# Approximate Periodic Solution for the Nonlinear Helmholtz-Duffing Oscillator via Analytical Approaches

Authors: A. Mirzabeigy; M. K. Yazdi; M. H. Nasehi
Journal: International Journal of Computational Mathematics
 (2014)

---

## Abstract

The conservative Helmholtz-Duffing oscillator is analyzed by means of three analytical techniques. The max-min, second-order of the Hamiltonian, and the global error minimization approaches are applied to achieve natural frequencies. The obtained results are compared with the homotopy perturbation method and numerical solutions. The results show that second-order of the global error minimization method is very accurate, so it can be widely applicable in engineering problems.


---

## Body

## 1. Introduction

Mathematical modeling and frequency analysis of the nonlinear vibrational systems are an important and interesting field of mechanics. A lot of researchers have worked in this field and have proposed a lot of methods for demonstrating the dynamics responses of these systems [1–4]. They have developed this field of science and have analyzed the responses of the nonlinear vibration problems such as Duffing oscillators [5–10], nonlinear dynamics of a particle on a rotating parabola [11], nonlinear oscillators with discontinuity [12], oscillators with noninteger order nonlinear connection [13], the plasma physics equation [14], and van der Pol oscillator [15, 16]. The Helmholtz-Duffing equation is a nonlinear problem with the quadratic and cubic nonlinear terms. Surveying the literature shows that this equation has wide applications in the engineering problems. For example, due to different vibration behavior of functionally graded materials (FGMs) at positive and negative amplitudes, the governing equations of FGM beams, plates, and shells are conduced to a second-order nonlinear ordinary equation with quadratic and cubic nonlinear terms [17–20]. Moreover, Sharabiani and Yazdi [21] obtained a Helmholtz-Duffing type equation within studying of nonlinear free vibrations of functionally graded nanobeams with surface effects. On the other hand, they revealed application of this equation in FG nanostructures.In this paper, the frequency-amplitude relationship of the conservative Helmholtz-Duffing oscillator is obtained by means of the max-min [22–26], Hamiltonian [27–31], and global error minimization methods [32–35]. The Hamiltonian approach is a kind of energy method and is proposed by He [27]. It is a simple method and can be used for the conservative nonlinear equations. Recently, it is applied for dynamic analysis of an electromechanical resonator [36]. Moreover, Akbarzade and Khan [37] employed the second-order Hamiltonian approach for nonlinear dynamic analysis of conservative coupled systems of mass-spring. The max-min approach is made on the base of Chengtian’s inequality [38]. It is a valuable method for obtaining the frequency responses of the nonlinear problems, and many researchers are attracted to use this method for studying the nonlinear systems. The global error minimization method is a modified type of variational approach and converts the nonlinear equation to an equivalent minimization problem.The Helmholtz-Duffing equation is analyzed by many researchers. For instance, Leung and Guo [39] have applied the homotopy perturbation method (HPM) to this equation and have obtained accurate responses. Askari et al. [40] studied approximate periodic solutions of this equation using He’s energy balance method (HEBM) and He’s frequency-amplitude formulation (HFAF). Akbarzade et al. [41] used the first-order of the Hamiltonian approach and coupled homotopy-variational formulation to study the periodic solutions of the Helmholtz-Duffing oscillator; they also discussed the stability of the system for selected constant parameters. Recently, Li et al. [42] determined limit cycles and homoclinic orbits of this oscillator via a generalized harmonic function perturbation method. In the next sections, the max-min, the Hamiltonian, and the global error minimization methods are applied for evaluating the dynamics responses of the Helmholtz-Duffing equation. The results of these methods are compared with the exact ones and HPM solutions.


## 2. Solution Procedure

Let us consider the Helmholtz-Duffing equation(1)u¨+u+(1-σ)u2+σu3=0,u(0)=A,u˙(0)=0.
The response oscillates between an asymmetric limit zone [-b,A], where both A and b are positive. One can determine b in the form of [39]
(2)b=19A(3Aσ+4-4A)+19AΔ1/3-29A(9A2σ2+6Aσ-6Aσ2+43σ-8-8σ2)Δ-1/3,
where
(3)Δ=270A2σ2(1+Aσ-σ)-72Aσ(1+σ2)-516σ(1-σ)+64(1-σ3)+630Aσ2+54σ[-12(1+σ2)+16A(a+1+Aσ4-σ3)fdfdfdfffff+78A(1+A3σ-A3σ2)-8A3σ(1-σ3)fdfdfdfffff-172A2σ(1+σ2)-120Aσ(1-σ)+9A4σ2ffdfdfdffff×(1+5Aσ+10σ+3A2σ2-5Aσ2+σ2)ffdfdfdffff+447A2σ2]1/2.


