑝𝑣(𝑢−1),𝑢(0)=1.3,𝑣(0)=0.6,(5.7) where 𝑝∈[0,1] is an embedding parameter. As in He's HPM, it is clear that when 𝑝=0, (5.7) becomes a linear system; when 𝑝=1, it becomes the original nonlinear one. We consider the imbedding parameter 𝑝 as a “small parameter". We assume the solutions of (5.7), (𝑢,𝑣) are expressed as power series: 𝑣𝑢==𝑢𝑣₀₀+𝑝𝑢₁+𝑝²𝑢₂+⋯,(5.8)+𝑝𝑣₁+𝑝²𝑣₂+⋯,(5.9) respectively. Substituting (5.8) and (5.9) into the system (5.7), and equating coefficients of like 𝑝, we obtain the following systems of differential equations: 𝑝⁰∶⎧⎪⎨⎪⎩𝑢^0𝑣=0_0𝑢=0₀𝑝(0)=1.3,𝑣₀(0)=0.6,₁∶⎧⎪⎨⎪⎩𝑢^1𝑣_1=𝑢₀1−𝑣₀𝑢=𝑣₀𝑢₀−1₁𝑝(0)=0,𝑣₁(0)=0,₂∶⎧⎪⎨⎪⎩𝑢^2𝑣_2=𝑢₁−𝑢₀𝑣₁+𝑢₁𝑣₀=𝑢₀𝑣₁+𝑢₁𝑣₀−𝑣₁𝑝𝑢₂(0)=0,𝑣₂(0)=0,₃∶⎧⎪⎨⎪⎩𝑢^3𝑣_3=𝑢₂−𝑢₀𝑣₂+𝑢₁𝑣₁+𝑢₂𝑣₀=𝑢₀𝑣₂+𝑢₁𝑣₁+𝑢₂𝑣₀−𝑣₂𝑝𝑢₃(0)=0,𝑣₃(0)=0,₄∶⎧⎪⎨⎪⎩𝑢^4𝑣_4=𝑢₃−𝑢₀𝑣₃+𝑢₁𝑣₂+𝑢₂𝑣₁+𝑢₃𝑣₀=𝑢₀𝑣₃+𝑢₁𝑣₂+𝑢₂𝑣₁+𝑢₃𝑣₀−𝑣₃⋮𝑢₄(0)=0,𝑣₄(0)=0,(5.10) where “primes" denote differentiation with respect to 𝜏. Thus, solving the above systems of equations yields 𝑢𝑢₀=1.3,𝑣₀=0.6,₁𝑢=0.52𝜏,𝑣₁=0.18𝜏,₂𝑢₃=−0.013𝜏²,𝑣₂=0.183𝜏²,𝑢₄=−0.1122𝜏³,𝑣₃=0.0469𝜏³,⋮=−0.0497𝜏⁴,𝑣₄=0.0099𝜏⁴,(5.11)
Substituting these 𝑢_𝑛,𝑣_𝑛, 𝑛≥0 into (5.8), and (5.9), respectively, we have 𝑣𝑢==10..36++00..5128𝑝𝑝𝜏𝜏−+00..011833𝑝𝑝²₂𝜏²−0.1122𝑝³𝜏³−0.0497𝑝⁴𝜏⁴−⋯,𝜏²+0.0469𝑝³𝜏³+0.0099𝑝⁴𝜏⁴+⋯.(5.12) Letting 𝑝→1 one obtains 𝑣𝑢==10..36++00..5128𝜏𝜏−+00..011833𝜏𝜏²₂−0.1122𝜏³−0.0497𝜏⁴−⋯,+0.0469𝜏³+0.0099𝜏⁴+⋯,(5.13) 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 𝑢_𝑡=(𝑢^𝑚𝑢_𝑥)_𝑥,(6.1) using modified HPM with initial condition: 𝑢(𝑥,0)=𝑓(𝑥).(6.2) Here 𝑢=𝑢(𝑥,𝑡). 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 3.5–3.7. In their work they construct the following homotopy with 𝐿(𝑢)=𝑢_𝑡 and 𝑁(𝑢)=−(𝑢^𝑚𝑢_𝑥)_𝑥, 𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝𝑁(𝑣)=0.(6.3) To deal with the nonlinear term, they employ He polynomials considered in [33, 34] which is given by 𝑁(𝑢)=𝑁𝑣₀+𝑁𝑣₀,𝑣₁𝑝+𝑁𝑣₀,𝑣₁,𝑣₂𝑝²+⋯+𝑁𝑣₀,𝑣₁,…,𝑣_𝑛𝑝^𝑛+⋯,(6.4) where 𝑁𝑣₀,𝑣₁,…,𝑣_𝑛=1𝑛!𝜕𝜕𝑝^𝑛𝑛𝑁^𝑛_𝑘=0𝑝^𝑘𝑣_𝑘_𝑝=0,𝑛=1,2,….(6.5) Substituting (6.4) into (6.3), and equating coefficients of like 𝑝, one obtains 𝑝𝑝⁰∶𝐿𝑣₀−𝐿𝑢₀=0,₁𝑝∶𝐿𝑣₁+𝐿𝑢₀+𝑁𝑣₀=0,₂𝑝∶𝐿𝑣₂+𝑁𝑣₀,𝑣₁=0,₃𝑝⋮∶𝐿𝑣₃+𝑁𝑣₀,𝑣₁,𝑣₂=0,_𝑛+1∶𝐿𝑣_𝑛+1+𝑁𝑣₀,𝑣₁,…,𝑣_𝑛=0,(6.6) and so on, which forms the basis of a complete determination of the components 𝑣₀,𝑣₁,…,𝑣_𝑛,…. They let 𝑢₀(𝑥,𝑡)=0 for simplicity. Therefore, they obtain the following linear equations for these components: (𝑣₁(𝑣₀)_𝑡=0,𝑣₀(𝑥,0)=𝑓(𝑥),(6.7)(𝑣₂)_𝑡−(𝑣^𝑚0𝑣_0𝑥)_𝑥=0,𝑣₁(𝑥,0)=0,(6.8))_𝑡+𝜕𝜕𝑝𝑁¹_𝑘∑₌₀𝑝^𝑘𝑣_𝑘_𝑝=0(𝑣₃=0,𝑣₂(𝑥,0)=0,(6.9))_𝑡+12!𝜕𝜕𝑝²²𝑁²_𝑘=0𝑝^𝑘𝑣_𝑘_𝑝=0=0,𝑣₃(𝑥,0)=0,(6.10) and so on.

