Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Review Article | Open Access

Volume 2009 |Article ID 202307 | 34 pages | https://doi.org/10.1155/2009/202307

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

Academic Editor: Ji Huan He
Received15 Jan 2009
Revised21 Apr 2009
Accepted29 Apr 2009
Published07 Jul 2009

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 [1โ€“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 [7โ€“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 [16โ€“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 ๐‘โˆˆ[0,1], which is considered as a โ€œsmall 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., [22โ€“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, 28โ€“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, 24โ€“26, 35โ€“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 ๐œ†, 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 [3โ€“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] ๐‘‘๐‘๐‘‘๐‘ก=๐‘Ÿ๐‘(1โˆ’๐‘/๐พ),(1.1) where ๐‘Ÿ and ๐พ are positive constants. Here ๐‘=๐‘(๐‘ก) represents the population of the species at time ๐‘ก, ๐‘Ÿ(1โˆ’๐‘/๐พ) is the per capita growth rate, and ๐พ is the carrying capacity of the environment. We nondimensionalize (1.1) by setting ๐‘ข(๐œ)=๐‘(๐‘ก)๐พ,๐œ=๐‘Ÿ๐‘ก,(1.2) and it becomes ๐‘‘๐‘ข๐‘‘๐œ=๐‘ข(1โˆ’๐‘ข).(1.3) If ๐‘(0)=๐‘0, then ๐‘ข(0)=๐‘0/๐พ. Therefore, the analytical solution of (1.3) is easily obtained: 1๐‘ข(๐œ)=๎€ท1+๐พ/๐‘0๎€ธ๐‘’โˆ’1โˆ’๐œ.(1.4)

Second, we consider the Predator-Prey Models: Lotka-Volterra systems as an interacting species model to be governed by [3, 43, 44] ๐‘‘๐‘๐‘‘๐‘ก=๐‘(๐‘Žโˆ’๐‘๐‘ƒ),๐‘‘๐‘ƒ๐‘‘๐‘ก=๐‘ƒ(๐‘๐‘โˆ’๐‘‘),(1.5) where ๐‘Ž,๐‘,๐‘, and ๐‘‘ are constants. Here ๐‘=๐‘(๐‘ก) is the prey population and ๐‘ƒ=๐‘ƒ(๐‘ก) that of the predator at time ๐‘ก. We nondimensionalize the system (1.5) [43] by setting ๐‘ข(๐œ)=๐‘๐‘(๐‘ก)๐‘‘,๐‘ฃ(๐œ)=๐‘๐‘ƒ(๐‘ก)๐‘Ž,๐œ=๐‘Ž๐‘ก,๐›ผ=๐‘‘/๐‘Ž,(1.6) and it becomes ๐‘‘๐‘ข๐‘‘๐œ=๐‘ข(1โˆ’๐‘ฃ),๐‘‘๐‘ฃ๐‘‘๐œ=๐›ผ๐‘ฃ(๐‘ขโˆ’1).(1.7)

Third, we present the heat equation [4]: ๐‘ข๐‘ก=๐‘ข๐‘ฅ๐‘ฅ+๐œ–๐‘ข๐‘š,(1.8) where ๐‘š=1,2,3,โ€ฆ, and ๐œ– is a parameter. Here, the indices ๐‘ก and ๐‘ฅ denote derivatives with respect to these variables. Unless ๐‘š=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 ๐‘ข๐‘ก=(๐ด(๐‘ข)๐‘ข๐‘ฅ)๐‘ฅ+๐ต(๐‘ข)๐‘ข๐‘ฅ+๐ถ(๐‘ข),(1.9) where ๐ด(๐‘ข), ๐ต(๐‘ข), and ๐ถ(๐‘ข) 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]: ๐œ•๐‘ข=๐œ•๐œ•๐‘ก๎‚€๐‘ข๐œ•๐‘ฅ๐‘š๐œ•๐‘ข๎‚๐œ•๐‘ฅ,(1.10) where ๐‘š 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 โ€œpopulation pressure" in biological systems. This equation is called a degenerate parabolic differential equation because the diffusion coefficient ๐ท(๐‘ข)=๐‘ข๐‘š does not satisfy the condition for classical diffusion equations, ๐ท(๐‘ข)>0 [49]. For the motion of thin viscous films, (1.10) with ๐‘š=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 ๐‘š can be determined by comparing known solutions with empirical data [49].

In the following section, we apply the ADM [7โ€“13] 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] ๐‘‘๐‘ข๐‘‘๐œ=๐‘ขโˆ’๐‘“(๐‘ข),๐‘ข(0)=๐›พ,(2.1) where ๐‘“ is a nonlinear function of ๐‘ข. We are looking for the solution ๐‘ข satisfying (2.1.1). The decomposition method consists of approximating the solution of (2.1.1) as an infinite series: ๐‘ข=โˆž๎“๐‘›=0๐‘ข๐‘›,(2.2) and decomposing ๐‘“ as ๐‘“(๐‘ข)=โˆž๎“๐‘›=0๐ด๐‘›,(2.3) where ๐ด๐‘›'s are the Adomian polynomials given by ๐ด๐‘›=1๐‘‘๐‘›!๐‘›๐‘‘๐œ†๐‘›๎ƒฌ๐‘“๎ƒฉโˆž๎“๐‘›=0๐œ†๐‘›๐‘ข๐‘›๎ƒช๎ƒญ๐œ†=0,๐‘›=0,1,2,โ€ฆ.(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 ๐ฟ๐‘ข=๐‘ขโˆ’๐‘“(๐‘ข),(2.5) where the notation ๐ฟ=๐œ•/๐œ•๐œ symbolizes the linear differential operator. We assume the integration inverse operators ๐ฟโˆ’1exist, and it is defined as ๐ฟโˆ’1=โˆซ๐œ0(โ‹…)๐‘‘๐œ. Therefore, applying on both sides of (2.1.5) with ๐ฟโˆ’1 yields ๐‘ข(๐œ)=๐‘ข(0)+๐ฟโˆ’1๐‘ข(๐œ)โˆ’๐ฟโˆ’1๐‘“(๐‘ข(๐œ)).(2.6) Using (2.1.2) and (2.1.3), it follows that โˆž๎“๐‘›=0๐‘ข๐‘›=๐‘ข(0)+๐ฟโˆžโˆ’1๎“๐‘›=0๐‘ข๐‘›โˆ’๐ฟโˆžโˆ’1๎“๐‘›=0๐ด๐‘›.(2.7) Therefore, one determines the iterates in the following recursive way: ๐‘ข0๐‘ข=๐‘ข(0)=๐›พ,๐‘›+1=๐ฟโˆ’1๐‘ข๐‘›โˆ’๐ฟโˆ’1๐ด๐‘›,๐‘›=0,1,2,โ€ฆ.(2.8) We then define the solution ๐‘ข as ๐‘ข=lim๐‘›๐‘›โ†’โˆž๎“๐‘˜=0๐‘ข๐‘˜.(2.9)