### 2.1. Max-Min Approach (MMA)

We can rewrite (1) in the following form:
(4)u¨+(1+(1-σ)u+σu2)u=0.By choosingu(t)=Acos⁡(ωt) as a trial function that satisfied the initial conditions, the maximum and minimum values of 1+(1-σ)u+σu2 can be calculated with Maple software as 1+A-σA+σA2 and -(σ2-6σ+1)/4σ, respectively, so we can write
(5)-σ2-6σ+14σ<ω2<1+A-σA+σA2.By using Chengtian interpolation [38], we have
(6)ω2=-m((σ2-6σ+1)/4σ)+n(1+A-σA+σA2)m+n=1-(1-k)(σ2-2σ+14σ)+k(A-σA+σA2),
where m, n are weighting factor, k=n/(m+n). Therefore the frequency can be approximated as
(7)ω=1+(1-k)(-σ2-2σ+14σ)+k(A-σA+σA2).
Then the solution of (1) can be written:
(8)u(t)=Acos⁡((1-(1-k)(σ2-2σ+14σ)dfdfdfdfdfff+k(A-σA+σA2)t(σ2-2σ+14σ))1/2).By using the approximate solution, (4) can be written in the following form:
(9)u¨+(1-(1-k)(σ2-2σ+14σ)+k(A-σA+σA2))u=-(1-k)(σ2-2σ+14σ)u+k(A-σA+σA2)u-(1-σ)u2-σu3.If by chance (8) is the exact solution of (1), then the right-hand side of (9) vanishes. According to [22], we set
(10)∫0T/4((k-1)(σ2-2σ+14σ)u+k(A-σA+σA2)udsdfdfff-(1-σ)u2-σu3(σ2-2σ+14σ))cos⁡(ωt)dt=0,
where T=2π/ω. By substituting (8) into (10), we obtain the following expression for k:
(11)k=133π-6πσ+3πσ2+32Aσ-32Aσ2+9πA2σ2π(1-2σ+σ2+4Aσ-4Aσ2+4A2σ2).Substituting (11) into (7) and after some simplification, we have
(12)ω=1+83πA-83πAσ+34A2σ.


### 2.2. Hamiltonian Approach

#### 2.2.1. The First-Order Hamiltonian Approach (FHA)

The Hamiltonian of (1) is constructed as
(13)H=12u˙2+12u2+13(1-σ)u3+14σu4.Assume the first approximate solution of (1) as
(14)u(t)=Acos⁡(ωt).
Then integrating (13) with respect to time from 0 to T/4, we have
(15)H~(u)=∫0T/4(12u˙2+12u2+13(1-σ)u3+14σu4)dt.
Substituting (14) in (15) yields
(16)H~(u)=∫0T/4(12A2ω2sin2(ωt)+12A2cos⁡2(ωt)fdfdfffff+13(1-σ)A3cos⁡3(ωt)+14σA4cos⁡4(ωt))dt=∫0π/2(12A2ωsin2(t)+12ωA2cos⁡2(t)dsdsdd+13ω(1-σ)A3cos⁡3(t)+14ωσA4cos⁡4(t))dt=1576A2(72π+128A-128Aσ+27πA2σ+72πω2)ω.Set(17)∂∂A(∂H~∂(1/ω))=148A(-12πω2+12π+32A-32Aσ+9πA2σ)=0.Finally,(18)ωFHA=1+83πA-83πAσ+34A2σ.
The result of first-order Hamiltonian and max-min approaches is alike.