Example 6.1. In this example we solve the initial value problem given by (3.30)-(3.31) using modified HPM. We simply take 𝑚=1 and 𝑓(𝑥)=𝑥 in (6.1) and (6.2), respectively, and use (6.7)–(6.10) to obtain the components 𝑣₀,𝑣₁,…,𝑣_𝑛,…. From (6.7)–(6.9), we easily obtain 𝑣₀(𝑥,𝑡)=𝑥, 𝑣₁(𝑥,𝑡)=𝑡, and 𝑣₂(𝑥,𝑡)=0. Therefore, one gets 𝑣_𝑛(𝑥,𝑡)=0, 𝑛≥2, which results in 𝑢(𝑥,𝑡)=𝑥+𝑡.(6.11) This is exactly the same solution we have obtained in (3.32).

Example 6.2. We now solve the initial value problem given by (3.33) and (3.36) using the same method. We now take 𝑚=−1 and 𝑓(𝑥)=1/𝑥 in (6.1) and (6.2), respectively, and use (6.7)–(6.10). After somewhat tedious computations we obtain 𝑣₀(𝑥,𝑡)=1/𝑥, 𝑣₁(𝑥,𝑡)=𝑡/𝑥², 𝑣₂(𝑥,𝑡)=𝑡²/𝑥³, 𝑣₃(𝑥,𝑡)=𝑡³/𝑥⁴, and so on. Therefore, one gets the solution to this problem as follows: 𝑢(𝑥,𝑡)=1/𝑥+𝑡/𝑥²+𝑡²/𝑥³+𝑡³/𝑥⁴+⋯,(6.12) which gives 𝑢(𝑥,𝑡)=1𝑥−𝑡,(6.13) in the closed form. This solution is exactly the same as we find in (3.35).

