Homotopy Perturbation Method with an Auxiliary Term

and Applied Analysis 3 Accordingly, we can construct a homotopy equation in the form ü ω2u p ( u3 −ω2u ) 0. 2.6 When p 0, we have ü ω2u 0, u 0 A, u′ 0 0, 2.7 which describes the basic solution property of the original nonlinear equation, 2.3 . When p 1, 2.6 becomes the original one. So the solution procedure is to deform from the initial solution, 2.4 , to the real one. Due to one unknown parameter in the initial solution, only one iteration is enough. For detailed solution procedure, refer to 5 . If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form ü 0 · u pu3 0. 2.8 We expand the solution and the coefficient, zero, of the linear term into a series in p: u u0 pu1 pu2 · · · , 2.9 0 ω2 pa1 pa2 · · · , 2.10 where the unknown constant, ai, is determined in the i 1 th iteration. The solution procedure is given in 5 . 3. Homotopy Equation with an Auxiliary Term In this paper, we suggest an alternative approach to construction of homotopy equation, which is ̃ Lu p ( Lu −  ̃ Lu Nu ) αp ( 1 − p)u 0, 3.1 where α is an auxiliary parameter. When α 0, 3.1 turns out to be that of the classical one, expressed in 2.2 . The auxiliary term, αp 1 − p u, vanishes completely when p 0 or p 1; so the auxiliary term will affect neither the initial solution p 0 nor the real solution p 1 . The homotopy perturbation method with an auxiliary term was first considered by Noor 15 . To illustrate the solution procedure, we consider a nonlinear oscillator in the form d2u dt2 bu cu3 0, u 0 A, u′ 0 0, 3.2 where b and c are positive constants. 4 Abstract and Applied Analysis Equation 3.2 admits a periodic solution, and the linearized equation of 3.2 is u′′ ω2u 0, u 0 A, u′ 0 0, 3.3 where ω is the frequency of 3.2 . We construct the following homotopy equation with an auxiliary term: u′′ ω2u p [( b −ω2 ) u cu3 ]


Introduction
The homotopy perturbation method 1-7 has been worked out over a number of years by numerous authors, and it has matured into a relatively fledged theory thanks to the efforts of many researchers, see Figure 1.For a relatively comprehensive survey on the concepts, theory, and applications of the homotopy perturbation method, the reader is referred to review articles 8-11 .
The homotopy perturbation method has been shown to solve a large class of nonlinear differential problems effectively, easily, and accurately; generally one iteration is enough for engineering applications with acceptable accuracy, making the method accessible to nonmathematicians.
In case of higher-order approximates needed, we can use parameter-expansion technology 12-14 ; in this paper, we suggest an alternative approach by adding a suitable term in the homotopy equation.

Homotopy Perturbation Method
Consider a general nonlinear equation where L and N are, respectively, the linear operator and nonlinear operator.The first step for the method is to construct a homotopy equation in the form 3-5 where L is a linear operator with a possible unknown constant and Lu 0 can approximately describe the solution property.The embedding parameter p monotonically increases from zero to unit as the trivial problem Lu 0 is continuously deformed to the original one Lu Nu 0 .
For example, consider a nonlinear oscillator For an oscillator, we can use sine or cosine function.We assume that the approximate solution of 2.3 is where ω is the frequency to be determined later.We, therefore, can choose Lu ü ω 2 u.

2.5
Accordingly, we can construct a homotopy equation in the form ü ω 2 u p u 3 − ω 2 u 0.

2.6
When p 0, we have which describes the basic solution property of the original nonlinear equation, 2.3 .When p 1, 2.6 becomes the original one.So the solution procedure is to deform from the initial solution, 2.4 , to the real one.Due to one unknown parameter in the initial solution, only one iteration is enough.For detailed solution procedure, refer to 5 .
If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form ü 0 • u pu 3 0.

2.8
We expand the solution and the coefficient, zero, of the linear term into a series in p: where the unknown constant, a i , is determined in the i 1 th iteration.The solution procedure is given in 5 .

Homotopy Equation with an Auxiliary Term
In this paper, we suggest an alternative approach to construction of homotopy equation, which is where α is an auxiliary parameter.When α 0, 3.1 turns out to be that of the classical one, expressed in 2.2 .The auxiliary term, αp 1 − p u, vanishes completely when p 0 or p 1; so the auxiliary term will affect neither the initial solution p 0 nor the real solution p 1 .The homotopy perturbation method with an auxiliary term was first considered by Noor 15 .To illustrate the solution procedure, we consider a nonlinear oscillator in the form where b and c are positive constants.Equation 3.2 admits a periodic solution, and the linearized equation of 3.2 is where ω is the frequency of 3.2 .We construct the following homotopy equation with an auxiliary term: Assume that the solution can be expressed in a power series in p as shown in 2.9 .Substituting 2.9 into 3.4 , and processing as the standard perturbation method, we have u 0 ω 2 u 0 0, u 0 0 A, u 0 0 0, 3.5 Solving 3.5 , we have u 0 A cos ωt.

3.9
Substituting u 0 into 3.6 results into Eliminating the secular term needs A special solution of 3.10 is If only a first-order approximate solution is enough, we just set α 0; this results in

3.13
The accuracy reaches 7.6% even for the case cA 2 → ∞.
The solution procedure continues by submitting u 1 into 3.7 ; after some simple calculation, we obtain

3.17
The exact period reads where k cA

3.20
Comparing between 3.17 and 3.20 , we find that the accuracy reaches 5.5%, while accuracy of the first-order approximate frequency is 7.6%.If a higher-order approximate solution is needed, we rewrite the homotopy equation in the form u ω 2 u p b − ω 2 u cu 3 1 • p 1 − p u 0.

3.21
The coefficient, 1, in the auxiliary term, is also expanded in a series in p in the form where α i is identified in the i 2 th iteration.The solution procedure is similar to that illustrated above.

Discussions and Conclusions
Generally the homotopy equation can be constructed in the form where f and g are functions of p, satisfying f 0 0 and g 1 0, and h can be generally expressed in the form For example, for the Blasius equation where the superscript denotes derivative with respect to η, we can construct a homotopy equation in the form where a and b are unknown constants to be determined.
Abstract and Applied Analysis 7 The operator L can be also a nonlinear one, for example, if we want to search for a solitonary solution, we can choose Lu u t 6uu x u xxx .
The homotopy equation can be easily constructed, and the solution procedure is simple.This paper can be considered a standard homotopy perturbation algorithm and can be used as a paradigm for many other applications.

2 AbstractFigure 1 :
Figure 1: Number of publications on homotopy perturbation according to web of science, August 20, 2011.