MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation45086210.1155/2009/450862450862Research ArticleHigher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational ApproachOrhan KayaM.1Altay DemirbağS.2Özen ZenginF.2PavlovskaiaEkaterina1Faculty of Aeronautics and AstronauticsIstanbul Technical UniversityMaslak 34469IstanbulTurkeyitu.edu.tr2Faculty of Science and LettersIstanbul Technical UniversityMaslak 34469IstanbulTurkeyitu.edu.tr20091907200920091610200812032009140520092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

He's variational approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). Three levels of approximation have been used. We obtained 1.6% relative error for the first approximate period, 0.3% relative error for the second-order approximate period. The third approximate solution has the accuracy as high as 0.1%.

1. Introduction

Considerableattention has been directed toward the solution of nonlinear equations since they play crucial role in applied mathematics, physics, and engineering problems. In general, the analytical approximationto solution of a given nonlinear problem ismoredifficult than the numerical solution approximation. During the past decades, several types of methods are proposed to obtain approximate solution of nonlinear equations of various types. Among them are variational iteration methods , homotopy perturbation method , modified Lindstedt-Poincare method , parameter expansion method [17, 18], and variational methods . The variational method is different from any other variational methods in open literature, and it is only valid for nonlinear oscillators . Paper  is an example of use of variational approach method in nonlinear oscillator problem.

When we examine the frequency amplitude relations of some nonlinear oscillators, it is seen that paper  focuses on only the first-order solutions.

Variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions .

In the present study, we have investigated the application of variational approach to nonlinear oscillator with discontinuity.

2. A Variational Method

Let us consider a general nonlinear oscillator in the form

u+f(u)=0.

He proposed a variational principle for (2.1) as follows :

J(u)=0T/4(-12u2+F(u))dt, where T is period of the nonlinear oscillator, F/u=f. Actually, the upper limit is originally T instead of T/4. Normally, it works in most of the cases. Let us suppose that f(u)=sgn(u) such

sgn(u)={-1,u<0,+1,u0,

therefore F(u)={-u,u<0,u,u0.

But this form is not suitable for discontinuity equation. Therefore, we propose the equation in the form of

J(u)=0T/2(-12u2+  |u|)dt.

Assume that its solution can be expressed as

u(t)=Acosωt+B(cosωt-cos3ωt)+C(cos3ωt-cos5ωt)+, where ω is the frequency of the oscillator.

Inserting (2.6) into (2.5) yields

J(A,B,C,,ω)=0T/2{-12ω2[Asinωt+B(sinωt-3sin3ωt)+C(3sin3ωt-5sin5ωt)+]2+12  |Acosωt+B(cosωt-cos3ωt)+C(cos3ωt-cos5ωt)+|}

Let us define τ=ωt. Then (2.7) becomes

J(A,B,C,..,ω)=0π{-12ω[Asinτ+B(sinτ-3sin3τ)+C(3sin3τ-5sin5τ)+]2  +1ω|Acosτ+B(cosτ-cos3τ)+C(cos3τ-cos5τ)+|}dτ.

Using the Ritz method, we require

Jω=0,JA=0,JB=0,JC=0,,Jω=-0π{12[Asinτ+B(sinτ-3sin3τ)+C(3sin3τ-5sin5τ)+]2        +1ω2|Acosτ+B(cosτ-cos3τ)+C(cos3τ-cos5τ)+|}dτ.

By a careful inspectation, we find that

Jω<0.

Thus, the conditions in (2.9) reduce to

JA=0,JB=0,JC=0,.

3. Application

Consider the following nonlinear oscillator with discontinuity:

u+sgn(u)=0, with initial conditions

u(0)=A,du(0)dt=0.

Its variational formulation can be written as follows:

J(u)=0π/2(-12u2+u)dt+π/2π(-12u2-u)dt.

For the first approximation assume that u(t) is in the following form:

u(t)=Acosωt.

Substitute this first approximation into  (3.3):

J(A,ω)=2A-14π  A2ω2.

The stationary condition with respect to A reads

JA=2-π  A2ω2=0, which leads to the result

ω=2Aπ=1.128379A, and the approximate period can be obtained as follows:

T1app(A)=ππ  A=5.568328A,relative  error=1.6%.

This solution agrees with Liu’s solution obtained by He’s modified Lindsted-Poincaré method , Rafei et al.’s solution obtained by He’s variational iteration method , Wu et al.’s solution obtained by the low-order harmonic balance method , and A. Belendéz et al.’s solution obtained by He’s homotopy perturbation method .

Secondly, to obtain a more accurate result, define u as follows:

u=Acosωt+B(cosωt-cos3ωt).

