Homotopy Perturbation for Excited Nonlinear Equations

The purpose of this paper is to apply a version of homotopy technique to excited nonlinear problems. A modulation for the homotopy perturbation is introduced in order to be successfully for nonlinear equations having periodic coefficients. The nonlinear damping Mathieu equation has been studied as a simplest example. The analysis proceeds without assuming weakly nonlinearity and without presence of small factor for the periodic term. In this analysis, two nonlinear solvability conditions are imposed. One of them imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The second level of the perturbation produces another solvability condition and used to determine the unknowns appear in solution for the first-order solvability condition. The method can be, also, used for excited linear equation. Stability conditions, and also the transition curves, are formulated independent of the small parameter i.e. in the unperturbed form as an alternative to classical methods.


Introduction
In [1], perturbation methods depend on small parameter and choose unsuitable small parameter can be lead to wrong solution. Homotopy is an important part of topology [2] and it can convert any non-linear problem in to a finite linear problems and it doesn't depend on small parameter.
The homotopy perturbation method, first proposed by Ji Huan He [3,4,5,6], has successfully been applied to solve many types of linear and nonlinear functional equations. This method, which is a combination of homotopy in topology and classic perturbation techniques, provides us with a convenient way to obtain analytic or approximate solutions to a wide variety of problems arising in different fields. He used the homotopy perturbation Mathieu equation which has been of great importance among researchers. The Mathieu equation serves as a useful model for many interesting problems in applied mathematics, engineering mechanics, etc.

Mathematical formulation
To explain the proposed technique, consider the following damping cubic nonlinear Mathieu equation as an illustrative example: This equation is without any restrictions on its coefficients. The homotopy perturbation method can be considered as combination of the classical perturbation technique and the homotopy (whose origin is in the topology [2]), but not restricted to the limitations of traditional perturbation methods. For example, this method does require neither small parameter nor linearization, and only requires little iteration to obtain accurate solutions [3] and [4].
We define the two parts of equation (1) It can be noticed that the homotopy function (6) is essentially the same as (3), except for , which contain embedded the homotopy parameter  . The introduction of that parameter within the differential equation is a strategy to redistribute the periodic part between the successive iterations of the homotopy method, and thus increase the probabilities of finding the sought solution. Thus, as  moves from 0 to 1, the function ) , Substituting (11) into equation (9) gets the for Before analyzed the first-order problem we must distinguish between two cases. The case of the frequency  is not equal the nature frequency  (which is known as the non-resonance case). The second one is the specific case when  approaches  (which is known as the resonance case).
For arbitrary frequency  , there are secular terms appears in equation (12). Elimination such secular terms requires that the arbitrary constant A be zero. This means that expansion (7) cannot be successfully to obtain valid solution for excited homotopy equation (6).

The modulation procedure
To obtain uniform expansions for problems of this kind, the expansion (7) needs to be modified. If we modulate the initial solution (5) Then (11) in the modulate case becomes Consequently, the homotopy state, equation (6), in the modulated form becomes ). , , It is convenient to choose the modulated function ) , , ( can be expanded as a power series in the small parameter  such that where ... , 3 , 2 , 1 ); (  n t u n are unknowns to be evaluated. If the expansion (18) is substituted into (17) then gets where cc. indicates to the complex conjugate for the preceding terms and It is noted that where dots indicate differentiation with respect to the time t , while dashes refer to the derivative with respect to the time modulate .
 Substituting (17) into equation (16) using (21) and (22) gives In the light of (18), the modulate homotopy equation (23) will be expanded as a power series in  so that the following non-homogenous harmonic equations are imposed: It is noted that equation (24) has been satisfied by (15) and the zero-order solution for equation (16) as approved in (14). Substituting (15) This equation contains secular terms at the non-resonance case and another secular terms when the applied frequency  approaches the natural frequency .
The analysis in this case concerned with the arbitrary chosen for the applied frequency  , in equation (27). At this stage, secular terms are removed when , 0 2 with its complex conjugate one. This leads to obtain the valid function ) Consequently, the solution at the first-order problem is formulated as Substituting (15) and (29) into (26), using (28), yields The valid solution requires to removing the terms that producing unbounded solution. These terms implies the following nonlinear solvability condition: The second-order solution is found to be If the accuracy to the second-order perturbation is enough, then the approximate solution at the non-resonance case is formulated by substituting (14), (15), (30) and (33) into (19), and setting 1 