#### 2.2.2. The Second-Order Hamiltonian Approach (SHA)

For the second-order Hamiltonian approach, we consider the following equation as the response of the system:(19)u=acos⁡(ωt)+bcos⁡(3ωt).
Whereas (19) must satisfy the initial condition we have
(20)A=a+b.
Substituting (19) into (15) yields
(21)H~(u)=∫0T/4[12ω2(asin(ωt)+3bsin(3ωt))2fdfdffdfff+12(acos⁡(ωt)+bcos⁡(3ωt))2fdfdffdfff+13(1-σ)(acos⁡⁡(ωt)+bcos⁡(3ωt))3fdfdffdfff+14σ(acos⁡(ωt)+bcos⁡(3ωt))4]dtH~(u)=∫0π/2[12ω(asin(t)+3bsin(3t))2fdfdffdfff+12ω(acos⁡(t)+bcos⁡(3t))2fdfdffdfff+13ω(1-σ)(acos⁡⁡(t)+bcos⁡(3t))3fdfdffdfff+14ωσ(acos⁡(t)+bcos⁡(3t))4]dtH~(u)=160480ω×(13440a3-4480b3-13440a3σdsdsd-8046a2bσ+2835πa4σ-31104ab2σdsdsd+8064a2b+2835πb4σ+4480b3σdsdsd+31104ab2+3780πa3bσ+11340πa2b2σdsdsd+7560πa2+7560πb2)+πω8(a2+9b2).Set(22)∂∂a(∂H~∂(1/ω))=-14πaω2+23a2-23a2σ-415abσ+316πa3σ-1835b2σ+415ab+1835b2+316πa2bσ+38πab2σ+14πa=0∂∂b(∂H~∂(1/ω))=-94πb2ω2-29b2-215a2σ-3635abσ+215a2+316πb3σ+29b2σ+3635ab+116πa3σ+38πa2bσ+14πb=0.
Since finding a relation between different parameters from (22) is not easy, the value of other parameters is determined in Table 1 for several values of σ.Table 1
Frequency-amplitude relationships for several values ofσ.
σ
a
b
ω
SHA
0.1
0.987791887
A
0.012208112
A
1
+
0.758436858
A
+
0.074106747
A
2
0.3
0.986559761
A
0.013440239
A
1
+
0.589470841
A
+
0.222057234
A
2
0.5
0.985302609
A
0.014697391
A
1
+
0.420742301
A
+
0.369650489
A
2
0.7
0.984019498
A
0.015980502
A
1
+
0.252257249
A
+
0.516878382
A
2
0.9
0.982709435
A
0.017290564
A
1
+
0.084021956
A
+
0.663732471
A
2
1
0.982043978
A
0.017956021
A
1
+
0.737016661
A
2


### 2.3. Global Error Minimization Method

#### 2.3.1. The First-Order Global Error Minimization Method (FGEM)

Based on standard procedure of modified variational approach, the minimization problem is(23)MinimizeE(u˙,u,t)=∫0T(u¨+u+(1-σ)u2+σu3)2dt,T=2πω.
For first-order approximation, consider a trial function as follows:
(24)u(t)=Acos⁡(ωt).
Substituting (24) into (23) yields
(25)MinimizeE(u˙,u,t)=πA28ω(5A4σ2-12A2σω2+6A2σ2+8ω4dsdsdsdsd-16ω2+6A2+8).
The solution of (25) could be found through
(26)∂E(u˙,u,t)∂ω=0.
By some simplifications, the following equation is obtained:
(27)-24ω4+12A2ω2σ+16ω2+5A4σ2+6A2σ2+6A2+8=0.
One may find the first-order approximation by solving (27) as
(28)ωFGEM=16(+36A2+64)1/29A2σ+12fdfdfdffd+3(39A4σ2+36A2σ2+24A2σfdfdffdfdff+36A2+64)1/2)1/2.


#### 2.3.2. The Second-Order Global Error Minimization Method (SGEM)

For the second-order approximation, consider a trial function as(29)u(t)=acos⁡(ωt)+bcos⁡(3ωt).
Whereas (29) must satisfy the initial condition, we have
(30)A=a+b.
Substituting (29) into (23) yields
(31)MinimizeE(u˙,u,t)=π8ω(5σ2a6-144b2ω2sdsdsdddd+45σ2a2b4+45σ2a4b2+24σ2a2b2sdsdsdddd+8σ2a3b+6σ2b4+5σ2b6+8a2ω4sdsdsdddd+30σ2a3b3+24a2b2-16a2ω2sdsdsdddd+8a2+8b2+6a4-240a2ω2σb2sdsdsdddd+15σ2a5b-12a4ω2σ+6σ2a4sdsdsdddd+648b2ω4+6b4-108b4ω2σsdsdsdddd+8a3b-48a3ω2σb).The solution of (31) could be found through
(32)∂E(u˙,u,t)∂ω=0,∂E(u˙,u,t)∂a=0,∂E(u˙,u,t)∂b=0,
which yield the following:
(33)∂E(u˙,u,t)∂ω=-π8ω2(5σ2a6+8a3b+45σ2a2b4dsdsdsdsdsd+8a2+8b2+5σ2b6+45σ2a4b2dsdsdsdsdsd-24a2ω4+16a2ω2+6b4dsdsdsdsdsd+6a4+24σ2a2b2+15σ2a5bdsdsdsdsdsd+30σ2a3b3-1944b2ω4dsdsdsdsdsd+144b2ω3+6σ2a4+6σ2b4dsdsdsdsdsd+48a3ω2σb+240a2ω2σb2dsdsdsdsdsd+8σ2a3b+12a4ω2σdsdsdsdsdsd+108b4ω2σ+24a2b2)=0∂E(u˙,u,t)∂a=π8ω(30σ2a5+90σ2ab4+180σ2a3b2gfgfgfg+24σ2a2b+16aω4+90σ2a2b3+48ab2gfgfgfg-32aω2+16a+24a3-480aω2σb2gfgfgfg+75σ2a4b-48a3ω2σ+24a2bgfgfgfg-144a2ω2σb+24σ2a3+48σ2ab2)=0∂E(u˙,u,t)∂b=π8ω(-288bω2+180σ2a2b3+90σ2a4bsdsdsdsd+48σ2a2b+8σ2a3+24σ2b3+30σ2b5sdsdsdsd+90σ2a3b2+48a2b+16b-480a2ω2σbsdsdsdsd+15σ2a5+1296bω4+24b3sdsdsdsd-432b3ω2σ+8a3-48a3ω2σ)=0.Finally, by simultaneously solving (30) and (33) unknown parameters are determined for different values of A and σ as the second-order global error minimization method (SGEM) solution.