7. Variational Iteration Method

To illustrate the basic idea of the variational iteration method (VIM), we consider the following general nonlinear system: 𝐿𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)=𝑔(𝑥,𝑡),(7.1) where 𝐿 is a linear operator, and 𝑁 is a nonlinear operator, and 𝑔(𝑥,𝑡) is the source inhomogeneous term.

According to the variational iteration method [23, 38], one can construct the following iteration formulation: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡0𝜆(𝑡,𝑠)𝐿𝑢_𝑛(𝑥,𝑠)+𝑁𝑢_𝑛̃(𝑥,𝑠)−𝑔(𝑥,𝑠)𝑑𝑠,(7.2) where 𝜆 is a general Lagrange's multiplier, which can be identified optimally via the variational theory, and 𝑢_𝑛̃ is a restricted variation which means 𝛿𝑢_𝑛̃=0.

It is obvious now that the main steps of variational iteration method require first the determination of the Lagrangian multiplier 𝜆 that will be identified optimally. Having determined the Lagrangian multiplier, the successive approximations 𝑢_𝑛+1, 𝑛≥0, of the solution 𝑢 will be readily obtained upon using any selective function 𝑢₀ [38]. Consequently, the solution is obtained as the limit of the resulting successive approximations, that is, 𝑢=lim_𝑛→∞𝑢_𝑛.(7.3)

8. Applications of VIM

Example 8.1. In this example we solve the initial value problem given by (3.12)-(3.13) using VIM. According to VIM described above, a correction functional for (3.12) can be constructed as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡0𝜆(𝑡,𝑠)𝜕𝑢^𝑛(𝑥,𝑠)𝜕𝑠−𝑢^𝑛̃(𝑥,𝑠)−𝜕²𝑢_𝑛̃(𝑥,𝑠)𝜕𝑥²𝑑𝑠,(8.1) where 𝜆 is a Lagrange multiplier, 𝑢_𝑛̃ is a restricted variation, that is, 𝛿𝑢_𝑛̃=0. To find the optimal value of 𝜆, we make (8.1) stationary with respect to 𝑢_𝑛, and obtain 𝜕𝜆(𝑡,𝑠)1+𝜆𝜕(𝑡𝑠,𝑠)|=_𝑠=𝑡0,=0.(8.2) The Lagrange multiplier can be identified as 𝜆=−1, submitting the result into (8.1) leads to the following iteration formula: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡0(−1)𝜕𝑢^𝑛(𝑥,𝑠)𝜕𝑠−𝑢^𝑛(𝑥,𝑠)−𝜕²𝑢_𝑛(𝑥,𝑠)𝜕𝑥²𝑑𝑠,𝑛≥0.(8.3) Beginning with an initial approximation 𝑢₀(𝑥,𝑡)=𝑢₀(𝑥,0)=cos(𝜋𝑥), we obtain the following successive approximations: 𝑢𝑢₁(𝑥,𝑡)=cos(𝜋𝑥)+1−𝜋²𝑡cos(𝜋𝑥),₂(𝑥,𝑡)=cos(𝜋𝑥)+1−𝜋²𝑡cos(𝜋𝑥)+12!(1−𝜋²𝑢)²𝑡²cos(𝜋𝑥),₃(𝑥,𝑡)=cos(𝜋𝑥)+1−𝜋²𝑡cos(𝜋𝑥)+12!1−𝜋²²𝑡²cos(𝜋𝑥)+13!(1−𝜋²𝑢⋮)³𝑡³cos(𝜋𝑥),_𝑛(𝑥,𝑡)=cos(𝜋𝑥)+1−𝜋²𝑡cos(𝜋𝑥)+12!1−𝜋²²𝑡²cos(𝜋𝑥)+⋯+1𝑛!(1−𝜋²)^𝑛𝑡^𝑛cos(𝜋𝑥).(8.4)
By the use of (7.3), the solution to (3.12)-(3.13) becomes 𝑢(𝑥,𝑡)=𝑒^{(1−𝜋2)𝑡}cos(𝜋𝑥),(8.5) which is exactly the same as we have obtained in (3.16).