Stability analysis at the non -resonance case
The stability criteria at the non-resonance case, can be obtained for solving equation (28). One may use the following polar form [1]:

The resonance case of  is near 
Return to the first-order problem equation (27) and re-analyzed it in view of the nearness of  to  . We express this approach by introducing the detuning parameter  [1] such that Elimination of secular terms from equation (27), in view of (37) and (38) yields The first-order solution at this case is Using (40) with equation (26) we obtain the uniform solution for the second-order problem, and the following solvability is presented: The approximate solution up to the second-order is formulated by substituting from (14), (15), (40)

Stability analysis for the linear Mathieu equation
In the limiting case as 0  k into equation (1), linear damping Mathieu equation arrived. In this case the two solvability conditions (39) and (41) that produced at the resonance case of  is near  having the following limit case , 0 2 The first-order solvability condition (44) can be used to find the stability picture at the resonance case. The second-order solvability condition (45) can be used to find the value of the detuning parameter .
It is easy to show that equation (44) can be satisfied by the form where, the parameter  must be positive, in order to find damping solution. The argument  is given by the following characteristic equation: The parameter  can be evaluated by substituting (46) Clearly, the stability criteria requires that the right-hand-side of (49) be positive, which implies that Stability condition (50) can be rearranged in powers of the applied frequency  as The transition curves separating stable state from unstable state corresponding to

Stability analysis for the nonlinear case
The first-order solvability condition (39) can be used to find the stability picture at the resonance case. The second-order solvability condition (41) can be used to find the value of the detuning parameter .
In order to relax the periodic term into equation (39) we let with real functions  and  . Insert (55) into (39), separating real and imaginary parts yields In order to solve the above coupled nonlinear equations (56) and (57), we may discuss the behavior at the steady-state response. This case is corresponding to the case of In addition the constants 0  and 0  may be chosen as , 2 and .
Combing (60)   and 0  . Then the system of (56) and (57) in the linearizing form becomes where relations (61) are used. This characteristic equation depends on the two related parameters  and 2 r . This relation between them is given in (60) or in (63). By help of the second-order solvability condition (41) one can find an expression for both the unknowns  and 2 r in terms of the frequency  . To accomplish this, one may substitute the steady-state solution into the second-order solvability condition (41). Separating the real and imaginary parts, produces the following relations, between the parameters ,  and 2 r : where relations (61) are used. Removing the parameter  from (72), by using its equivalent in (63), gives a polynomial of second-order in 2 0  2  2  4  3  8  2  3  3  6   2  2  3  36  2  4  3  8  3   144  2  4  3  3  2  4  8  3  6   3  2  4  3  3  8  2   6  8  1  8  6   2  2  2  2  2  2  2  2   3  2  2  5  2  2  2  2  2  2   6  2  2  2  2  2  4  2  2  2   2  2  2  2  2  2   2  2  2  Clearly, the stabilization for the problem requires that the right-hand side of (69) be positive provided that the exponential in (67) and (68) has positive values. It is noted that the stability reveals as the coefficient of the periodic term in (1) tends to zero. The instability arrived as the parameter q going away the zero value. Thus, the stability conditions are found as , 0 .

Conclusion
In this study we proposes a variation of the homotopy perturbation method, by using a modulation technique, which allows to find solutions for ordinary differential equations with periodic coefficients. This work has been employed to analyze of parametrically excited oscillator without smallness the cubic nonlinearity. The simplest equation of this type is the Mathieu equation which usually contains a small parameter [14,15]. As in the homotopy perturbation [3], the analysis has no dependence on equations having a small parameter. Due to this modulation technique a solvability condition at each level of perturbation is imposed. Solving these solvability conditions leads to studying the stability behavior. Stability conditions, in both resonance and non-resonance cases are derived.