Stochastic Oscillators with Quadratic Nonlinearity Using WHEP and HPM Methods

In this paper, quadratic nonlinear oscillators under stochastic excitation are considered. The Wiener-Hermite expansion with perturbation (WHEP) method and the homotopy perturbation method (HPM) are used and compared. Different approximation orders are considered and statistical moments are computed in the two methods. The two methods show efficiency in estimating the stochastic response of the nonlinear differential equations.


Introduction
Quadrate oscillation arises through many applied models in applied sciences and engineering when studying oscillatory systems [1].These systems can be exposed to a lot of uncertainties through the external forces, the damping coefficient, the frequency and/or the initial or boundary conditions.These input uncertainties cause the output solution process to be also uncertain.For most of the cases, getting the probability density function (p.d.f.) of the solution process may be impossible.So, developing approximate techniques through which approximate statistical moments can be obtained, is an important and necessary work.
In HPM technique [31][32][33][34], the response of nonlinear differential equations can be obtained analytically as a series solution.The basic idea of the homotopy method is to deform continuously a simple problem (and easy to solve) into the difficult problem under study [35].The HPM method is a special case of homotopy analysis method (HAM) propounded by Liao in 1992 [36].The HAM was systematically described in Liao's book in 2003 [37] and was applied by many authors in [38][39][40][41].The HAM method possesses auxiliary parameters and functions which can control the convergence of the obtained series solution.
The stochastic oscillator with cubic nonlinearity (Duffing oscillator) was considered in [17,42].The nonlinear term is due to the restoring nonlinear force.In some applications, the restoring force is quadratic and it is required to estimate the response in this case.The main goal of this paper is to consider the quadratic nonlinear oscillator under stochastic excitation.The WHEP and HPM methods are used and compared.
This paper is organized as follows.The problem formulation is outlined in Section 2. The WHEP technique is described and applied to the stochastic quadratic oscillator in Section 3. The HPM is outlined in Section 4 and applied also to the quadratic oscillator.A comparison between the two methods is shown in Section 5.

 
under stochastic excitation

 
; where w: frequency of oscillation, : a triple probability space with  as the sample space,  is a  -algebra on events in  and P is a probability measure.

WHEP Technique
The application of the WHE aims at finding a truncated series solution to the solution process of differential equations.The truncated series composes of two major parts; the first is the Gaussian part which consists of the first two terms, while the rest of the series constitute the non-Gaussian part.In nonlinear cases, there exists always difficulties of solving the resultant set of deterministic integro-differential equations got from the applications of a set of comprehensive averages on the stochastic integrodifferential equation obtained after the direct application of WHE.Many authors introduced different methods to face these obstacles.Among them, the WHEP technique was introduced in [22] using the perturbation technique to solve perturbed nonlinear problems.
The WHE method utilizes the Wiener-Hermite polynomials which are the elements of a complete set of statistically orthogonal random functions [30].The Wiener-Hermite polynomial satisfies the following recurrence relation: where in which n(t) is the white noise with the following statistical properties where   .


is the Dirac delta function and E denotes the ensemble average operator.
The Wiener-Hermite set is a statistically orthogonal set, i.e.
The average of almost all H functions vanishes, particularly,   0 for 1.
Due to the completeness of the Wiener-Hermite set, any random function where the first two terms are the Gaussian part of G(t; ω).The rest of the terms in the expansion represent the non-Gaussian part of G(t; ω).The average of G(t; ω) is The covariance of

 
; The WHE method can be elementary used in solving stochastic differential equations by expanding the solution process as well as the stochastic input processes via the WHE.The resultant equation is more complex than the original one due to being a stochastic integro-differential equation.Taking a set of ensemble averages together with using the statistical properties of the WH po lynomials, a set of deterministic integro-differential equations are obtained in the deterministic kernels To obtain an approximate solutions for these deterministic kernels, one can use perturbation theory in the case of having a perturbed system depending on, say,  .Expanding the kernels as a power series of  , another set of simpler iterative equations in the kernel series components are obtained.This is the main algorithm of the WHEP technique.The technique ied to several nonlinear stochastic equations; see [20,22,23,25].
The WHEP technique can be applied on linear or nonlinear perturbed systems described by ordinary or partial dif rential equations.The solution can be modified in the sense that additional parts of the Wiener-was successfully appl fe Hermite expansion can always be taken into considerations and the required order of approximatio ways be made.It can be even run through a package if it is coded in some sort of symbolic languages.