Example 8.2. In this example we solve the initial value problem given by (3.19) and (3.24) using VIM. For this problem, a correction functional for (3.19) can be constructed as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡0𝜆(𝑡,𝑠)𝜕𝑢^𝑛(𝑥,𝑠)𝜕𝑠+2𝑢^3𝑛̃(𝑥,𝑠)−𝜕²𝑢_𝑛̃(𝑥,𝑠)𝜕𝑥²𝑑𝑠,(8.6) where 𝜆 is a Lagrange multiplier, 𝑢_𝑛̃ is a restricted variation, that is, 𝛿𝑢_𝑛̃=0. The optimal value of the Lagrange multiplier is calculated to be 𝜆=−1 as done in the above example (see (8.2). Submitting this 𝜆 into (8.6) leads to the following iteration formula: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡0(−1)𝜕𝑢^𝑛(𝑥,𝑠)𝜕𝑠+2𝑢^3𝑛(𝑥,𝑠)−𝜕²𝑢_𝑛(𝑥,𝑠)𝜕𝑥²𝑑𝑠,𝑛≥0.(8.7)
Beginning with an initial approximation 𝑢₀(𝑥,𝑡)=𝑢₀(𝑥,0)=(1+2𝑥)/(𝑥²+𝑥+1), we obtain the following successive approximations: 𝑢₁(𝑥,𝑡)=1+2𝑥𝑥²+𝑥+1−(𝑥6²(+1+𝑥+2𝑥1))²𝑢𝑡,₂(𝑥,𝑡)=1+2𝑥𝑥²+𝑥+1−(𝑥6²(+1+𝑥+2𝑥1))²𝑡+36(1+2𝑥)(𝑥²+𝑥+1)³𝑢₃𝑡²,(𝑥,𝑡)=1+2𝑥𝑥²+𝑥+1−(𝑥6²(+1+𝑥+2𝑥1))²𝑡+36(1+2𝑥)(𝑥²+𝑥+1)³𝑡²−216(1+2𝑥)(𝑥²+𝑥+1)⁴𝑢_𝑛⋮𝑡³,(𝑥,𝑡)=1+2𝑥𝑥²+𝑥+1−(𝑥6²(+1+𝑥+2𝑥1))²𝑡+36(1+2𝑥)(𝑥²+𝑥+1)³𝑡²−⋯+(−1)^𝑛6^𝑛(1+2𝑥)(𝑥²+𝑥+1)^𝑛+1𝑡^𝑛.(8.8)
The VIM admits the use of (7.3), therefore, one obtains the exact solution 𝑢(𝑥,𝑡)=1+2𝑥6𝑡+𝑥²+𝑥+1,(8.9) which is exactly the same as we have obtained in (3.23).

9. Conclusion and Results

In this paper we present a review of some recent results for the approximate analytical solutions of nonlinear differential equations. To do this, we introduce and compare three methods, namely, ADM, HPM, and VIM, which are recently studied by the researchers in various fields of science and engineering.

In Section 2, we work out a detailed analysis of ADM for logistic growth model, predator-prey model, for nonlinear heat equation of the form 𝑢_𝑡=𝑢_𝑥𝑥+𝑢^𝑚, and for the porous media equation, respectively.

In Section 3, we provide several applications of ADM to some nonlinear differential equations of the form mentioned in the last paragraph. For example, Figure 1 shows a very good approximation to the analytical solution of logistic growth model in the time interval [0,0.4] by using only 10 terms of the decomposition series, which indicates that the speed of convergence of this method is very fast. In addition, the overall errors can be made pretty small, and a good approximation to the analytical solution for 𝜏≥0.4 can be achieved by adding new terms to the partial sums of the decomposition series given by (3.5).

Figure 2 shows the numerical solutions of the system given by (1.7) with initial data 𝑢(0)=𝛿=1.3, 𝑣(0)=𝛽=0.6. We obtain these solutions using ode23, an ordinary differential equation solver found in the MATLAB package. Also, Figure 3 shows the approximate solutions to the same system using only 5 terms of the decomposition series. It is clear from the comparison between two figures that there is a very close agreement between the two solutions for both 𝑢 (prey population) and 𝑣 (predator population) in the time interval [0,1.45]. As mentioned above for the logistic growth model, a very good approximation to the numerical solution for 𝜏≥1.45 can be achieved by adding new terms to the decomposition series.

Figure 2 

Numerical solutions to the system (1.7) with 𝛼=1, 𝑢(0)=1.3, 𝑣(0)=0.6.