2.2. Analysis of the Method for Interacting Species

In this section, we consider the system of the form [3] ๐‘‘๐‘ข๐‘‘๐œ=๐‘ขโˆ’๐‘“(๐‘ข,๐‘ฃ),๐‘‘๐‘ฃ[],๐‘‘๐œ=๐›ผ๐‘”(๐‘ข,๐‘ฃ)โˆ’๐‘ฃ(2.10) with initial data ๐‘ข(0)=๐›ฟ,๐‘ฃ(0)=๐›ฝ.(2.11) Here, ๐‘“ and ๐‘” are nonlinear functions of ๐‘ข and ๐‘ฃ. We are looking for the solutions (๐‘ข,๐‘ฃ) satisfying (2.2.1)-(2.2.2). The decomposition method consists of approximating the solutions of the above system as an infinite series: ๐‘ข=โˆž๎“๐‘›=0๐‘ข๐‘›,๐‘ฃ=โˆž๎“๐‘›=0๐‘ฃ๐‘›,(2.12) and decomposing ๐‘“ and ๐‘” as [6] ๐‘“(๐‘ข,๐‘ฃ)=โˆž๎“๐‘›=0๐ต๐‘›,๐‘”(๐‘ข,๐‘ฃ)=โˆž๎“๐‘›=0๐ถ๐‘›,(2.13) where ๐ต๐‘› and ๐ถ๐‘› 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 [๐‘”],๐ฟ๐‘ข=๐‘ขโˆ’๐‘“(๐‘ข,๐‘ฃ),๐ฟ๐‘ฃ=๐›ผ(๐‘ข,๐‘ฃ)โˆ’๐‘ฃ(2.14) where the notation ๐ฟ=๐œ•/๐œ•๐œ again symbolizes the linear differential operator. Therefore, applying on both sides of the equations of the system (2.2.5) with ๐ฟโˆ’1 yields [6] ๐‘ข(๐œ)=๐‘ข(0)+๐ฟโˆ’1๐‘ข(๐œ)โˆ’๐ฟโˆ’1๎€บ๐ฟ๐‘“(๐‘ข(๐œ),๐‘ฃ(๐œ)),๐‘ฃ(๐œ)=๐‘ฃ(0)+๐›ผโˆ’1๐‘”(๐‘ข(๐œ),๐‘ฃ(๐œ))โˆ’๐ฟโˆ’1๎€ป.๐‘ฃ(๐œ)(2.15) Using (2.2.3) and (2.2.4), it follows that โˆž๎“๐‘›=0๐‘ข๐‘›=๐‘ข(0)+๐ฟโˆžโˆ’1๎“๐‘›=0๐‘ข๐‘›โˆ’๐ฟโˆžโˆ’1๎“๐‘›=0๐ต๐‘›,โˆž๎“๐‘›=0๐‘ฃ๐‘›๎ƒฌ๐ฟ=๐‘ฃ(0)+๐›ผโˆžโˆ’1๎“๐‘›=0๐ถ๐‘›โˆ’๐ฟโˆžโˆ’1๎“๐‘›=0๐‘ฃ๐‘›๎ƒญ.(2.16) Therefore, one determines the iterates in the following recursive way: ๐‘ข0๐‘ข=๐‘ข(0)=๐›ฟ,๐‘›+1=๐ฟโˆ’1๐‘ข๐‘›โˆ’๐ฟโˆ’1๐ต๐‘›๐‘ฃ,๐‘›=0,1,2,โ€ฆ,0๐‘ฃ=๐‘ฃ(0)=๐›ฝ,๐‘›+1๎€บ๐ฟ=๐›ผโˆ’1๐ถ๐‘›โˆ’๐ฟโˆ’1๐‘ฃ๐‘›๎€ป,๐‘›=0,1,2โ€ฆ.(2.17)

We then define the solutions of the initial value problem (2.2.1)-(2.2.2) as ๎ƒฉ(๐‘ข,๐‘ฃ)=lim๐‘›๐‘›โ†’โˆž๎“๐‘˜=0๐‘ข๐‘˜,lim๐‘›๐‘›โ†’โˆž๎“๐‘˜=0๐‘ฃ๐‘˜๎ƒช.(2.18)

2.3. Analysis of the Method for the Heat Equation ๐‘ข๐‘ก=๐‘ข๐‘ฅ๐‘ฅ+๐œ–๐‘ข๐‘š

In this section, we consider (1.8) in an operator form [4] ๐ฟ๐‘ก(๐‘ข(๐‘ฅ,๐‘ก))โˆ’๐ฟ๐‘ฅ(๐‘ข(๐‘ฅ,๐‘ก))โˆ’๐œ–๐‘ข๐‘š=0,(2.19) with the initial and boundary conditions, where the notations ๐ฟ๐‘ก=๐œ•/๐œ•๐‘ก and ๐ฟ๐‘ฅ=๐œ•2/๐œ•๐‘ฅ2 symbolize the linear differential operators. We assume the integration inverse operators ๐ฟ๐‘กโˆ’1 and ๐ฟ๐‘ฅโˆ’1 exist, and they are defined as ๐ฟ๐‘กโˆ’1=โˆซ๐‘ก0(โ‹…)๐‘‘๐‘ก and ๐ฟ๐‘ฅโˆ’1=โˆฌ๐‘ฅ0(โ‹…)๐‘‘๐‘ฅ๐‘‘๐‘ฅ, respectively. Therefore, we can write the solutions in ๐‘ก and ๐‘ฅ directions as [1, 2, 7] ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข(๐‘ฅ,0)+๐ฟ๐‘กโˆ’1๎€บ๐ฟ๐‘ฅ๎€ป,๐‘ข(๐‘ข(๐‘ฅ,๐‘ก))+ฮฆ(๐‘ข)(๐‘ฅ,๐‘ก)=๐‘ข(0,๐‘ก)+๐‘ฅ๐‘ข๐‘ฅ(0,๐‘ก)+๐ฟ๐‘ฅโˆ’1๎€บ๐ฟ๐‘ก๎€ป,(๐‘ข(๐‘ฅ,๐‘ก))โˆ’ฮฆ(๐‘ข)(2.20) respectively, where ฮฆ(๐‘ข)=๐œ–๐‘ข๐‘š. By ADM [7], one can write the solution in series form as ๐‘ข(๐‘ฅ,๐‘ก)=โˆž๎“๐‘›=0๐‘ข๐‘›(๐‘ฅ,๐‘ก).(2.21)

To find the solutions in ๐‘ก and ๐‘ฅ directions, one solves the recursive relations:

๐‘ข0=๐‘ข(๐‘ฅ,0),๐‘ข๐‘›+1=๐ฟ๐‘กโˆ’1๎€บ๐ฟ๐‘ฅ๎€ท๐‘ข๐‘›๎€ธ+๐ด๐‘›๎€ป๐‘ข,๐‘›โ‰ฅ0,(2.22)0=๐‘ข(0,๐‘ก)+๐‘ฅ๐‘ข๐‘ฅ(0,๐‘ก),๐‘ข๐‘›+1=๐ฟ๐‘ฅโˆ’1๎€บ๐ฟ๐‘ก๎€ท๐‘ข๐‘›๎€ธโˆ’๐ด๐‘›๎€ป,๐‘›โ‰ฅ0,(2.23) respectively, where the Adomian polynomials are [1, 2, 7] ๐ด๐‘›=1๐‘‘๐‘›!๐‘›๐‘‘๐œ†๐‘›๎ƒฌฮฆ๎ƒฉโˆž๎“๐‘›=0๐œ†๐‘›๐‘ข๐‘›๎ƒช๎ƒญ๐œ†=0,๐‘›โ‰ฅ0.(2.24) We obtain the first few Adomian polynomials for ฮฆ(๐‘ข)=๐œ–๐‘ข๐‘š as ๐ด0=๐œ–๐‘ข๐‘š0, ๐ด1=๐‘š๐œ–๐‘ข0๐‘šโˆ’1๐‘ข1, ๐ด2=(๐‘š๐œ–/2)[(๐‘šโˆ’1)๐‘ข0๐‘šโˆ’2๐‘ข21+2๐‘ข2๐‘ข0๐‘šโˆ’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 |๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐œ™๐‘›(๐‘ฅ,๐‘ก)| in Tables 1โ€“4, where ๐‘ข(๐‘ฅ,๐‘ก) is the particular analytical solution and ๐œ™๐‘›(๐‘ฅ,๐‘ก) is the partial sum: ๐œ™๐‘›(๐‘ฅ,๐‘ก)=๐‘›๎“๐‘˜=0๐‘ข๐‘˜(๐‘ฅ,๐‘ก),๐‘›โ‰ฅ0.(2.25) As it is clear from (2.3.4) and (2.3.8), we have ๐‘ข(๐‘ฅ,๐‘ก)=lim๐‘›โ†’โˆž๐œ™๐‘›(๐‘ฅ,๐‘ก).(2.26)


๐‘ก ๐‘– โˆฃ ๐‘ฅ ๐‘– 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5

0 . 1 1 . 9 9 4 2 ๐ธ โˆ’ 2 1 1 . 6 9 7 3 ๐ธ โˆ’ 2 1 1 . 2 3 2 2 ๐ธ โˆ’ 2 1 6 . 4 7 9 0 ๐ธ โˆ’ 2 2 0 . 0 0 0 0 ๐ธ + 0 0
0 . 2 4 . 3 7 6 5 ๐ธ โˆ’ 2 0 3 . 7 2 2 8 ๐ธ โˆ’ 2 0 2 . 7 0 4 8 ๐ธ โˆ’ 2 0 1 . 4 2 2 0 ๐ธ โˆ’ 2 0 0 . 0 0 0 0 ๐ธ + 0 0
0 . 3 8 . 5 9 5 5 ๐ธ โˆ’ 2 0 7 . 3 1 1 6 ๐ธ โˆ’ 2 0 5 . 3 1 2 3 ๐ธ โˆ’ 2 0 2 . 7 9 2 8 ๐ธ โˆ’ 2 0 0 . 0 0 0 0 ๐ธ + 0 0
0 . 4 1 . 0 1 4 8 ๐ธ โˆ’ 1 6 8 . 6 3 2 9 ๐ธ โˆ’ 1 7 6 . 2 7 2 2 ๐ธ โˆ’ 1 7 3 . 2 9 7 5 ๐ธ โˆ’ 1 7 0 . 0 0 0 0 ๐ธ + 0 0
0 . 5 7 . 9 8 8 9 ๐ธ โˆ’ 1 4 6 . 7 9 5 8 ๐ธ โˆ’ 1 4 4 . 9 3 7 4 ๐ธ โˆ’ 1 4 2 . 5 9 5 7 ๐ธ โˆ’ 1 4 0 . 0 0 0 0 ๐ธ + 0 0


