Removal of Noise Oscillation Term Appearing in the Nonlinear Equation Solution

Abstract

This paper suggests a novel modified Laplace method for removal of noise oscillation term appearing in the nonlinear equation solutions. The modified method overcomes the noise oscillation during the iteration procedure by suitable choice of an initial solution. Several examples are tested, and the obtained results suggest that this newly developed technique could lead to a promising tool and powerful improvement for many applications in differential and integral equations.

1. Introduction

Nonlinear science emerged in its present form following a series of decisive analytic, numerical, and experimental development took place in close interaction in the last three decades. And the application of many nonlinear science partial differential equations has attracted a great deal of attention in recent years. For example, partial differential equations are increasing used to model many problems in biology, chemistry, economic, engineering, physics, and other areas of applications. Many analytical and numerical techniques have been developed by various scientists to cope with the nonlinearity of such problems. In reality, the Laplace transform is one of only few methods that can be useful to linear systems with periodic or discontinuous driving inputs. In spite of its great usefulness in solving linear problems, however, the Laplace transform is totally incapable of handling nonlinear equations because of the difficulties that are caused by the nonlinear terms. Various ways have been proposed recently to deal with these nonlinearities such as the Adomian decomposition method [1–3], the Laplace decomposition method [4, 5], the homotopy perturbation method [6–14], variational approach [15, 16], Hamiltonian approach [17, 18], Boubaker polynomials expansion scheme [19, 20], and recently the homotopy perturbation transform method was proposed by Khan and Wu [21].

In this paper, we proposed a new method to solve any kind of nonlinear equations. This method is called modified homotopy perturbation transform method (MHPTM). It is important to note that the modified technique works effectively independent of other phenomena in some cases, or it may be combined with the powerful phenomena of the self-canceling “noise terms’’. We aim to extend the works of [21] and make further progress beyond the achievements made so far in this regard. The proposed modification will accelerate the rapid convergence of the series solution if compared with the standard homotopy perturbation transform method (HPTM), and therefore provides a major progress. The modified technique has been shown to be computationally efficient in several examples that are important to researchers in applied fields. In addition, the modified technique may give the exact solution for nonlinear equations without any need of the so-called Adomian polynomials. Although the modified technique needs only a slight variation from the standard homotopy perturbation transform method [21], the results are promising in that it minimizes the size of calculations needed. While this slight variation is rather simple, it does demonstrate the reliability and the power of the proposed modification. The modified homotopy perturbation transform method is much easier to implement as compared with the Adomian decomposition method and the Laplace decomposition method, where huge complexities are involved. Several examples are given to reconfirm the efficiency and accuracy of the proposed modified homotopy perturbation transform method.

2. Description of the Method

To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous partial differential equation with initial condition of the following form: ()()()(),𝑢()(𝑥)𝐷𝑢𝑥,𝑡+𝑅𝑢𝑥,𝑡+𝑁𝑢𝑥,𝑡=𝑔𝑥,𝑡𝑥,0=ℎ,𝑢_𝑡(𝑥,0)=𝑓(𝑥), (2.1) where 𝑓(𝑥),ℎ(𝑥)∈𝐶(𝑅),𝐷 is the second order linear differential operator 𝐷=𝜕²/𝜕𝑡², 𝑅 the linear differential operator of less order than 𝐷, 𝑁 representing the general nonlinear differential operator, and 𝑔(𝑥,𝑡) as the source term.