## 2.1. Max-Min Approach (MMA)

We can rewrite (1) in the following form:
(4)u¨+(1+(1-σ)u+σu2)u=0.By choosingu(t)=Acos⁡(ωt) as a trial function that satisfied the initial conditions, the maximum and minimum values of 1+(1-σ)u+σu2 can be calculated with Maple software as 1+A-σA+σA2 and -(σ2-6σ+1)/4σ, respectively, so we can write
(5)-σ2-6σ+14σ<ω2<1+A-σA+σA2.By using Chengtian interpolation [38], we have
(6)ω2=-m((σ2-6σ+1)/4σ)+n(1+A-σA+σA2)m+n=1-(1-k)(σ2-2σ+14σ)+k(A-σA+σA2),
where m, n are weighting factor, k=n/(m+n). Therefore the frequency can be approximated as
(7)ω=1+(1-k)(-σ2-2σ+14σ)+k(A-σA+σA2).
Then the solution of (1) can be written:
(8)u(t)=Acos⁡((1-(1-k)(σ2-2σ+14σ)dfdfdfdfdfff+k(A-σA+σA2)t(σ2-2σ+14σ))1/2).By using the approximate solution, (4) can be written in the following form:
(9)u¨+(1-(1-k)(σ2-2σ+14σ)+k(A-σA+σA2))u=-(1-k)(σ2-2σ+14σ)u+k(A-σA+σA2)u-(1-σ)u2-σu3.If by chance (8) is the exact solution of (1), then the right-hand side of (9) vanishes. According to [22], we set
(10)∫0T/4((k-1)(σ2-2σ+14σ)u+k(A-σA+σA2)udsdfdfff-(1-σ)u2-σu3(σ2-2σ+14σ))cos⁡(ωt)dt=0,
where T=2π/ω. By substituting (8) into (10), we obtain the following expression for k:
(11)k=133π-6πσ+3πσ2+32Aσ-32Aσ2+9πA2σ2π(1-2σ+σ2+4Aσ-4Aσ2+4A2σ2).Substituting (11) into (7) and after some simplification, we have
(12)ω=1+83πA-83πAσ+34A2σ.


## 2.2. Hamiltonian Approach

### 2.2.1. The First-Order Hamiltonian Approach (FHA)

The Hamiltonian of (1) is constructed as
(13)H=12u˙2+12u2+13(1-σ)u3+14σu4.Assume the first approximate solution of (1) as
(14)u(t)=Acos⁡(ωt).
Then integrating (13) with respect to time from 0 to T/4, we have
(15)H~(u)=∫0T/4(12u˙2+12u2+13(1-σ)u3+14σu4)dt.
Substituting (14) in (15) yields
(16)H~(u)=∫0T/4(12A2ω2sin2(ωt)+12A2cos⁡2(ωt)fdfdfffff+13(1-σ)A3cos⁡3(ωt)+14σA4cos⁡4(ωt))dt=∫0π/2(12A2ωsin2(t)+12ωA2cos⁡2(t)dsdsdd+13ω(1-σ)A3cos⁡3(t)+14ωσA4cos⁡4(t))dt=1576A2(72π+128A-128Aσ+27πA2σ+72πω2)ω.Set(17)∂∂A(∂H~∂(1/ω))=148A(-12πω2+12π+32A-32Aσ+9πA2σ)=0.Finally,(18)ωFHA=1+83πA-83πAσ+34A2σ.
The result of first-order Hamiltonian and max-min approaches is alike.