๐‘ก ๐‘– โˆฃ ๐‘ฅ ๐‘– 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5

0 . 1 4 . 5 9 7 5 ๐ธ โˆ’ 2 1 3 . 9 1 9 8 ๐ธ โˆ’ 2 1 5 . 5 0 4 4 ๐ธ โˆ’ 2 1 4 . 8 4 2 6 ๐ธ โˆ’ 2 1 2 . 9 2 5 3 ๐ธ โˆ’ 2 0
0 . 2 2 . 0 9 5 7 ๐ธ โˆ’ 2 1 4 . 4 9 6 4 ๐ธ โˆ’ 2 1 2 . 5 0 7 6 ๐ธ โˆ’ 2 1 2 . 7 6 2 9 ๐ธ โˆ’ 2 2 3 . 1 2 4 1 ๐ธ โˆ’ 2 2
0 . 3 2 . 7 3 9 4 ๐ธ โˆ’ 2 2 3 . 5 3 4 3 ๐ธ โˆ’ 2 2 6 . 4 0 0 7 ๐ธ โˆ’ 2 2 4 . 6 8 8 3 ๐ธ โˆ’ 2 2 3 . 0 3 0 7 ๐ธ โˆ’ 2 1
0 . 4 2 . 2 3 4 9 ๐ธ โˆ’ 2 2 9 . 1 1 4 1 ๐ธ โˆ’ 2 3 4 . 2 3 8 2 ๐ธ โˆ’ 2 2 1 . 9 2 7 8 ๐ธ โˆ’ 2 2 5 . 7 2 2 3 ๐ธ โˆ’ 2 2
0 . 5 1 . 1 3 8 8 ๐ธ โˆ’ 2 3 9 . 3 8 4 0 ๐ธ โˆ’ 2 3 2 . 0 9 0 5 ๐ธ โˆ’ 2 2 2 . 9 3 5 8 ๐ธ โˆ’ 2 2 6 . 2 5 9 1 ๐ธ โˆ’ 2 2


๐‘ก ๐‘– โˆฃ ๐‘ฅ ๐‘– 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5

0 . 0 1 3 . 0 0 0 0 ๐ธ โˆ’ 2 0 4 . 0 0 0 0 ๐ธ โˆ’ 2 0 3 . 0 0 0 0 ๐ธ โˆ’ 2 0 2 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0
0 . 0 2 2 . 8 0 0 0 ๐ธ โˆ’ 1 9 2 . 8 0 0 0 ๐ธ โˆ’ 1 9 1 . 7 0 0 0 ๐ธ โˆ’ 1 9 1 . 4 0 0 0 ๐ธ โˆ’ 1 9 1 . 3 0 0 0 ๐ธ โˆ’ 1 9
0 . 0 3 8 . 9 3 6 7 ๐ธ โˆ’ 1 6 7 . 3 7 5 3 ๐ธ โˆ’ 1 6 6 . 0 7 1 7 ๐ธ โˆ’ 1 6 4 . 9 8 8 1 ๐ธ โˆ’ 1 6 4 . 0 8 8 8 ๐ธ โˆ’ 1 6
0 . 0 4 2 . 6 8 2 8 ๐ธ โˆ’ 1 3 2 . 2 1 4 8 ๐ธ โˆ’ 1 3 1 . 8 2 4 2 ๐ธ โˆ’ 1 3 1 . 4 9 9 2 ๐ธ โˆ’ 1 3 1 . 2 2 9 5 ๐ธ โˆ’ 1 3
0 . 0 5 2 . 2 2 0 4 ๐ธ โˆ’ 1 1 1 . 8 3 3 7 ๐ธ โˆ’ 1 1 1 . 5 1 0 9 ๐ธ โˆ’ 1 1 1 . 2 4 2 2 ๐ธ โˆ’ 1 1 1 . 0 1 9 1 ๐ธ โˆ’ 1 1


๐‘ก ๐‘– โˆฃ ๐‘ฅ ๐‘– 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5

0 . 0 1 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0
0 . 0 2 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 0 . 0 0 0 0 ๐ธ + 0 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 0 . 0 0 0 0 ๐ธ + 0 0
0 . 0 3 0 . 0 0 0 0 ๐ธ + 0 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 2 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0
0 . 0 4 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0 3 . 0 0 0 0 ๐ธ โˆ’ 2 0 2 . 0 0 0 0 ๐ธ โˆ’ 2 0 2 . 0 0 0 0 ๐ธ โˆ’ 2 0
0 . 0 5 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 0 . 0 0 0 0 ๐ธ + 0 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0 0 . 0 0 0 0 ๐ธ + 0 0 1 . 0 0 0 0 ๐ธ โˆ’ 2 0

2.4. Analysis of the Method for the Porous Media Equation

Equation (1.10) can be written in an operator form [5] ๐ฟ๐‘ก(๐‘ข)=๐ฟ๐‘ฅ๎€ท๐‘ข๐‘š๐ฟ๐‘ฅ๐‘ข๎€ธ,(2.27) with the initial and boundary conditions, where the notations ๐ฟ๐‘ก=๐œ•/๐œ•๐‘ก and ๐ฟ๐‘ฅ=๐œ•/๐œ•๐‘ฅ symbolize the linear differential operators. We assume the integration inverse operators ๐ฟ๐‘กโˆ’1 and ๐ฟ๐‘ฅโˆ’1 exist, and they are defined as ๐ฟ๐‘กโˆ’1=โˆซ๐‘ก0(โ‹…)๐‘‘๐‘ก and ๐ฟ๐‘ฅโˆ’1=โˆซ๐‘ฅ0(โ‹…)๐‘‘๐‘ฅ, respectively. Therefore, one can write the solution in ๐‘ก direction as [7] ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข(๐‘ฅ,0)+๐ฟ๐‘กโˆ’1๎€บ๐ฟ๐‘ฅ๎€ป(ฮฆ(๐‘ข)),(2.28) where ฮฆ(๐‘ข)=๐‘ข๐‘š๐‘ข๐‘ฅ. By ADM [7] one can write the solution in series form as ๐‘ข(๐‘ฅ,๐‘ก)=โˆž๎“๐‘›=0๐‘ข๐‘›(๐‘ฅ,๐‘ก).(2.29)

To find the solutions in ๐‘ก direction, one solves the recursive relations: ๐‘ข0=๐‘ข(๐‘ฅ,0),๐‘ข๐‘›+1=๐ฟ๐‘กโˆ’1๎€บ๐ฟ๐‘ฅ๎€ท๐ด๐‘›๎€ธ๎€ป,๐‘›โ‰ฅ0,(2.30) respectively, where the Adomian polynomials are [1, 2, 7] ๐ด๐‘›=1๐‘‘๐‘›!๐‘›๐‘‘๐œ†๐‘›๎ƒฌฮฆ๎ƒฉโˆž๎“๐‘›=0๐œ†๐‘›๐‘ข๐‘›๎ƒช๎ƒญ๐œ†=0,๐‘›โ‰ฅ0.(2.31) We obtain the first few Adomian polynomials for ฮฆ(๐‘ข)=๐‘ข๐‘š๐‘ข๐‘ฅ as ๐ด0=๐‘ข๐‘š0(๐‘ข0)๐‘ฅ,๐ด1=๐‘š๐‘ข0๐‘šโˆ’1๐‘ข1(๐‘ข0)๐‘ฅ+๐‘ข๐‘š0(๐‘ข1)๐‘ฅ,๐ด2=๐‘š๐‘ข0๐‘šโˆ’1๐‘ข2(๐‘ข0)๐‘ฅ+๐‘š๐‘ข0๐‘šโˆ’1๐‘ข1(๐‘ข1)๐‘ฅ+๐‘ข๐‘š0๎€ท๐‘ข2๎€ธ๐‘ฅ+๐‘š2(๐‘šโˆ’1)๐‘ข0๐‘šโˆ’2๐‘ข21(๐‘ข0)๐‘ฅ,๐ด3=๐‘š๐‘ข0๐‘šโˆ’1๐‘ข3(๐‘ข0)๐‘ฅ+๐‘š(๐‘šโˆ’1)๐‘ข0๐‘šโˆ’2๐‘ข1๐‘ข2๎€ท๐‘ข0๎€ธ๐‘ฅ+๐‘š2(๐‘šโˆ’1)๐‘ข0๐‘šโˆ’2๐‘ข31(๐‘ข0)๐‘ฅ+๐‘š๐‘ข0๐‘šโˆ’1๐‘ข2(๐‘ข1)๐‘ฅ+๐‘š๐‘ข0๐‘šโˆ’1๐‘ข1(๐‘ข2)๐‘ฅ+๐‘ข๐‘š0(๐‘ข3)๐‘ฅ,โ‹ฎ(2.32)