According to the homotopy perturbation transform method [21], we apply the Laplace transform (denoted throughout this paper by 𝐿) on both sides of (2.1) as follows: 𝐿[𝐷𝑢(𝑥,𝑡)]+𝐿[𝑅𝑢(𝑥,𝑡)]+𝐿[𝑁𝑢(𝑥,𝑡)]=𝐿𝑔(𝑥,𝑡). (2.2) Using the differentiation property of Laplace transform, we have𝑠²𝐿[𝑢(𝑥,𝑡)]−𝑠𝑢(𝑥,0)−𝑢_𝑡()[([(𝑔(,𝑠𝑥,0+𝐿𝑅𝑢𝑥,𝑡)]+𝐿𝑁𝑢𝑥,𝑡)]=𝐿𝑥,𝑡)₂𝐿[𝑢((𝑥)(𝑥)[([(𝑔(,𝐿[𝑢(=ℎ(𝑥)𝑥,𝑡)]−𝑠ℎ−𝑓+𝐿𝑅𝑢𝑥,𝑡)]+𝐿𝑁𝑢𝑥,𝑡)]=𝐿𝑥,𝑡)𝑥,𝑡)]𝑠+𝑓(𝑥)𝑠²+1𝑠²𝐿𝑔(−1𝑥,𝑡)𝑠²𝐿[𝑅𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)]. (2.3)(2.4) Operating with Laplace inverse on both sides of (2.4) gives 𝑢(𝑥,𝑡)=𝐺(𝑥,𝑡)−𝐿⁻¹1𝑠²𝐿[𝑅𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)], (2.5) where 𝐺(𝑥,𝑡) represents the term arising from the source term and prescribed initial condition. The homotopy perturbation transform method [21] admits a solution in the form 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡). (2.6) The nonlinear term is decomposed as 𝑁𝑢(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑢), (2.7) where 𝐻_𝑛 are He’s polynomials [22, 23] of 𝑢₀,𝑢₁,𝑢₂,𝑢₃,…,𝑢_𝑛 which can be calculated by the following formula: 𝐻_𝑛=1𝑑𝑛!𝑑𝑝^𝑛𝑛𝑁^∞_𝑖=0𝑝^𝑖𝑢_𝑖𝑝=0,𝑛=0,1,2,3,…. (2.8) Using (2.7) and (2.6) in (2.5), we get ^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝐺(𝑥,𝑡)−𝑝𝐿⁻¹1𝑠²𝐿𝑅^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)+^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑢). (2.9) Comparing the coefficient of like powers of 𝑝, the following approximations are obtained: 𝑝⁰∶𝑢₀()(),𝑝𝑥,𝑡=𝐺𝑥,𝑡₁∶𝑢₁(𝑥,𝑡)=−𝐿⁻¹1𝑠²𝐿𝑅𝑢₀(𝑥,𝑡)+𝐻₀(𝑢,𝑝₂)∶𝑢₂(𝑥,𝑡)=−𝐿⁻¹1𝑠²𝐿𝑅𝑢₁(𝑥,𝑡)+𝐻₁(𝑢,𝑝₃)∶𝑢₃(𝑥,𝑡)=−𝐿⁻¹1𝑠²𝐿𝑅𝑢₂(𝑥,𝑡)+𝐻₂(𝑢,⋮) (2.10)(2.11) Clearly, the general recursive relation is given by 𝑝^𝑛∶𝑢_𝑛+1(𝑥,𝑡)=−𝐿⁻¹1𝑠²𝐿𝑅𝑢_𝑛(𝑥,𝑡)+𝐻_𝑛(𝑢),𝑛≥0. (2.12) The initial solution is important, and the choice of (2.10) as the initial solution always leads to noise oscillation/term during the iteration procedure. It is important to note that these terms may appear for inhomogeneous problems, whereas homogeneous problems do not generate noise terms. A necessary condition for the generation of the noise terms for inhomogeneous problems is that the zeroth component 𝑢₀ must contain the exact solution 𝑢 among other terms. The noise terms are defined as the identical terms, with opposite signs, that may appear in various components 𝑢_𝑛,𝑛≥1. It was formally shown that by canceling the noise terms that appear in 𝑢₀ and 𝑢₁ from 𝑢₀, even though 𝑢₁ contains further terms, the remaining noncancelled terms of 𝑢₀ may give the exact solution of the inhomogeneous problem. In order to overcome this shortcoming, we decompose 𝐺(𝑥,𝑡) into two parts as 𝐺(𝑥,𝑡)=𝐺₀(𝑥,𝑡)+𝐺₁(𝑥,𝑡). (2.13) Instead of the iteration procedure, (2.10), (2.11), and (2.12), we suggest the following modification: 𝑝⁰∶𝑢₀(𝑥,𝑡)=𝐺₀(),𝑝𝑥,𝑡₁∶𝑢₁(𝑥,𝑡)=𝐺₁(𝑥,𝑡)−𝐿⁻¹1𝑠²𝐿𝑅𝑢₀(𝑥,𝑡)+𝐻₀(𝑢,𝑝₂)∶𝑢₂(𝑥,𝑡)=−𝐿⁻¹1𝑠²𝐿𝑅𝑢₁(𝑥,𝑡)+𝐻₁(𝑢,𝑝₃)∶𝑢₃(𝑥,𝑡)=−𝐿⁻¹1𝑠²𝐿𝑅𝑢₂(𝑥,𝑡)+𝐻₂(𝑢,⋮) (2.14) The solution through the modified homotopy perturbation transform method highly depends upon the choice of 𝐺₀(𝑥,𝑡) and 𝐺₁(𝑥,𝑡). We will show how to suitably choose 𝐺₀(𝑥,𝑡) and 𝐺₁(𝑥,𝑡) through the use of examples.