Figure 3 

Approximate solutions to the system (1.7) by Adomian's decomposition method with 𝛼=1, 𝑢(0)=1.3, 𝑣(0)=0.6.

Tables 1-2 show the absolute errors |𝑢(𝑥,𝑡)−𝜙_𝑛(𝑥,𝑡)|, where 𝑢(𝑥,𝑡) is the analytical solution of the linear initial-boundary value problem given by (3.12)–(3.14), and 𝜙_𝑛(𝑥,𝑡) are the 𝑛th partial sums given by (2.3.8) in 𝑡 and 𝑥 directions, respectively. Also, Tables 3-4 show the absolute errors for the nonlinear initial-boundary value problem given by (3.19), (3.24)-(3.25) in 𝑡 and 𝑥 directions, respectively. For numerical purposes, we take 𝑛=30 for the linear problem, and take 𝑛=20 for the nonlinear problem. As seen from Tables 1-2, the absolute errors are very small. The same is also true for the nonlinear problem as it is clear from Tables 3-4. For the nonlinear problem, we achieve a very good approximation to the partial analytical solution by using only 20 terms of the decomposition series, which shows that the speed of convergence of this method is very fast, and the overall errors can be made very small by adding new terms to the series given by (2.3.8).

In Example 3.5, we directly obtain the analytical solution of the porous media equation (1.10) for 𝑚=1 using ADM. In Examples 3.6-3.7, the approximate analytical solutions of the porous media equation are obtained for 𝑚=−1 and 𝑚=−4/3, respectively. The absolute errors have been calculated for these values of 𝑚 in Tables 5 and 6, respectively. For numerical purposes, we take 𝑛=50 for 𝑚=−1, and take 𝑛=6 for 𝑚=−4/3. As seen from Tables 5 and 6, the absolute errors in both cases are very small. We do achieve very good approximations to the analytical solutions by using only 50 terms of the decomposition series for the case 𝑚=−1, and by using only 6 terms for the case 𝑚=−4/3. Also, the first pictures of Figures 4 and 5 show the particular analytical solutions for 𝑚=−1 and 𝑚=−4/3, respectively, whereas the second pictures of Figures 4 and 5 show the corresponding approximate analytical solutions obtained using ADM. In both figures the pictures look almost identical.

(a)

(b)

Figure 4 

The numerical results for (3.33), two pictures are almost identical.

(a)

(b)

Figure 5 

The numerical results for (3.38), two pictures are almost identical.

On the other hand, in Section 4 we present the idea of HPM which was proposed by Ji-Huan He in 1999. According to this method, the solution is obtained as the summation of an infinite series by constructing a homotopy with an embedding parameter 𝑝∈[0,1], which is considered as a “small parameter".

In Section 5, we provide a couple of applications of HPM. For example, in Example 5.1 we apply this method to logistic growth model, and in Example 5.2 we apply it to the Predator-Prey Models. As the reader remembers, we have solved the same problems in Examples 3.1 and 3.2, respectively, using ADM. Even though we obtain the same solutions for both methods, the HPM needs not to calculate Adomian polynomials, and it is very straightforward, and the solution procedure is very simple. Therefore, one clearly can conclude that HPM and He polynomials can completely replace the Adomian method and Adomian polynomials.

Also, in Section 6, we present two applications (Examples 6.1-6.2) of modified HPM to porous media equations. We obtain again exactly the same solutions as we have obtained in Examples 3.5-3.6 using ADM. However, as we have faced in the HPM, the modified HPM is very simple, and we do not have to compute Adomian polynomials in this method either.

In Section 7, we introduce the basic idea of VIM which was proposed by Ji-Huan He. This method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Moreover, we provide a couple of applications of VIM in Section 8. In Examples 8.1 and 8.2, we apply the VIM to the initial value problems which we have solved in Examples 3.3 and 3.4 using ADM. We see that the VIM uses only the prescribed initial conditions and does not require a specific treatment, whereas we do have to compute Adomian polynomials in Examples 3.3 and 3.4 which are tedious (see also Section 2.3).

As a result, although the numerical results are almost the same, HPM is much easier, and more convenient and efficient than ADM and VIM.

Acknowledgment

The author would like to thank the referees for their valuable suggestions, which greatly improved the paper.