In Section 3, we provide some examples and demonstrate the absolute errors |๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐œ™๐‘›(๐‘ฅ,๐‘ก)| in Tables 5-6, where ๐‘ข(๐‘ฅ,๐‘ก) is the particular analytical solution and ๐œ™๐‘›(๐‘ฅ,๐‘ก) is the partial sum: ๐œ™๐‘›(๐‘ฅ,๐‘ก)=๐‘›๎“๐‘˜=0๐‘ข๐‘˜(๐‘ฅ,๐‘ก),๐‘›โ‰ฅ0.(2.33) Equations of the form (1.10) admit traveling-wave solutions ๐‘ข=๐‘ข(๐‘˜๐‘ฅ+๐œ†๐‘ก) where ๐‘˜ and ๐œ† are constants [48].


๐‘ก ๐‘– โˆฃ ๐‘ฅ ๐‘– 0 . 5 0 . 6 0 . 6 5 0 . 7 0 . 8

0 . 1 1 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0
0 . 2 0 . 0 0 0 0 ๐ธ + 0 0 1 . 3 3 2 2 ๐ธ โˆ’ 1 5 1 . 3 3 2 2 ๐ธ โˆ’ 1 5 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0
0 . 3 4 . 0 4 1 3 ๐ธ โˆ’ 1 1 2 . 6 6 4 5 ๐ธ โˆ’ 1 5 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0
0 . 4 1 . 4 2 7 2 ๐ธ โˆ’ 4 7 . 8 4 1 6 ๐ธ โˆ’ 9 1 . 1 4 6 5 ๐ธ โˆ’ 1 0 2 . 3 5 1 0 ๐ธ โˆ’ 1 2 2 . 2 2 0 4 ๐ธ โˆ’ 1 5
0 . 4 5 0 . 1 0 3 0 ๐ธ + 0 0 3 . 7 7 5 4 ๐ธ โˆ’ 6 5 . 1 7 5 2 ๐ธ โˆ’ 8 1 . 0 1 8 0 ๐ธ โˆ’ 9 9 . 1 5 7 1 ๐ธ โˆ’ 1 3


๐‘ก ๐‘– โˆฃ ๐‘ฅ ๐‘– 1 2 . 5 4 . 8 8 1 0 . 8

0 . 0 1 5 . 1 0 7 0 ๐ธ โˆ’ 1 4 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0 0 . 0 0 0 0 ๐ธ + 0 0
0 . 1 5 . 8 8 5 7 ๐ธ โˆ’ 7 4 . 4 0 0 9 ๐ธ โˆ’ 1 0 2 . 7 2 4 6 ๐ธ โˆ’ 1 2 5 . 1 3 4 8 ๐ธ โˆ’ 1 4 5 . 0 2 3 8 ๐ธ โˆ’ 1 5
0 . 2 9 . 0 8 1 2 ๐ธ โˆ’ 5 6 . 0 0 3 9 ๐ธ โˆ’ 8 3 . 5 9 9 9 ๐ธ โˆ’ 1 0 6 . 6 9 7 3 ๐ธ โˆ’ 1 2 6 . 4 8 0 2 ๐ธ โˆ’ 1 3
0 . 3 1 . 9 5 3 9 ๐ธ โˆ’ 3 1 . 0 9 8 2 ๐ธ โˆ’ 6 6 . 3 5 5 6 ๐ธ โˆ’ 9 1 . 1 6 6 3 ๐ธ โˆ’ 1 0 1 . 1 2 2 7 ๐ธ โˆ’ 1 1
0 . 4 5 0 . 0 5 4 8 ๐ธ + 0 0 2 . 0 9 8 4 ๐ธ โˆ’ 5 1 . 1 4 3 1 ๐ธ โˆ’ 7 2 . 0 5 1 9 ๐ธ โˆ’ 9 1 . 9 5 9 5 ๐ธ โˆ’ 1 0