3. Nonlinear Klein-Gordan Equations

In order to elucidate the solution procedure of the modified method and how to choose a suitable initial solution, we consider first the nonlinear Klein-Gordon equations.

Example 3.1. Consider the following nonlinear Klein-Gordon equation: 𝑢_𝑡𝑡(𝑥,𝑡)−𝑢_𝑥𝑥(𝑥,𝑡)+𝑢²(𝑥,𝑡)=𝑥²𝑡²,𝑢(𝑥,0)=0,𝑢_𝑡(𝑥,0)=𝑥. (3.1)(3.2) Taking Laplace transform of both sides of (3.1) gives 𝑠²𝑢(𝑥,𝑠)−𝑠𝑢(𝑥,0)−𝑢_𝑡(𝑥,0)=2!𝑥²𝑠³+𝐿𝑢_𝑥𝑥−𝑢². (3.3) The initial conditions (3.2) imply 𝑢()=𝑥𝑥,𝑠𝑠²+2!𝑥²𝑠⁵+1𝑠²𝐿𝑢_𝑥𝑥−𝑢². (3.4) The inverse Laplace transform applied to (3.4) results in 𝑢()𝑥𝑥,𝑡=𝑥𝑡+²𝑡⁴12+𝐿⁻¹1𝑠²𝐿𝑢_𝑥𝑥−𝑢². (3.5) Applying the homotopy perturbation transform method [21], we get ^∞_𝑛=0𝑝^𝑛𝑢_𝑛()𝑥𝑥,𝑡=𝑥𝑡+²𝑡⁴𝐿12+𝑝⁻¹1𝑠²𝐿^∞_𝑛=0𝑝^𝑛𝑢_𝑛𝑥𝑥(𝑥,𝑡)−^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑢), (3.6) where 𝐻_𝑛(𝑢) are He’s polynomials [22, 23] that represent the nonlinear terms. In view of (3.6), recursive relation is given as follows: 𝑝⁰∶𝑢₀()𝑥𝑥,𝑡=𝑥𝑡+²𝑡⁴,𝑝12₁∶𝑢₁(𝑥,𝑡)=𝐿⁻¹1𝑠²𝐿𝑢_0𝑥𝑥−𝐻₀(𝑢)=𝐿⁻¹1𝑠²𝐿𝑢_0𝑥𝑥−𝑢²₀𝑥=−²𝑡⁴−𝑥12⁴𝑡¹⁰−𝑥1080³𝑡⁷+𝑡252⁶𝑝180₂∶𝑢₂(𝑥,𝑡)=𝐿⁻¹1𝑠²𝐿𝑢_1𝑥𝑥−𝐻₁(𝑢)=𝐿⁻¹1𝑠²𝐿𝑢_1𝑥𝑥−2𝑢₀𝑢₁𝑡=−⁶+𝑥180³𝑡⁷−25211𝑥⁴𝑡¹⁰−𝑥45360²𝑡¹²+7128037𝑥⁵𝑡¹³+𝑥7076160⁶𝑡¹⁶.18662400 (3.7) It is important to recall here that the noise terms appear between the two adjacent components, for example, 𝑥²𝑡⁴/12 in 𝑢₀(𝑥,𝑡) and 𝑢₁(𝑥,𝑡). The noise terms between the adjacent components can be cancelled using our modified version. The exact solution is 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑢_𝑛(𝑥,𝑡)=𝑥𝑡. (3.8) According to the modified homotopy perturbation transform method, we first set 𝐺()𝑥𝑥,𝑡=𝑥𝑡+²𝑡⁴.12 (3.9) As suggested before, we split 𝐺(𝑥,𝑡) into two parts as 𝐺₀(𝑥,𝑡)=𝑥𝑡,𝐺₁()=𝑥𝑥,𝑡²𝑡⁴.12 (3.10) According to the iteration algorithm equation (2.14), we obtain 𝑝⁰∶𝑢₀()𝑝𝑥,𝑡=𝑥𝑡,₁∶𝑢₁()=𝑥𝑥,𝑡²𝑡⁴12+𝐿⁻¹1𝑠²𝐿𝑢_0𝑥𝑥−𝐻₀(𝑢=𝑥₂)𝑡⁴12+𝐿⁻¹1𝑠²𝐿𝑢_0𝑥𝑥−𝑢²₀=𝑥²𝑡⁴−𝑥12²𝑡⁴𝑝12=0_𝑖∶𝑢_𝑖(𝑥,𝑡)=0,𝑖=2,3,4,…. (3.11) One iteration leads to the exact solution 𝑢(𝑥,𝑡)=𝑥𝑡. It is obvious from this example that “noise terms” appeared in the standard homotopy perturbation transform method can be completely eliminated in our solution procedure.