### 2.2.2. The Second-Order Hamiltonian Approach (SHA)

For the second-order Hamiltonian approach, we consider the following equation as the response of the system:(19)u=acos⁡(ωt)+bcos⁡(3ωt).
Whereas (19) must satisfy the initial condition we have
(20)A=a+b.
Substituting (19) into (15) yields
(21)H~(u)=∫0T/4[12ω2(asin(ωt)+3bsin(3ωt))2fdfdffdfff+12(acos⁡(ωt)+bcos⁡(3ωt))2fdfdffdfff+13(1-σ)(acos⁡⁡(ωt)+bcos⁡(3ωt))3fdfdffdfff+14σ(acos⁡(ωt)+bcos⁡(3ωt))4]dtH~(u)=∫0π/2[12ω(asin(t)+3bsin(3t))2fdfdffdfff+12ω(acos⁡(t)+bcos⁡(3t))2fdfdffdfff+13ω(1-σ)(acos⁡⁡(t)+bcos⁡(3t))3fdfdffdfff+14ωσ(acos⁡(t)+bcos⁡(3t))4]dtH~(u)=160480ω×(13440a3-4480b3-13440a3σdsdsd-8046a2bσ+2835πa4σ-31104ab2σdsdsd+8064a2b+2835πb4σ+4480b3σdsdsd+31104ab2+3780πa3bσ+11340πa2b2σdsdsd+7560πa2+7560πb2)+πω8(a2+9b2).Set(22)∂∂a(∂H~∂(1/ω))=-14πaω2+23a2-23a2σ-415abσ+316πa3σ-1835b2σ+415ab+1835b2+316πa2bσ+38πab2σ+14πa=0∂∂b(∂H~∂(1/ω))=-94πb2ω2-29b2-215a2σ-3635abσ+215a2+316πb3σ+29b2σ+3635ab+116πa3σ+38πa2bσ+14πb=0.
Since finding a relation between different parameters from (22) is not easy, the value of other parameters is determined in Table 1 for several values of σ.Table 1
Frequency-amplitude relationships for several values ofσ.
σ
a
b
ω
SHA
0.1
0.987791887
A
0.012208112
A
1
+
0.758436858
A
+
0.074106747
A
2
0.3
0.986559761
A
0.013440239
A
1
+
0.589470841
A
+
0.222057234
A
2
0.5
0.985302609
A
0.014697391
A
1
+
0.420742301
A
+
0.369650489
A
2
0.7
0.984019498
A
0.015980502
A
1
+
0.252257249
A
+
0.516878382
A
2
0.9
0.982709435
A
0.017290564
A
1
+
0.084021956
A
+
0.663732471
A
2
1
0.982043978
A
0.017956021
A
1
+
0.737016661
A
2


## 2.2.1. The First-Order Hamiltonian Approach (FHA)

The Hamiltonian of (1) is constructed as
(13)H=12u˙2+12u2+13(1-σ)u3+14σu4.Assume the first approximate solution of (1) as
(14)u(t)=Acos⁡(ωt).
Then integrating (13) with respect to time from 0 to T/4, we have
(15)H~(u)=∫0T/4(12u˙2+12u2+13(1-σ)u3+14σu4)dt.
Substituting (14) in (15) yields
(16)H~(u)=∫0T/4(12A2ω2sin2(ωt)+12A2cos⁡2(ωt)fdfdfffff+13(1-σ)A3cos⁡3(ωt)+14σA4cos⁡4(ωt))dt=∫0π/2(12A2ωsin2(t)+12ωA2cos⁡2(t)dsdsdd+13ω(1-σ)A3cos⁡3(t)+14ωσA4cos⁡4(t))dt=1576A2(72π+128A-128Aσ+27πA2σ+72πω2)ω.Set(17)∂∂A(∂H~∂(1/ω))=148A(-12πω2+12π+32A-32Aσ+9πA2σ)=0.Finally,(18)ωFHA=1+83πA-83πAσ+34A2σ.
The result of first-order Hamiltonian and max-min approaches is alike.


## 2.2.2. The Second-Order Hamiltonian Approach (SHA)

For the second-order Hamiltonian approach, we consider the following equation as the response of the system:(19)u=acos⁡(ωt)+bcos⁡(3ωt).
Whereas (19) must satisfy the initial condition we have
(20)A=a+b.
Substituting (19) into (15) yields
(21)H~(u)=∫0T/4[12ω2(asin(ωt)+3bsin(3ωt))2fdfdffdfff+12(acos⁡(ωt)+bcos⁡(3ωt))2fdfdffdfff+13(1-σ)(acos⁡⁡(ωt)+bcos⁡(3ωt))3fdfdffdfff+14σ(acos⁡(ωt)+bcos⁡(3ωt))4]dtH~(u)=∫0π/2[12ω(asin(t)+3bsin(3t))2fdfdffdfff+12ω(acos⁡(t)+bcos⁡(3t))2fdfdffdfff+13ω(1-σ)(acos⁡⁡(t)+bcos⁡(3t))3fdfdffdfff+14ωσ(acos⁡(t)+bcos⁡(3t))4]dtH~(u)=160480ω×(13440a3-4480b3-13440a3σdsdsd-8046a2bσ+2835πa4σ-31104ab2σdsdsd+8064a2b+2835πb4σ+4480b3σdsdsd+31104ab2+3780πa3bσ+11340πa2b2σdsdsd+7560πa2+7560πb2)+πω8(a2+9b2).Set(22)∂∂a(∂H~∂(1/ω))=-14πaω2+23a2-23a2σ-415abσ+316πa3σ-1835b2σ+415ab+1835b2+316πa2bσ+38πab2σ+14πa=0∂∂b(∂H~∂(1/ω))=-94πb2ω2-29b2-215a2σ-3635abσ+215a2+316πb3σ+29b2σ+3635ab+116πa3σ+38πa2bσ+14πb=0.
Since finding a relation between different parameters from (22) is not easy, the value of other parameters is determined in Table 1 for several values of σ.Table 1
Frequency-amplitude relationships for several values ofσ.
σ
a
b
ω
SHA
0.1
0.987791887
A
0.012208112
A
1
+
0.758436858
A
+
0.074106747
A
2
0.3
0.986559761
A
0.013440239
A
1
+
0.589470841
A
+
0.222057234
A
2
0.5
0.985302609
A
0.014697391
A
1
+
0.420742301
A
+
0.369650489
A
2
0.7
0.984019498
A
0.015980502
A
1
+
0.252257249
A
+
0.516878382
A
2
0.9
0.982709435
A
0.017290564
A
1
+
0.084021956
A
+
0.663732471
A
2
1
0.982043978
A
0.017956021
A
1
+
0.737016661
A
2