3. Applications of ADM

Example 3.1. We first consider (1.3) with initial data ๐‘ข(0)=๐‘0/๐พ. We proceed as in Section 2.1. We take ๐‘“(๐‘ข)=๐‘ข2 and ๐›พ=๐‘0/๐พ. Adomian polynomials can be derived as follows: ๐‘“(๐‘ข)=๐‘ข2=โˆž๎“๐‘›=0๐ด๐‘›=(๐‘ข0+๐‘ข1+๐‘ข2+โ‹ฏ)2=๎€ท๐‘ข20๎€ธ+๎€ท2๐‘ข0๐‘ข1๎€ธ+๎€ท2๐‘ข0๐‘ข2+๐‘ข21๎€ธ+๎€ท2๐‘ข0๐‘ข3+2๐‘ข1๐‘ข2๎€ธ+๎€ท2๐‘ข0๐‘ข4+2๐‘ข1๐‘ข3+๐‘ข22๎€ธ+๎€ท2๐‘ข0๐‘ข5+2๐‘ข1๐‘ข4+2๐‘ข2๐‘ข3๎€ธ+โ‹ฏ.(3.1) Therefore, we get the following Adomian polynomials [14]: ๐ด0=๐‘ข20,๐ด1=2๐‘ข0๐‘ข1,๐ด2=2๐‘ข0๐‘ข2+๐‘ข21,๐ด3=2๐‘ข0๐‘ข3+2๐‘ข1๐‘ข2,๐ด4=2๐‘ข0๐‘ข4+2๐‘ข1๐‘ข3+๐‘ข22,๐ด5=2๐‘ข0๐‘ข5+2๐‘ข1๐‘ข4+2๐‘ข2๐‘ข3,โ‹ฎ(3.2) For numerical purposes we take ๐‘0=2 and ๐พ=1. Therefore, ๐‘ข0=๐‘ข(0)=๐‘0๐‘ข/๐พ=2,1=๐ฟโˆ’1๐‘ข0โˆ’๐ฟโˆ’1๐ด0๐‘ข=โˆ’2๐œ,2=๐ฟโˆ’1๐‘ข1โˆ’๐ฟโˆ’1๐ด1=3๐œ2,๐‘ข3=๐ฟโˆ’1๐‘ข2โˆ’๐ฟโˆ’1๐ด2=โˆ’13/3๐œ3,๐‘ข4=๐ฟโˆ’1๐‘ข3โˆ’๐ฟโˆ’1๐ด3=25/4๐œ4,๐‘ข5=๐ฟโˆ’1๐‘ข4โˆ’๐ฟโˆ’1๐ด4=โˆ’541/60๐œ5,โ‹ฎ(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 ๐‘ข(๐œ)=๐‘ข0(๐œ)+๐‘ข1(๐œ)+๐‘ข2(๐œ)+๐‘ข3(๐œ)+๐‘ข4(๐œ)+๐‘ข5(๐œ)+โ‹ฏ=2โˆ’2๐œ+3๐œ2โˆ’13/3๐œ3+25/4๐œ4โˆ’541/60๐œ5+โ‹ฏ,(3.4) which gives the analytical solution obtained in (1.4) in the closed form, with ๐‘0=2,๐พ=1. We let ๐œ™๐‘› be the ๐‘›th partial sums of the series in (2.1.2), that is, ๐œ™๐‘›=๐‘›๎“๐‘˜=0๐‘ข๐‘˜,๐‘›โ‰ฅ0,(3.5) 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 ๐‘ข(0)=๐›ฟ=1.3, ๐‘ฃ(0)=๐›ฝ=0.6. We proceed as in Section 2.2. We take ๐›ผ=1, ๐‘“(๐‘ข,๐‘ฃ)=๐‘”(๐‘ข,๐‘ฃ)=๐‘ข๐‘ฃ. Therefore, from (2.2.4) we obtain ๐ต๐‘›=๐ถ๐‘›, ๐‘›=0,1,2,โ€ฆ, and Adomian polynomials can be derived as follows: =๐‘“(๐‘ข,๐‘ฃ)=๐‘”(๐‘ข,๐‘ฃ)=๐‘ข๐‘ฃโˆž๎“๐‘›=0๐ต๐‘›=๎ƒฉโˆž๎“๐‘›=0๐‘ข๐‘›๎ƒช๎ƒฉโˆž๎“๐‘›=0๐‘ฃ๐‘›๎ƒช=โˆž๎“๐‘›=0๎ƒฉ๐‘›๎“๐‘˜=0๐‘ข๐‘˜๐‘ฃ๐‘›โˆ’๐‘˜๎ƒช.(3.6) Hence, we get the following Adomian polynomials: ๐ต๐‘›=๐ถ๐‘›=๐‘›๎“๐‘˜=0๐‘ข๐‘˜๐‘ฃ๐‘›โˆ’๐‘˜,๐‘›=0,1,2,โ€ฆ.(3.7) From this equality, we have ๐ต0=๐ถ0=๐‘ข0๐‘ฃ0,๐ต1=๐ถ1=๐‘ข0๐‘ฃ1+๐‘ข1๐‘ฃ0,๐ต2=๐ถ2=๐‘ข0๐‘ฃ2+๐‘ข1๐‘ฃ1+๐‘ข2๐‘ฃ0,๐ต3=๐ถ3=๐‘ข0๐‘ฃ3+๐‘ข1๐‘ฃ2+๐‘ข2๐‘ฃ1+๐‘ข3๐‘ฃ0,โ‹ฎ(3.8) Let us now compute the ๐‘ข๐‘˜ and ๐‘ฃ๐‘˜ from (2.2.8): ๐‘ข0๐‘ฃ=๐‘ข(0)=๐›ฟ=1.3,0๐‘ข=๐‘ฃ(0)=๐›ฝ=0.6,1=๐ฟโˆ’1๐‘ข0โˆ’๐ฟโˆ’1๐ต0๐‘ฃ=0.52๐œ,1=๐ฟโˆ’1๐ถ0โˆ’๐ฟโˆ’1๐‘ฃ0๐‘ข=0.18๐œ,2=๐ฟโˆ’1๐‘ข1โˆ’๐ฟโˆ’1๐ต1=โˆ’0.013๐œ2,๐‘ฃ2=๐ฟโˆ’1๐ถ1โˆ’๐ฟโˆ’1๐‘ฃ1=0.1830๐œ2,๐‘ข3=๐ฟโˆ’1๐‘ข2โˆ’๐ฟโˆ’1๐ต2=โˆ’0.1122๐œ3,๐‘ฃ3=๐ฟโˆ’1๐ถ2โˆ’๐ฟโˆ’1๐‘ฃ2=0.0469๐œ3,๐‘ข4=๐ฟโˆ’1๐‘ข3โˆ’๐ฟโˆ’1๐ต3=โˆ’0.0497๐œ4,๐‘ฃ4=๐ฟโˆ’1๐ถ3โˆ’๐ฟโˆ’1๐‘ฃ3=0.0099๐œ4,โ‹ฎ(3.9) and so on, in this manner the rest of the terms of the decomposition series have been calculated using ๐‘€๐‘Ž๐‘กโ„Ž๐‘๐‘Ž๐‘‘7. 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 ๐‘ข(0)=๐›ฟ=1.3, ๐‘ฃ(0)=๐›ฝ=0.6: ๐‘ข(๐œ)=๐‘ข0(๐œ)+๐‘ข1(๐œ)+๐‘ข2(๐œ)+๐‘ข3(๐œ)+๐‘ข4(๐œ)+โ‹ฏ=1.3+0.52๐œโˆ’0.013๐œ2โˆ’0.1122๐œ3โˆ’0.0497๐œ4โˆ’โ‹ฏ,(3.10)๐‘ฃ(๐œ)=๐‘ฃ0(๐œ)+๐‘ฃ1(๐œ)+๐‘ฃ2(๐œ)+๐‘ฃ3(๐œ)+๐‘ฃ4(๐œ)+โ‹ฏ=0.6+0.18๐œ+0.1830๐œ2+0.0469๐œ3+0.0099๐œ4+โ‹ฏ.(3.11)

Example 3.3. If we take ๐œ–=1 and ๐‘š=1 in the (1.8), we obtain the linear heat equation, namely, ๐‘ข๐‘ก=๐‘ข๐‘ฅ๐‘ฅ+๐‘ข.(3.12) We impose the initial condition ๐‘ข(๐‘ฅ,0)=cos(๐œ‹๐‘ฅ),(3.13) and boundary conditions ๐‘ข(0,๐‘ก)=๐‘’(1โˆ’๐œ‹2)๐‘ก,๐‘ข๐‘ฅ(0,๐‘ก)=0.(3.14)
To obtain the solution in ๐‘ก direction, we use the recursive relation (2.3.5) by simply taking ๐‘ข0=cos(๐œ‹๐‘ฅ). In this case the Adomian Polynomials are ๐ด0=๐‘ข0,๐ด1=๐‘ข1,๐ด2=๐‘ข2, and so on. Therefore, we have ๐‘ข1=๎€ท1โˆ’๐œ‹2๎€ธ๐‘กcos(๐œ‹๐‘ฅ),๐‘ข2=1๎€ท2!1โˆ’๐œ‹2๎€ธ2๐‘ก2cos(๐œ‹๐‘ฅ),๐‘ข3=13!(1โˆ’๐œ‹2)3๐‘ก3cos(๐œ‹๐‘ฅ),(3.15) and so on, in this manner the rest of the components of the series (2.3.4) have been calculated using ๐‘€๐‘Ž๐‘กโ„Ž๐‘๐‘Ž๐‘‘7. Putting these individual terms in (2.3.4) one gets the analytical solution: ๎€ท๐‘ข(๐‘ฅ,๐‘ก)=cos(๐œ‹๐‘ฅ)+1โˆ’๐œ‹2๎€ธ1๐‘กcos(๐œ‹๐‘ฅ)+๎€ท2!1โˆ’๐œ‹2๎€ธ๐‘ก21cos(๐œ‹๐‘ฅ)+3!(1โˆ’๐œ‹2)3๐‘ก3cos(๐œ‹๐‘ฅ)+โ‹ฏ=๐‘’(1โˆ’๐œ‹2)๐‘กcos(๐œ‹๐‘ฅ),(3.16) which can be verified through substitution.
Similarly, to obtain the solution in ๐‘ฅ direction, we use the recursive relation (2.3.6) by taking ๐‘ข0=๐‘’(1โˆ’๐œ‹2)๐‘ก, where the ๐ด๐‘› are the same as above. We therefore have ๐‘ข1=โˆ’(๐œ‹๐‘ฅ)2๐‘’2!(1โˆ’๐œ‹2)๐‘ก,๐‘ข2=(๐œ‹๐‘ฅ)4๐‘’4!(1โˆ’๐œ‹2)๐‘ก,๐‘ข3=โˆ’(๐œ‹๐‘ฅ)6๐‘’6!(1โˆ’๐œ‹2)๐‘ก,(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: ๐‘ข(๐‘ฅ,๐‘ก)=๐‘’(1โˆ’๐œ‹2)๐‘กโˆ’(๐œ‹๐‘ฅ)2๐‘’2!(1โˆ’๐œ‹2)๐‘ก+(๐œ‹๐‘ฅ)4๐‘’4!(1โˆ’๐œ‹2)๐‘กโˆ’(๐œ‹๐‘ฅ)6๐‘’6!(1โˆ’๐œ‹2)๐‘ก+โ‹ฏ=๐‘’(1โˆ’๐œ‹2)๐‘กcos(๐œ‹๐‘ฅ).(3.18)

