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.

Keywords: Homotopy perturbation method; nonlinear Mathieu equation; modulation method; parametric resonance

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 [3-6], 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 method to solve Lighthill equation [3], Duffing equation [7] and Blasius equation [8], then the idea go through and has been used to solve nonlinear wave equations [9], boundary value problems [10]. E. Babolian, J. Saeidian and A. Azizi [11] apply homotopy perturbation method to Burgers equation, the regularized long wave equation and the modified Korteweg-de Vries equation. M. Rabbani [12], introduce a new homotopy perturbation method for solving non-linear problems through projection method. A new homotopy perturbation method for solving linear and nonlinear Schrödinger equations has been addressed by Z.Ayati1, J. Biazar and S. Ebrahimi [13].

In this work, we apply the homotopy perturbation method to linear or nonlinear equations having periodic coefficients such 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.

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) as L( y) and N( y) , where

Construct the homotopy statement as

As in He’s homotopy perturbation method, it is obvious that when ρ=0, Eq. (3) becomes the harmonic equation; L( y) =0. Thus,

According to linear differential equations theory, the general solution of (4) is expressed in terms of two linearly independent solutions, say, e^iωt and e^-iωt . Thus, the composite solutions may be in the form

where A and its complex conjugate Ā are arbitrary constants of integration. Eq. (3) becomes the original nonlinear Mathieu equation (1) as ρ=1 . For arbitrary the small parameter ρ, the solution of equation (3) can be sought in terms of ρ so that the function y(t) becomes y(t,ρ) . Accordingly, equation (3) can be rewrite as

It can be noticed that the homotopy function (6) is essentially the same as (3), except for function y(t,ρ), which contain embedded the homotopy parameter ρ . 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 ρ moves from 0 to 1, the function y(t, ρ) moves from y₀(t) to y_app(t). Expand the function y(t,ρ) as a power series in the small parameter ρ such that

where y_n(t);n =1,2,3,... n are unknowns in needs to determined. Substituting this expansion into the homotopy equation (6) and equating terms with identical powers of ρ , leads to

Equation (8) can be satisfied by

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 Ω is not equal the nature frequency ω (which is known as the non-resonance case). The second one is the specific case when Ω approaches ω (which is known as the resonance case).

For arbitrary frequency Ω , 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).

TTo obtain uniform expansions for problems of this kind, the expansion (7) needs to be modified. If we modulate the initial solution (5) so that the constant A becomes A(τ) with τ=ρt, such that

Then (11) in the modulate case becomes

where

Consequently, the homotopy state, equation (6), in the modulated form becomes

It is convenient to choose the modulated function Y(t,τ , ρ) in separated variables as

The function u(t,ρ) can be expanded as a power series in the small parameter ρ such that

where u_n(t);n =1,2,3,... 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

and

where dots indicate differentiation with respect to the time t , while dashes refer to the derivative with respect to the time modulate τ. Substituting (17) into equation (16) using (21) and (22) gives

Equation (23) remains obey the same homotopy concept, because it’s become the same harmonic equation (4) as ρ → 0 . In addition consequently the original equation (1) is found.

In the light of (18), the modulate homotopy equation (23) will be expanded as a power series in ρ 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) into (25) becomes

This equation contains secular terms at the non-resonance case and another secular terms when the applied frequency Ω approaches the natural frequency ω.

The analysis in this case concerned with the arbitrary chosen for the applied frequency Ω , in equation (27). At this stage, secular terms are removed when

with its complex conjugate one. This leads to obtain the valid function u₁(t) as

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 ρ=1 , gets

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

with real the unknowns functions ξ(τ) and η(τ) . Insert (35) into the first-order solvability condition (27) which will separate into real and imaginary parts and gives

where, ξ₀ and &eta₀ are integration constants. Clearly, the stability criteria at the non-resonance case requires that μ> 0.

The resonance case of Ω is near ω

Return to the first-order problem equation (27) and re-analyzed it in view of the nearness of Ω to ω . We express this approach by introducing the detuning parameter σ [1] such that

Accordingly, we have

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:

with its complex conjugate. The valid function Y₂(t,τ) is given by

The approximate solution up to the second-order is formulated by substituting from (14), (15), (40) and (42) into (15) gets

Stability analysis for the linear Mathieu equation

In the limiting case as k → 0 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 Ω is near ω having the following limit case

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 σ .