## 2.3. Global Error Minimization Method

### 2.3.1. The First-Order Global Error Minimization Method (FGEM)

Based on standard procedure of modified variational approach, the minimization problem is(23)MinimizeE(u˙,u,t)=∫0T(u¨+u+(1-σ)u2+σu3)2dt,T=2πω.
For first-order approximation, consider a trial function as follows:
(24)u(t)=Acos⁡(ωt).
Substituting (24) into (23) yields
(25)MinimizeE(u˙,u,t)=πA28ω(5A4σ2-12A2σω2+6A2σ2+8ω4dsdsdsdsd-16ω2+6A2+8).
The solution of (25) could be found through
(26)∂E(u˙,u,t)∂ω=0.
By some simplifications, the following equation is obtained:
(27)-24ω4+12A2ω2σ+16ω2+5A4σ2+6A2σ2+6A2+8=0.
One may find the first-order approximation by solving (27) as
(28)ωFGEM=16(+36A2+64)1/29A2σ+12fdfdfdffd+3(39A4σ2+36A2σ2+24A2σfdfdffdfdff+36A2+64)1/2)1/2.


### 2.3.2. The Second-Order Global Error Minimization Method (SGEM)

For the second-order approximation, consider a trial function as(29)u(t)=acos⁡(ωt)+bcos⁡(3ωt).
Whereas (29) must satisfy the initial condition, we have
(30)A=a+b.
Substituting (29) into (23) yields
(31)MinimizeE(u˙,u,t)=π8ω(5σ2a6-144b2ω2sdsdsdddd+45σ2a2b4+45σ2a4b2+24σ2a2b2sdsdsdddd+8σ2a3b+6σ2b4+5σ2b6+8a2ω4sdsdsdddd+30σ2a3b3+24a2b2-16a2ω2sdsdsdddd+8a2+8b2+6a4-240a2ω2σb2sdsdsdddd+15σ2a5b-12a4ω2σ+6σ2a4sdsdsdddd+648b2ω4+6b4-108b4ω2σsdsdsdddd+8a3b-48a3ω2σb).The solution of (31) could be found through
(32)∂E(u˙,u,t)∂ω=0,∂E(u˙,u,t)∂a=0,∂E(u˙,u,t)∂b=0,
which yield the following:
(33)∂E(u˙,u,t)∂ω=-π8ω2(5σ2a6+8a3b+45σ2a2b4dsdsdsdsdsd+8a2+8b2+5σ2b6+45σ2a4b2dsdsdsdsdsd-24a2ω4+16a2ω2+6b4dsdsdsdsdsd+6a4+24σ2a2b2+15σ2a5bdsdsdsdsdsd+30σ2a3b3-1944b2ω4dsdsdsdsdsd+144b2ω3+6σ2a4+6σ2b4dsdsdsdsdsd+48a3ω2σb+240a2ω2σb2dsdsdsdsdsd+8σ2a3b+12a4ω2σdsdsdsdsdsd+108b4ω2σ+24a2b2)=0∂E(u˙,u,t)∂a=π8ω(30σ2a5+90σ2ab4+180σ2a3b2gfgfgfg+24σ2a2b+16aω4+90σ2a2b3+48ab2gfgfgfg-32aω2+16a+24a3-480aω2σb2gfgfgfg+75σ2a4b-48a3ω2σ+24a2bgfgfgfg-144a2ω2σb+24σ2a3+48σ2ab2)=0∂E(u˙,u,t)∂b=π8ω(-288bω2+180σ2a2b3+90σ2a4bsdsdsdsd+48σ2a2b+8σ2a3+24σ2b3+30σ2b5sdsdsdsd+90σ2a3b2+48a2b+16b-480a2ω2σbsdsdsdsd+15σ2a5+1296bω4+24b3sdsdsdsd-432b3ω2σ+8a3-48a3ω2σ)=0.Finally, by simultaneously solving (30) and (33) unknown parameters are determined for different values of A and σ as the second-order global error minimization method (SGEM) solution.