Example 3.4. In the second example, we consider the nonlinear heat equation (1.8) with ๐œ–=โˆ’2 and ๐‘š=3, that is, ๐‘ข๐‘ก=๐‘ข๐‘ฅ๐‘ฅโˆ’2๐‘ข3.(3.19) In [46] the authors solve (3.19) using antireduction method, and give the solution by means of ansatz (๐œ‘๐‘–,๐‘–=1,2) as follows: ๎€ท๐œ‘๐‘ข(๐‘ฅ,๐‘ก)=1(๐‘ก)+2๐œ‘2๎€ธ(๐‘ก)๐‘ฅ(1+๐œ‘1(๐‘ก)๐‘ฅ+๐œ‘2(๐‘ก)๐‘ฅ2)โˆ’1,(3.20) where ๐œ‘1 and ๐œ‘2 satisfy the ordinary differential equations: ๐œ‘๎…ž1=โˆ’6๐œ‘1๐œ‘2,๐œ‘๎…ž2=โˆ’6๐œ‘22.(3.21) We impose ๐œ‘1(0)=๐œ‘2(0)=1, and solve the above ordinary differential equations, and obtain ๐œ‘1(๐‘ก)=๐œ‘21(๐‘ก)=6๐‘ก+1.(3.22) Therefore, we have the partial analytical solution of (3.19) as ๐‘ข(๐‘ฅ,๐‘ก)=1+2๐‘ฅ๐‘ฅ2+๐‘ฅ+6๐‘ก+1.(3.23) We now solve (3.19) using ADM with the initial condition: ๐‘ข(๐‘ฅ,0)=1+2๐‘ฅ๐‘ฅ2+๐‘ฅ+1,(3.24) and the boundary conditions: 1๐‘ข(0,๐‘ก)=6๐‘ก+1,๐‘ข๐‘ฅ(0,๐‘ก)=12๐‘ก+1(6๐‘ก+1)2.(3.25) For the solution of this equation in the ๐‘ก 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 ๐ด0=โˆ’2๐‘ข30,๐ด1=โˆ’6๐‘ข20๐‘ข1,๐ด2=โˆ’6(๐‘ข0๐‘ข21+๐‘ข20๐‘ข2), and so on. Therefore, we obtain ๐‘ข0=1+2๐‘ฅ๐‘ฅ2,๐‘ข+๐‘ฅ+11=๐ฟ๐‘กโˆ’1๎€ท๐ฟ๐‘ฅ๎€ท๐‘ข0๎€ธ๎€ธโˆ’2๐ฟ๐‘กโˆ’1๎€ท๐‘ข30๎€ธ=โˆ’6(1+2๐‘ฅ)(๐‘ฅ2+๐‘ฅ+1)2๐‘ข๐‘ก,2=๐ฟ๐‘กโˆ’1๎€ท๐ฟ๐‘ฅ๎€ท๐‘ข1๎€ธ๎€ธโˆ’6๐ฟ๐‘กโˆ’1๎€ท๐‘ข20๐‘ข1๎€ธ=36(1+2๐‘ฅ)(๐‘ฅ2+๐‘ฅ+1)3๐‘ก2,๐‘ข3=๐ฟ๐‘กโˆ’1๎€ท๐ฟ๐‘ฅ๎€ท๐‘ข2๎€ธ๎€ธโˆ’6๐ฟ๐‘กโˆ’1๎€ท๐‘ข20๐‘ข2+๐‘ข21๐‘ข0๎€ธ=โˆ’216(1+2๐‘ฅ)(๐‘ฅ2+๐‘ฅ+1)4๐‘ก3,(3.26) and so on, in this manner the rest of the terms of the decomposition series have been calculated using ๐‘€๐‘Ž๐‘กโ„Ž๐‘๐‘Ž๐‘‘7. Substituting (3.26) into the decomposition series (2.3.8), we obtain ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข0(๐‘ฅ,๐‘ก)+๐‘ข1(๐‘ฅ,๐‘ก)+๐‘ข2(๐‘ฅ,๐‘ก)+๐‘ข3=(๐‘ฅ,๐‘ก)โ‹ฏ1+2๐‘ฅ๐‘ฅ2โˆ’+๐‘ฅ+16(1+2๐‘ฅ)(๐‘ฅ2+๐‘ฅ+1)2๐‘ก+36(1+2๐‘ฅ)(๐‘ฅ2+๐‘ฅ+1)3๐‘ก2โˆ’216(1+2๐‘ฅ)(๐‘ฅ2+๐‘ฅ+1)4๐‘ก3โ‹ฏ,(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 ๐‘ฅ direction, we use the recursive relation given by (2.3.6) to determine the individual terms of the series (2.3.8): ๐‘ข0=1๎‚ต6๐‘ก+1+๐‘ฅ12๐‘ก+1(6๐‘ก+1)2๎‚ถ,๐‘ข1=๐ฟ๐‘ฅโˆ’1๎€ท๐ฟ๐‘ก๎€ท๐‘ข0๎€ธ๎€ธ+2๐ฟ๐‘ฅโˆ’1๎€ท๐‘ข30๎€ธ=๐‘ฅ2๎‚ตโˆ’18๐‘กโˆ’2(6๐‘ก+1)3๎‚ถ+๐‘ฅ3๎‚ตโˆ’72๐‘ก2+1(6๐‘ก+1)4๎‚ถ+๐‘ฅ4๎‚ต180๐‘ก2+30๐‘ก+1(6๐‘ก+1)5๎‚ถ+๐‘ฅ5๎‚ต432๐‘ก3โˆ’108๐‘ก2โˆ’36๐‘กโˆ’2(6๐‘ก+1)6๎‚ถ,(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 ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข0(๐‘ฅ,๐‘ก)+๐‘ข1=1(๐‘ฅ,๐‘ก)+โ‹ฏ๎‚ต6๐‘ก+1+๐‘ฅ12๐‘ก+1(6๐‘ก+1)2๎‚ถ+๐‘ฅ2๎‚ตโˆ’18๐‘กโˆ’2(6๐‘ก+1)3๎‚ถ+๐‘ฅ3๎‚ตโˆ’72๐‘ก2+1(6๐‘ก+1)4๎‚ถ+๐‘ฅ4๎‚ต180๐‘ก2+30๐‘ก+1(6๐‘ก+1)5๎‚ถ+๐‘ฅ5๎‚ต432๐‘ก3โˆ’108๐‘ก2โˆ’36๐‘กโˆ’2(6๐‘ก+1)6๎‚ถโ‹ฏ,(3.29) which again gives the analytical solution given by (3.23) in the closed form.

Example 3.5. Let us take ๐‘š=1 in (1.10). We obtain ๐œ•๐‘ข=๐œ•๐œ•๐‘ก๎‚€๐‘ข๐œ•๐‘ฅ๐œ•๐‘ข๎‚๐œ•๐‘ฅ.(3.30) We impose the initial condition ๐‘ข(๐‘ฅ,0)=๐‘ฅ.(3.31)
To obtain the solution, we use the recursive relation (2.4.4) by taking ๐‘ข0=๐‘ฅ. In this case the first Adomian Polynomial is ๐ด0=๐‘ฅ. Therefore, we have ๐‘ข1=๐‘ก and ๐ด1=๐‘ก. Finally, ๐‘ข2=0 which follows that ๐‘ข๐‘›(๐‘ฅ,๐‘ก)=0 for ๐‘›โ‰ฅ2. Putting these individual terms in (2.4.3), one gets the analytical solution ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ฅ+๐‘ก,(3.32) which can be verified through substitution.