It is easy to show that equation (44) can be satisfied by the form

where, the parameter μ must be positive, in order to find damping solution. The argument Θ is given by the following characteristic equation:

The parameter σ can be evaluated by substituting (46) into the second-order solvability condition (45) to gets

The use of the first value of σ with (47) yields a zero solution for equation (44). For non-zero solution, the other values for σ are conforms. Inserting (48) into (47) gets

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 Ω 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 σ .

In order to relax the periodic term into equation (39) we let

with real functions α and β . 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 . If the solutions of (56) and (57), at the steady-state, are represented by α₀ and β₀, which are given by

where r²=α₀²+β₀² is used. Equations (58) and (59) are two coupled algebraic equations in α₀ and β₀. For nontrivial solutions in α₀ and β₀, we obtain

In addition the constants α₀ and β₀ may be chosen as

Squaring both equations in (61) and adding we get

Combing (60) with (62) yields

In order to find a constrain for bounded solution we may modulate the functions α and β as

where the functions α₁(τ) and β₁(τ) refer to a small deviation from the steady-state solution α₀ and β₀ . Then the system of (56) and (57) in the linearizing form becomes

The above system are two coupled linear differential equations of first-order in the two functions α₁ and β₁ . This system can be satisfied by

where Θ is given by the following characteristic equation:

where relations (61) are used. This characteristic equation depends on the two related parameters σ and r². 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 σ and r² in terms of the frequency Ω . 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 σ ,Ω and r² :

where relations (61) are used. Removing the parameter σ from (72), by using its equivalent in (63), gives a polynomial of second-order in r²

Replacing r² and r² into (71) with their equivalents in (63) and (72) leads to the following quadratic equation in the detuning parameter σ :

This equation gives two values σ₁ and σ₂ for the detuning parameter σ which makes the solution (55) without unknowns.

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

Removing the parameter r² from the above stability conditions by using (72) yields the following two conditions for stability:

The transition curves separating stable state from unstable one are corresponding to

Using the definition (37) the above transition curves can be sought within the parameter ρ

In order to obtain the transition curves independent of the parameter ρ , we may inserting (80) as well as (81), into the relation (74) , then the following transition curves are imposed:

It is noted that the instability state lies between the above transition curves. The constant coefficients a_j and , b_j j = 0,1,2,3 are given below:

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.

1. Nayfeh Ali.H, Mook D.T., "Non-Linear Oscillations, John Wily & Sons, New York (1979).
   Return to citation in text: [1] [2] [3]
2. Nash C, Sen S., "Topology and Geometry for physicists," Academic Press. rnc, London (1983).
   Return to citation in text: [1] [2]
3. J.H. He, Homotopy perturbation technique, Comp. Meth. Appl. Mech.Eng. 178, 1999, 257- 262.
   Return to citation in text: [1] [2] [3] [3]
4. J.H. He, A coupling method of homotopy technique and a perturbation technique for non- linear problems, Int. J. Non-Linear. Mech. 35, 2000, 37-43.
   Return to citation in text: [1] [2]
5. J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals. 26, 2005, 295-300.
   Return to citation in text: [1]
6. J.H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350, 2006, 87-88
   Return to citation in text: [1]
7. J.H. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput. 135, 2003, 73-79.
   Return to citation in text: [1]
8. J.H. He, A simple perturbation approach to Blasius equation, Appl. Math. Comput. 140, 2003, 217-222.
   Return to citation in text: [1]
9. J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals. 26, 2005, 295-300.
   Return to citation in text: [1]
10. J.H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350, 2006, 87-88.
    Return to citation in text: [1]
11. E. Babolian, J. Saeidian, and A. Azizi, Applied Mathematical Sciences, Vol. 3, 2009, no. 45, 2215 – 2226.
    Return to citation in text: [1]
12. M. Rabbani, Journal of mathematics and computer Science, 7, 2013, 272 – 275.
    Return to citation in text: [1]
13. Z.Ayati, J. Biazar and S. Ebrahimi, Journal of Interpolation and Approximation in Scientific Computing, 2014, 1-8.
    Return to citation in text: [1]
14. Y.O. El-Dib, Chaos, Solitons and Fractals 12, 2001, 705-720
    Return to citation in text: [1]
15. Y.O. El-Dib, Science and Engineering Applications 1(5), 2016, 76-88.
    Return to citation in text: [1]