Notice that (3.9) satisfies the initial conditions (3.2).

Substituting (3.9) into (3.3), we obtain

J(A,B,ω)=0T/4(-12(A+B)2ω2sin2ωt+3B(A+B)ω2sinωtsin3ωt)dt+0T/4(-92B2ω2sin23ωt+(A+B)cosωt-Bcos3ωt)dt+T/4T/2(-12(A+B)2ω2sin2ωt+3B(A+B)ω2sinωtsin3ωt)dt+T/4T/2(-92B2ω2sin23ωt-(A+B)cosωt+Bcos3ωt)dt. The J(A,B,ω) in (3.10) can be obtained as follows:

J(A,B,ω)=2A+83B-14πA2ω2-12πABω2-52πB2ω2.

The stationary condition with respect to A and B reads

JA=2-12Aπω2-12Bπω2=0,A=10427πω2,  JB=83-12Aπω2-5Bπω2=0,B=427πω2, from which the relationship between oscillator frequency and amplitude can be determined.

From (3.12) we have

ω=23263Aπ=1.10729A, and the approximate period can be obtained as follows:

T2app(A)=27Aπ326=5.67440A,relative  error=0.31  %.

In this study, we obtained the relative error as 1.6% for the first-order approximation while the other researchers [2, 16] obtained the relative error as 1.8%. The reason for the difference in the relative error is that the other researchers take less precision in the decimal numbers during calculations. In , the frequency ω=1.107452/A and the period T2app(A)=5.67440A were found for the same problem by second-order approximation and the relative error was calculated as %0.30.

Equation (3.1) was approximately solved in  using an improved harmonic balance method that incorporates salient features of both Newtons’s method and the harmonic balance method. In , the following results for the first and second-order approximations were obtained:

TWSL1(A)=5.568328A,relativeerror=1.6%TWSL2(A)=5.67440A,relativeerror=0.31%.,

To obtain a more accurate result, define u as follows:

u=Acosωt+B(cosωt-cos3ωt)+C(cos3ωt-cos5ωt).

Notice that (3.17) satisfies the initial conditions (3.2).

Substituting (3.17) into (3.3) gives

J(A,B,C,ω)=2A+83B-1615C-14πω2(A2+2AB+10B2-18BC+34C2).

The stationary condition with respect to A, B, and C reads

JA=2-12Aπω2-12Bπω2=0,A=131083375πω2,JB=83-12Aπω2-5Bπω2+92Cπω2=0,B=3923375πω2,JC=-1615-17Cπω2+92Bπω2=0,C=-4125πω2.

Hence the approximate frequency is

ω=131083375Aπ=1.11188A.

Therefore, approximate period of the nonlinear oscillator can be obtained as follows:

T3app(A)=23375π3A13108=5.65098A,relative  error=0.1%.

For this nonlinear problem in (3.1), the exact period is given as follows :

Te(A)=42A=5.656854A.

The period values and these relative errors obtained in this method for nonlinear oscillator with discontinuity are the following:

T1app(A)=5.568328A,relativeerror=1.6%,T2app(A)=5.67440A,relativeerror=0.31%,T3app(A)=5.65098A,relativeerror=0.10%.

Equation (3.1) was approximately solved in  using an improved harmonic balance method that incorporates salient features of both Newtons’s method and the harmonic balance method. In , the following result for the third-order approximations was obtained:

TWSL3(A)=5.650976A,relativeerror=0.10%.

Equation (3.1) was approximately solved in  using a homotopy perturbation method. In , the following result for the third-order approximations was obtained:

T3(A)=5.653609A,relativeerror=0.057%.

By using above values, the periodic function u(t) can be written for three levels of approximation as follows:

ua1(t)=Acos(ω1t),ua2(t)=1.03846Acos(ω2t)-0.0384615Acos(3ω2t),ua3(t)=1.02991Acos(ω3t)-0.0381446Acos(3ω3t)+0.00823924Acos(5ω3t).

The normalized exact periodic solution uex/A has been obtained by numerically integrating (3.1) and (3.2) and compared with approximate solutions (3.26) in Figure 1. Here nondimensional time h is defined as follows: h=tTe.

Comparison of approximate normalized functions (dashed line) with exact normalized functions (continuous line) for (a) first approximation (b) second approximation (c) third approximation.

4. Conclusions

He’s variational approach is modified for nonlinear oscillator with discontinuities. The method has been applied to obtain three levels of approximation of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn(u). We reached 1.6%, 0.31%, and 0.1% relative errors for the first, second, and third approximate periods, respectively. One can obtain higher-order accuracy by extending the idea given in this paper.

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