## 2.3.1. The First-Order Global Error Minimization Method (FGEM)

Based on standard procedure of modified variational approach, the minimization problem is(23)MinimizeE(u˙,u,t)=∫0T(u¨+u+(1-σ)u2+σu3)2dt,T=2πω.
For first-order approximation, consider a trial function as follows:
(24)u(t)=Acos⁡(ωt).
Substituting (24) into (23) yields
(25)MinimizeE(u˙,u,t)=πA28ω(5A4σ2-12A2σω2+6A2σ2+8ω4dsdsdsdsd-16ω2+6A2+8).
The solution of (25) could be found through
(26)∂E(u˙,u,t)∂ω=0.
By some simplifications, the following equation is obtained:
(27)-24ω4+12A2ω2σ+16ω2+5A4σ2+6A2σ2+6A2+8=0.
One may find the first-order approximation by solving (27) as
(28)ωFGEM=16(+36A2+64)1/29A2σ+12fdfdfdffd+3(39A4σ2+36A2σ2+24A2σfdfdffdfdff+36A2+64)1/2)1/2.


## 2.3.2. The Second-Order Global Error Minimization Method (SGEM)

For the second-order approximation, consider a trial function as(29)u(t)=acos⁡(ωt)+bcos⁡(3ωt).
Whereas (29) must satisfy the initial condition, we have
(30)A=a+b.
Substituting (29) into (23) yields
(31)MinimizeE(u˙,u,t)=π8ω(5σ2a6-144b2ω2sdsdsdddd+45σ2a2b4+45σ2a4b2+24σ2a2b2sdsdsdddd+8σ2a3b+6σ2b4+5σ2b6+8a2ω4sdsdsdddd+30σ2a3b3+24a2b2-16a2ω2sdsdsdddd+8a2+8b2+6a4-240a2ω2σb2sdsdsdddd+15σ2a5b-12a4ω2σ+6σ2a4sdsdsdddd+648b2ω4+6b4-108b4ω2σsdsdsdddd+8a3b-48a3ω2σb).The solution of (31) could be found through
(32)∂E(u˙,u,t)∂ω=0,∂E(u˙,u,t)∂a=0,∂E(u˙,u,t)∂b=0,
which yield the following:
(33)∂E(u˙,u,t)∂ω=-π8ω2(5σ2a6+8a3b+45σ2a2b4dsdsdsdsdsd+8a2+8b2+5σ2b6+45σ2a4b2dsdsdsdsdsd-24a2ω4+16a2ω2+6b4dsdsdsdsdsd+6a4+24σ2a2b2+15σ2a5bdsdsdsdsdsd+30σ2a3b3-1944b2ω4dsdsdsdsdsd+144b2ω3+6σ2a4+6σ2b4dsdsdsdsdsd+48a3ω2σb+240a2ω2σb2dsdsdsdsdsd+8σ2a3b+12a4ω2σdsdsdsdsdsd+108b4ω2σ+24a2b2)=0∂E(u˙,u,t)∂a=π8ω(30σ2a5+90σ2ab4+180σ2a3b2gfgfgfg+24σ2a2b+16aω4+90σ2a2b3+48ab2gfgfgfg-32aω2+16a+24a3-480aω2σb2gfgfgfg+75σ2a4b-48a3ω2σ+24a2bgfgfgfg-144a2ω2σb+24σ2a3+48σ2ab2)=0∂E(u˙,u,t)∂b=π8ω(-288bω2+180σ2a2b3+90σ2a4bsdsdsdsd+48σ2a2b+8σ2a3+24σ2b3+30σ2b5sdsdsdsd+90σ2a3b2+48a2b+16b-480a2ω2σbsdsdsdsd+15σ2a5+1296bω4+24b3sdsdsdsd-432b3ω2σ+8a3-48a3ω2σ)=0.Finally, by simultaneously solving (30) and (33) unknown parameters are determined for different values of A and σ as the second-order global error minimization method (SGEM) solution.