Example 3.2. Consider the following nonlinear Klein-Gordon equation: 𝑢_𝑡𝑡(𝑥,𝑡)−𝑢_𝑥𝑥(𝑥,𝑡)+𝑢²(𝑥,𝑡)=2𝑥²−2𝑡²+𝑥⁴𝑡⁴,𝑢(𝑥,0)=0,𝑢_𝑡(𝑥,0)=0. (3.12) By applying the aforesaid method subject to the initial conditions, we have 𝑢(𝑥,𝑠)=𝑥²2!𝑠³−4𝑠⁵+𝑥⁴4!𝑠⁷+1𝑠²𝐿𝑢_𝑥𝑥−𝑢². (3.13) The inverse of Laplace transform implies that 𝑢(𝑥,𝑡)=𝑥²𝑡²−𝑡⁴6+𝑥⁴𝑡⁶30+𝐿⁻¹1𝑠²𝐿𝑢_𝑥𝑥−𝑢². (3.14) In view of the homotopy perturbation transform method [21], we have ^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝑥²𝑡²−𝑡⁴6+𝑥⁴𝑡⁶𝐿30+𝑝⁻¹1𝑠²𝐿^∞_𝑛=0𝑝^𝑛𝑢_𝑛𝑥𝑥(𝑥,𝑡)−^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑢), (3.15) where the nonlinear operator is decomposed as in (2.8) in terms of the He’s polynomials. Matching both sides of (3.15), the components of 𝑢 can be defined as follows: 𝑝⁰∶𝑢₀(𝑥,𝑡)=𝑥²𝑡²,𝑝¹∶𝑢₁()𝑡𝑥,𝑡=−⁴6+𝑥⁴𝑡⁶30+𝐿⁻¹1𝑠²𝐿𝑢_0𝑥𝑥−𝐻₀(𝑢)=0. (3.16) The solution procedure can continue, and we find the remaining components are all zero. We, therefore, obtain a closed-form solution which is 𝑢(𝑥,𝑡)=𝑥²𝑡².

4. Conclusion

This paper suggests an effective modification of the Laplace transform method. A suitable choice of 𝐺₀(𝑥,𝑡) and 𝐺₁(𝑥,𝑡) results in an extremely simple solution procedure. The modified version is also valid for other nonlinear equations, and this paper can be used as a standard paradigm for other applications.

Acknowledgment

The authors wish to express their cordial thanks to the anonymous referees for useful suggestions and comments.