Example 3.6. When ๐‘š=โˆ’1, (1.10) becomes ๐œ•๐‘ข=๐œ•๐œ•๐‘ก๎‚€1๐œ•๐‘ฅ๐‘ข๐œ•๐‘ข๎‚๐œ•๐‘ฅ.(3.33) In [48] the authors give a particular analytical solution to (3.33) as follows: ๐‘ข(๐‘ฅ,๐‘ก)=(๐‘1๐‘ฅโˆ’๐‘21๐‘ก+๐‘2)โˆ’1,(3.34) where ๐‘1 and ๐‘2 are arbitrary constants. We take ๐‘1=1 and ๐‘2=0 for simplicity. Therefore, with these choices of ๐‘1 and ๐‘2 their solution becomes 1๐‘ข(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก.(3.35) We now solve (3.33) using ADM with the initial condition: 1๐‘ข(๐‘ฅ,0)=๐‘ฅ.(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 ๐‘ข0=1/๐‘ฅ, ๐ด0=โˆ’1/๐‘ฅ, ๐‘ข1=๐‘ก/๐‘ฅ2, ๐ด1=โˆ’๐‘ก/๐‘ฅ2, ๐‘ข2=๐‘ก2/๐‘ฅ3, ๐ด2=โˆ’๐‘ก2/๐‘ฅ3, ๐‘ข3=๐‘ก3/๐‘ฅ4, and so on. In this manner the rest of the terms of the decomposition series have been calculated using ๐‘€๐‘Ž๐‘กโ„Ž๐‘๐‘Ž๐‘‘7. Substituting these individual terms in (2.4.3), we obtain ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข0(๐‘ฅ,๐‘ก)+๐‘ข1(๐‘ฅ,๐‘ก)+๐‘ข2(๐‘ฅ,๐‘ก)+๐‘ข3=1(๐‘ฅ,๐‘ก)+โ‹ฏ๐‘ฅ+๐‘ก๐‘ฅ2+๐‘ก2๐‘ฅ3+๐‘ก3๐‘ฅ4+โ‹ฏ,(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 ๐‘š=โˆ’4/3, (1.10) reads ๐œ•๐‘ข=๐œ•๐œ•๐‘ก๎‚€๐‘ข๐œ•๐‘ฅโˆ’4/3๐œ•๐‘ข๎‚๐œ•๐‘ฅ.(3.38) In [48], a particular analytical solution to (3.38) is given as follows: ๐‘ข(๐‘ฅ,๐‘ก)=(2๐‘1๐‘ฅโˆ’3๐‘21๐‘ก+๐‘2)โˆ’3/4,(3.39) where ๐‘1 and ๐‘2 are arbitrary constants, and we take ๐‘1=1 and ๐‘2=0 for simplicity. Therefore, one has ๐‘ข(๐‘ฅ,๐‘ก)=(2๐‘ฅโˆ’3๐‘ก)โˆ’3/4.(3.40) We now solve (3.38) using ADM with the initial condition: ๐‘ข(๐‘ฅ,0)=(2๐‘ฅ)โˆ’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 ๐‘ข0=(2๐‘ฅ)โˆ’3/4, ๐ด0=โˆ’3ร—2โˆ’7/4๐‘ฅโˆ’3/4, ๐‘ข1=9ร—2โˆ’15/4๐‘ฅโˆ’7/4๐‘ก, ๐ด1=โˆ’27ร—2โˆ’19/4๐‘ฅโˆ’7/4๐‘ก, ๐‘ข2=189ร—2โˆ’31/4๐‘ฅโˆ’11/4๐‘ก2, ๐ด2=โˆ’567ร—2โˆ’35/4๐‘ฅโˆ’11/4๐‘ก2, ๐‘ข3=2079ร—2โˆ’43/4๐‘ฅโˆ’15/4๐‘ก3, and so on. In this manner the rest of the terms of the decomposition series have been calculated using ๐‘€๐‘Ž๐‘กโ„Ž๐‘๐‘Ž๐‘‘7. Substituting these individual terms in (2.4.3), one obtains ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข0(๐‘ฅ,๐‘ก)+๐‘ข1(๐‘ฅ,๐‘ก)+๐‘ข2(๐‘ฅ,๐‘ก)+๐‘ข3(๐‘ฅ,๐‘ก)+โ‹ฏ=(2๐‘ฅ)โˆ’3/4+9ร—2โˆ’15/4๐‘ฅโˆ’7/4๐‘ก+189ร—2โˆ’31/4๐‘ฅโˆ’11/4๐‘ก2+2079ร—2โˆ’43/4๐‘ฅโˆ’15/4๐‘ก3+โ‹ฏ,(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: ๐ด(๐‘ข)โˆ’๐‘“(๐ซ)=0,๐ซโˆˆฮฉ,(4.1) with boundary conditions: ๐ต(๐‘ข,๐œ•๐‘ข/๐œ•๐‘›)=0,๐ซโˆˆฮ“,(4.2) where ๐ด is a general differential operator, ๐ต is a boundary operator, ๐‘“(๐ซ) is a known analytic function, and ฮ“ is the boundary of the domain ฮฉ.

In general, one divides the operator ๐ด into two parts ๐ฟ and ๐‘, where ๐ฟ is linear, while ๐‘ is nonlinear. Therefore, (4.1) is written as follows: ๐ฟ(๐‘ข)+๐‘(๐‘ข)โˆ’๐‘“(๐ซ)=0.(4.3)

By the homotopy technique [19, 51], one constructs a homotopy ๐‘ฃ(๐ซ,๐‘)โˆถฮฉร—[0,1]โ†’๐‘ which satisfies ๐ป๎€บ๐ฟ๎€ท๐‘ข(๐‘ฃ,๐‘)=(1โˆ’๐‘)(๐‘ฃ)โˆ’๐ฟ0[๐ด][]๎€ธ๎€ป+๐‘(๐‘ฃ)โˆ’๐‘“(๐ซ)=0,๐‘โˆˆ0,1,๐ซโˆˆฮฉ,(4.4) or ๐ป๎€ท๐‘ข(๐‘ฃ,๐‘)=๐ฟ(๐‘ฃ)โˆ’๐ฟ0๎€ธ๎€ท๐‘ข+๐‘๐ฟ0๎€ธ[๐‘]+๐‘(๐‘ฃ)โˆ’๐‘“(๐ซ)=0,(4.5) where ๐‘โˆˆ[0,1] is an embedding parameter, ๐‘ข0 is an initial approximation of (4.1), which satisfies the boundary conditions. It is clear that ๐ป๎€ท๐‘ข(๐‘ฃ,0)=๐ฟ(๐‘ฃ)โˆ’๐ฟ0๎€ธ=0,๐ป(๐‘ฃ,1)=๐ด(๐‘ฃ)โˆ’๐‘“(๐ซ)=0,(4.6) the changing process of ๐‘ from zero to unity is just that of ๐‘ฃ(๐ซ,๐‘) from ๐‘ข0(๐ซ) to ๐‘ข(๐ซ).

According to the HPM, we can first use the embedding parameter ๐‘ as a โ€œsmall parameter", and assume that the solution of (4.4) and (4.5) can be written as a power series in ๐‘: ๐‘ฃ=๐‘ฃ0+๐‘๐‘ฃ1+๐‘2๐‘ฃ2+โ‹ฏ.(4.7) Setting ๐‘=1 results in the approximate solution of (4.1): ๐‘ข=lim๐‘โ†’1๐‘ฃ=๐‘ฃ0+๐‘ฃ1+๐‘ฃ2+โ‹ฏ.(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 ๐ด(๐‘ฃ) (the following opinions are suggested by He [19]).

(1) The second derivative of ๐‘(๐‘ฃ) with respect to ๐‘ฃ must be small because the parameter may be relatively large, that is, ๐‘โ†’1.

(2) The norm of ๐ฟโˆ’1๐œ•๐‘/๐œ•๐‘ฃ 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 ๐‘ข(0)=2, as chosen in Example 3.1. We rewrite (1.3) in the form [52] ๐‘‘๐‘ข๐‘‘๐œ=๐‘๐‘ข(1โˆ’๐‘ข),๐‘ข(0)=2,(5.1) where ๐‘โˆˆ[0,1] is an embedding parameter. As in He's HPM, it is clear that when ๐‘=0, (5.1) becomes a linear equation; when ๐‘=1, it becomes the original nonlinear one. We consider the imbedding parameter ๐‘ as a โ€œsmall 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 ๐‘, we obtain the following differential equations: ๐‘0โˆถ๎€ฝ๐‘ฃ๎…ž0=0,๐‘ฃ0๐‘(0)=2,1โˆถ๎€ฝ๐‘ฃ๎…ž1=๐‘ฃ0โˆ’๐‘ฃ20,๐‘ฃ1๐‘(0)=0,2โˆถ๎€ฝ๐‘ฃ๎…ž2=๐‘ฃ1โˆ’2๐‘ฃ0๐‘ฃ1,๐‘ฃ2๐‘(0)=0,3โˆถ๎€ฝ๐‘ฃ๎…ž3=๐‘ฃ2โˆ’๐‘ฃ21โˆ’2๐‘ฃ0๐‘ฃ2,๐‘ฃ3(๐‘0)=0,4โˆถ๎€ฝ๐‘ฃ๎…ž4=๐‘ฃ3โˆ’2๐‘ฃ0๐‘ฃ3โˆ’2๐‘ฃ1๐‘ฃ2,๐‘ฃ4๐‘(0)=0,5โˆถ๎€ฝ๐‘ฃ๎…ž5=๐‘ฃ4โˆ’๐‘ฃ22โˆ’2๐‘ฃ1๐‘ฃ3โˆ’2๐‘ฃ0๐‘ฃ4,๐‘ฃ5โ‹ฎ(0)=0,(5.2) where โ€œprimes" denote differentiation with respect to ๐œ. Thus, solving the equations above yields ๐‘ฃ0๐‘ฃ=2,1๐‘ฃ=โˆ’2๐œ,2=3๐œ2,๐‘ฃ3=โˆ’13/3๐œ3,๐‘ฃ4=25/4๐œ4,๐‘ฃ5=โˆ’541/60๐œ5,โ‹ฎ(5.3)
Substituting these in (4.7) gives ๐‘ฃ=2โˆ’2๐‘๐œ+3๐‘2๐œ2โˆ’13/3๐‘3๐œ3+25/4๐‘4๐œ4โˆ’541/60๐‘5๐œ5+โ‹ฏ.(5.4) Hence, by (4.8) one has ๐‘ข=2โˆ’2๐œ+3๐œ2โˆ’13/3๐œ3+25/4๐œ4โˆ’541/60๐œ5+โ‹ฏ,(5.5) which is exactly the same solution obtained in (3.4). Also, the solution in (5.5) is equal to 2๐‘ข=2โˆ’๐‘’โˆ’๐œ,(5.6) in the closed form which is exactly the same as in (1.4) with ๐พ=1,๐‘0=2 (see Example 3.1).