## 3. Results

The relative errors of the max-min, first- and second-orders of the Hamiltonian, and the global error minimization methods are shown in Tables2, 3, and 4 for σ=0.5, σ=0.9, and σ=1, respectively. It is seen that the second-order global error minimization method results are closer to the exact ones than the other mentioned methods for large initial amplitudes.Table 2
Comparison between MMA, FHA, SHA, FGEM, and SGEM results with exact ones and HPM responses (σ=0.5).
A
T
exact
T
MMA
=
T
FHA (relative error %)
T
FGEM (relative error %)
T
SHA (relative error %)
T
SGEM (relative error %)
T
HPM [39] (relative error %)
0.01
6.283133
6.26977721(0.2125657)
6.28305278(0.0012767)
6.2698934(0.2107154)
6.28305278(0.0012767)
6.283132(0.0000159)
0.05
6.281851
6.21470929(1.0688203)
6.27987482(0.0314585)
6.21530801(1.0592895)
6.27987515(0.0314630)
6.281831(0.0003184)
0.1
6.277721
6.14291226(2.1474152)
6.26997826(0.1233367)
6.14414741(2.1277402)
6.26998356(0.1234043)
6.277642(0.0012584)
10
1.004172
0.9610403(4.2952461)
0.99830911(0.5838523)
0.96753201(3.6487768)
1.02917191(2.4291285)
0.98321194(2.087298)
20
0.5143428
0.4975256(3.2696323)
0.50439032(1.9349883)
0.50101412(2.5914005)
0.52050519(1.1839256)
0.50328477(2.14993)
50
0.2082766
0.2028184(2.6206285)
0.20235575(2.8427777)
0.20426634(1.9254484)
0.20888151(0.2895968)
0.20375994(2.168587)
100
0.1045213
0.1020148(2.3980756)
0.10122095(3.1575832)
0.10274684(1.6976947)
0.10448959(0.0303460)
0.1022518(2.171328)Table 3
Comparison between MMA, FHA, SHA, FGEM, and SGEM results with exact ones and HPM responses (σ=0.9).
A
T
exact
T
MMA
=
T
FHA (relative error %)
T
FGEM (relative error %)
T
SHA (relative error %)
T
SGEM (relative error %)
T
HPM [39] (relative error %)
0.01
6.2829757
6.28030856(0.0424502)
6.28297267(0.0000482)
6.28033909(0.0419642)
6.28297267(0.0000482)
6.2829757(0.00000005)
0.1
6.2622091
6.23585315(0.42087294)
6.26202447(0.0029481)
6.23646361(0.4111248)
6.26203439(0.0027898)
6.26220029(0.0001407)
10
0.7727359
0.75450185(2.3596739)
0.74930385(3.032348)
0.76075498(1.5504538)
0.77284969(0.0147255)
0.7564511(2.1074)
50
0.1561833
0.15271588(2.2200941)
0.15087022(3.4018226)
0.15400462(1.3949487)
0.15573888(0.2845491)
0.1527948(2.1696)
100
0.0781394
0.07642280(2.1968399)
0.07545104(3.4404631)
0.07706833(1.3707164)
0.07788792(0.3218291)
0.07644255(2.1716)Table 4
Comparison between MMA, FHA, SHA, FGEM, and SGEM results with exact ones (σ=1).
A
T
exact
T
MMA
=
T
FHA (relative error %)
T
FGEM (relative error %)
T
SHA (relative error %)
T
SGEM (relative error %)
0.01
6.282949702
6.282949704(0.000000032)
6.282949698(0.000000063)
6.282953782(0.000064937)
6.282949704(0.000000032)
0.1
6.259762302
6.259755078(0.000115404)
6.259750255(0.0001924513)
6.260158455(0.006328562)
6.259762297(0.00000008)
10
0.736288961
0.720730784(2.113053139)
0.7113585054(3.385960801)
0.7269670459(1.266067481)
0.7337722018(0.3418167776)
50
0.148282774
0.1450652702(2.169843275)
0.1431321874(3.473489510)
0.1463367452(1.312376851)
0.1477516413(0.3581890773)
100
0.074157585
0.0725471382(2.171654781)
0.07157966046(3.476278981)
0.0731832605(1.313856823)
0.07389156930(0.3587167786)The differences of obtained responses and velocities with exact ones are plotted in Figures1 and 2, respectively. It is found that the second-order global error minimization method illustrates a better accuracy than the aforementioned techniques.Figure 1
Difference between obtained responses and the numerical solution (A=50,  σ=0.5).Figure 2
Difference between obtained velocities and the numerical solution (A=50,  σ=0.5).


## 4. Conclusion

The Helmholtz-Duffing equation is investigated via the analytical approaches; the accuracy and validity of the obtained results have been examined by comparing to the exact ones and HPM solutions. The second-order of the global error minimization method achieved better approximate solutions for this equation. In present study, it is demonstrated that higher order of the modified variational approach is accurate and simple for solving asymmetric nonlinear conservative oscillatory systems.



---

*Source: 939623-2014-09-29.xml*