Case-Study ns can al-
The quadratic nonlinear oscillatory problem, Equation (1) under stochastic excitation   ; F t  with deterministic initial conditions is solved using WHEP technique.The solution process takes the following form: Applying the WHEP technique, the following equations in the deterministic kernels are obtained:  , , , d d

d x t t t t x t t t t t t t
Let us take the simple case of evaluating the only Gaussian part (first order approximation) of the solution process of the previous case study, mainly  , , d

t x t t t x t t t t t x t t t x t t t t t x t t t x t t t t t x t t t x t t t t t x t t t x t t t t t G t t t t G t t t t G t t
In this case, the governing equations are 0 The ensemble average is It has to be noticed that all the previous equations are deterministic linear ones in the general form In which we have When adding the first term in the non-Gaussian part (the second approximation) of the solution pro inly cess of the previous case study, ma ; , d

Lx t t w x t x t t w x t t x t t t t G t t
, , ; , ; , d , , The ensemble average is still got by Equation ( 19) 2 while the variance is got as The WHEP technique uses the following expansion for its deterministic kernels as corrections made under each approximation order.
, 0,1,2,3,. Example: Let us take in the previous case-study and then solving using the WHEP technique.The following results are obtained, see Figures 1-3.

The Homotopy Perturbation Method (HPM)
In this technique, a parameter where is an initial approximation to the solution of the equ with boundary conditions in which A is a nonlinear differential operator which can be decompose into a linear operator L and a nonlinear operator N, B is a boundary operator, f(r) is a known analytic function and is the boundary of   .The homotopy introduces ontinuously deform ution for the case of p = 0, , to the case of p = 1, inal ation (30) motopy ethod which is to deform continuously a simple problem (and easy to solve) into the difficult problem under study [35].
The basic assumption of the HPM method is that the solution of the original Equation ( 29) can be expanded as a power series in p as: ich is the orig ea the ho . This is the ba of Now, setting p = 1, the approximate solution of Equation ( 23) is obtained as: The rate of convergence of the method depends greatly on the initial approximation The idea of the imbedded rameter can be utilized to solve nonlinear problems by imbedding this parameter to the problem and then forcing it to be unity in the obtained approximate solution if converge can be assured.A simple technique enables the extension of th applicability of the perturbation ods from small valued ap- The homotopy function takes the following form: or equivalently, , substituting in Equation ( 34) and equating the equal powers of p in both sides of the equation, one can get the following results: , in which one may consider the fol- wing simple solution: The approximate solution is which can be considered to any approximation order.On can notice that the algorithm of the solution is straigh t a lot of flexibilities can be made.For any choices in guessing the initial approximation together with its initial condition zero initial conditions, we can choose e t forward and tha example, we have m s.For 0 v  0 which leads to: Figures 4-7 are obtained for 0.5   : [42].

Comparisons between WHEP and HPM Methods
Figure [8] shows comparisons between the WHEP and HPM methods for different values of the nonlinearity strength,  .As the nonlinearity strength increases, the deviation between the two methods is also increasing. it will diverge.The M is more accurate for HP higher values of  .The HPM has advantages when used in solving differential equations with large nonlinearities.

Conclusion
The quadratic nonlinear oscillator with stochastic excitation is considered.The solution was obtained using the WHEP technique with different orders and different number of corrections.The HPM is used also with different approximations.The WHEP technique is more efficient but it converges only for certain limit of the nonlinearity strength.The HPM is more difficult in the stochastic differential equations but it is more preferable for high values of the nonlinearity st th.The two methods are shown to be efficient in estimating the stochastic response of the quadratic nonlinear oscillators.

Figure 1 .Figure 2 .Figure 3 .
Figure 1.(a) The first order a ximation of the mean at ε correction for different correcti ; (b) The first order approximation of the mean at ε 2 correction for different correction levels; (c) The first order approximation of the mean at ε 3 correction; (d) The first order approximation of the mean at ε, ε 2 , ε 3 correction; (e) The first order approximation of the mean at ε, ε 2 , ε 3 correction; (f) The first order approximation of the mean at ε, ε 2 , ε 3 correction.ppro on levels

Example
Considering the same previous example of Sub-Sect 3.1.1,one can get the following results w.r.t.homo ati ion topy perturb on: different correction levels; (b) The first and second levels.

Figure 4 .Figure 5 .
Figure 4. (a) The first and second order approximation of th order approximation of the variance at for different correction

Figure 6 .Figure 7 .
Figure 6.(a) A comparison between first, second order and th first, second order and the, third order o ariance at ε = 0 e n f the v .1.third order of the mean at ε = 0.1; (b) Comparison betwee

Figure 8 .
Figure 8.(a) A comparison between homotopy perturbation and Wiener-Hermite of the mean at ε = 0.1; (b) A comparison between homotopy perturbation and Wiener-Hermite of the variance at ε = 0.1; (c) A comparison between homotopy perturbation and Wiener-Hermite of the mean at ε = 0.3; (d) A comparison between homotopy perturbation and Wiener-Hermite of the variance at ε = 0.3; (e) A comparison between homotopy perturbation and Wiener-Hermite of the mean at ε = 0.7; (f) A comparison between homotopy perturbation and Wiener-Hermite of the variance at ε = 0.7.This is due to the convergence condition of the WHEP technique which depends on  .For small values of  , e the WHEP technique conver but after a certain val of ges u