Example 5.2. We now solve (1.7) using HPM with ๐›ผ=1,๐‘ข(0)=1.3,๐‘ฃ(0)=0.6 as taken in Example 3.2. We rewrite (1.7) in the form [52] ๐‘‘๐‘ข๐‘‘๐œ=๐‘๐‘ข(1โˆ’๐‘ฃ),๐‘‘๐‘ฃ๐‘‘๐œ=๐‘๐‘ฃ(๐‘ขโˆ’1),๐‘ข(0)=1.3,๐‘ฃ(0)=0.6,(5.7) where ๐‘โˆˆ[0,1] is an embedding parameter. As in He's HPM, it is clear that when ๐‘=0, (5.7) becomes a linear system; when ๐‘=1, it becomes the original nonlinear one. We consider the imbedding parameter ๐‘ as a โ€œsmall parameter". We assume the solutions of (5.7), (๐‘ข,๐‘ฃ) are expressed as power series: ๐‘ข=๐‘ข0+๐‘๐‘ข1+๐‘2๐‘ข2+โ‹ฏ,(5.8)๐‘ฃ=๐‘ฃ0+๐‘๐‘ฃ1+๐‘2๐‘ฃ2+โ‹ฏ,(5.9) respectively. Substituting (5.8) and (5.9) into the system (5.7), and equating coefficients of like ๐‘, we obtain the following systems of differential equations: ๐‘0โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ข๎…ž0๐‘ฃ=0๎…ž0๐‘ข=00(0)=1.3,๐‘ฃ0๐‘(0)=0.6,1โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ข๎…ž1=๐‘ข0๎€ท1โˆ’๐‘ฃ0๎€ธ๐‘ฃ๎…ž1=๐‘ฃ0๎€ท๐‘ข0๎€ธ๐‘ขโˆ’11(0)=0,๐‘ฃ1๐‘(0)=0,2โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ข๎…ž2=๐‘ข1โˆ’๎€ท๐‘ข0๐‘ฃ1+๐‘ข1๐‘ฃ0๎€ธ๐‘ฃ๎…ž2=๐‘ข0๐‘ฃ1+๐‘ข1๐‘ฃ0โˆ’๐‘ฃ1๐‘ข2(0)=0,๐‘ฃ2(๐‘0)=0,3โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ข๎…ž3=๐‘ข2โˆ’๎€ท๐‘ข0๐‘ฃ2+๐‘ข1๐‘ฃ1+๐‘ข2๐‘ฃ0๎€ธ๐‘ฃ๎…ž3=๐‘ข0๐‘ฃ2+๐‘ข1๐‘ฃ1+๐‘ข2๐‘ฃ0โˆ’๐‘ฃ2๐‘ข3(0)=0,๐‘ฃ3๐‘(0)=0,4โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ข๎…ž4=๐‘ข3โˆ’๎€ท๐‘ข0๐‘ฃ3+๐‘ข1๐‘ฃ2+๐‘ข2๐‘ฃ1+๐‘ข3๐‘ฃ0๎€ธ๐‘ฃ๎…ž4=๐‘ข0๐‘ฃ3+๐‘ข1๐‘ฃ2+๐‘ข2๐‘ฃ1+๐‘ข3๐‘ฃ0โˆ’๐‘ฃ3๐‘ข4(0)=0,๐‘ฃ4โ‹ฎ(0)=0,(5.10) where โ€œprimes" denote differentiation with respect to ๐œ. Thus, solving the above systems of equations yields ๐‘ข0=1.3,๐‘ฃ0๐‘ข=0.6,1=0.52๐œ,๐‘ฃ1๐‘ข=0.18๐œ,2=โˆ’0.013๐œ2,๐‘ฃ2=0.183๐œ2,๐‘ข3=โˆ’0.1122๐œ3,๐‘ฃ3=0.0469๐œ3,๐‘ข4=โˆ’0.0497๐œ4,๐‘ฃ4=0.0099๐œ4,โ‹ฎ(5.11)
Substituting these ๐‘ข๐‘›,๐‘ฃ๐‘›, ๐‘›โ‰ฅ0 into (5.8), and (5.9), respectively, we have ๐‘ข=1.3+0.52๐‘๐œโˆ’0.013๐‘2๐œ2โˆ’0.1122๐‘3๐œ3โˆ’0.0497๐‘4๐œ4โˆ’โ‹ฏ,๐‘ฃ=0.6+0.18๐‘๐œ+0.183๐‘2๐œ2+0.0469๐‘3๐œ3+0.0099๐‘4๐œ4+โ‹ฏ.(5.12) Letting ๐‘โ†’1 one obtains ๐‘ข=1.3+0.52๐œโˆ’0.013๐œ2โˆ’0.1122๐œ3โˆ’0.0497๐œ4โˆ’โ‹ฏ,๐‘ฃ=0.6+0.18๐œ+0.183๐œ2+0.0469๐œ3+0.0099๐œ4+โ‹ฏ,(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 ๐‘ข๐‘ก=(๐‘ข๐‘š๐‘ข๐‘ฅ)๐‘ฅ,(6.1) using modified HPM with initial condition: ๐‘ข(๐‘ฅ,0)=๐‘“(๐‘ฅ).(6.2) Here ๐‘ข=๐‘ข(๐‘ฅ,๐‘ก). 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 3.5โ€“3.7. In their work they construct the following homotopy with ๐ฟ(๐‘ข)=๐‘ข๐‘ก and ๐‘(๐‘ข)=โˆ’(๐‘ข๐‘š๐‘ข๐‘ฅ)๐‘ฅ, ๐ฟ๎€ท๐‘ข(๐‘ฃ)โˆ’๐ฟ0๎€ธ๎€ท๐‘ข+๐‘๐ฟ0๎€ธ+๐‘๐‘(๐‘ฃ)=0.(6.3) To deal with the nonlinear term, they employ He polynomials considered in [33, 34] which is given by ๎€ท๐‘ฃ๐‘(๐‘ข)=๐‘0๎€ธ๎€ท๐‘ฃ+๐‘0,๐‘ฃ1๎€ธ๎€ท๐‘ฃ๐‘+๐‘0,๐‘ฃ1,๐‘ฃ2๎€ธ๐‘2๎€ท๐‘ฃ+โ‹ฏ+๐‘0,๐‘ฃ1,โ€ฆ,๐‘ฃ๐‘›๎€ธ๐‘๐‘›+โ‹ฏ,(6.4) where ๐‘๎€ท๐‘ฃ0,๐‘ฃ1,โ€ฆ,๐‘ฃ๐‘›๎€ธ=1๐œ•๐‘›!๐‘›๐œ•๐‘๐‘›๎ƒฌ๐‘๎ƒฉ๐‘›๎“๐‘˜=0๐‘๐‘˜๐‘ฃ๐‘˜๎ƒช๎ƒญ๐‘=0,๐‘›=1,2,โ€ฆ.(6.5) Substituting (6.4) into (6.3), and equating coefficients of like ๐‘, one obtains ๐‘0๎€ท๐‘ฃโˆถ๐ฟ0๎€ธ๎€ท๐‘ขโˆ’๐ฟ0๎€ธ๐‘=0,1๎€ท๐‘ฃโˆถ๐ฟ1๎€ธ๎€ท๐‘ข+๐ฟ0๎€ธ๎€ท๐‘ฃ+๐‘0๎€ธ๐‘=0,2๎€ท๐‘ฃโˆถ๐ฟ2๎€ธ๎€ท๐‘ฃ+๐‘0,๐‘ฃ1๎€ธ๐‘=0,3๎€ท๐‘ฃโˆถ๐ฟ3๎€ธ๎€ท๐‘ฃ+๐‘0,๐‘ฃ1,๐‘ฃ2๎€ธโ‹ฎ๐‘=0,๐‘›+1๎€ท๐‘ฃโˆถ๐ฟ๐‘›+1๎€ธ๎€ท๐‘ฃ+๐‘0,๐‘ฃ1,โ€ฆ,๐‘ฃ๐‘›๎€ธ=0,(6.6) and so on, which forms the basis of a complete determination of the components