# An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

Authors: B. M. Ikramul Haque; M. M. Ayub Hossain
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.


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## Body

## 1. Introduction

Nonlinear systems are widespread around us. Nonlinear systems are widely involved in science, engineering, medical science, etc. So, research on the nonlinear systems has enriched science, engineering, medical science, etc. Research on nonlinear systems is complex and sensitive because most of the nonlinear systems suddenly change their characteristics due to some small changes of some valid parameters. The development of the theorems of dynamical systems has been derived by the modeling and formulating of nonlinear oscillators. Thus, nonlinear oscillators are one of the most important parts of nonlinear dynamical systems. Recently, many scientists have made significant improvement in finding a new mathematical tool which would be related to nonlinear dynamical systems whose understanding will rely not on analytic techniques but also on numerical and asymptotic methods. They set up many fruitful and potential methods to operate the nonlinear systems. There are many analytical methods to solve nonlinear oscillators such as perturbation [1–3]; standard and modified Poincare–Linstedt [4]; harmonic balance [5–7]; multiple scale [8]; homotopy perturbation [9–14]; modified He’s homotopy perturbation [15]; He’s frequency-amplitude formulation [16, 17]; cubication method [18, 19]; energy balance method [20]; He’s energy balance method [21]; Mickens iteration method [7, 22–25]; Hu’s iteration method [26]; Haque’s iteration method [27–29]; Haque’s extended iteration method [30–32]; variation iteration method [33]; homotopy analysis method [34]; finite element method and Akbari Ganji method [35]; and Taguchi method [36]. Besides, a few numbers of researchers have done work on the cube-root nonlinear oscillator using different methods such as the methods by Beléndez [9], Beléndez et al. [15], Ganji et al. [21], Mickens [25], Zhang [17], and Lim and Wu [6]. Among them, the proposed method is more easy, simple, and also valid for higher order approximation solutions. Perturbation method is the most widely utilized established method in which the nonlinear terms are considered as small. Mickens has developed a technique named harmonic balance (HB) method and further work has been done by Lim, Hu, Wu, and Alam, and so forth, for handling the oscillators in which nonlinear terms are strong. Mickens has developed another update technique named iteration method. The method of iteration is applicable for small as well as strong nonlinearities. Mickens has also developed the iteration method named extended iteration method; sometimes, this is faster than direct or simple iteration method. Further, the method of iteration and extended iteration has been developed by Lim, Hu, and Haque. Ikramul Haque et al. applied the proposed method to obtain the suitable periodic solution of nonlinear oscillators [30–32].The purpose of this article is to exhibit an extended iteration method which gives us a simple path line for obtaining the approximate angular frequencies and corresponding analytic solutions and a significantly improved results of the “cube-root truly nonlinear oscillator (TNL),” “ the inverse cube-root TNL oscillator,” and “the pendulum equation.”


## 2. The Method

The fundamental base of the method of extended iteration is to reconvey the prestage’s solution to obtain the present stage’s solution and its sequence of extensions. The most important concern is(i)
How to rearrange the preliminary shape of the given nonlinear oscillator so that it will help us to handle step by step iterations with simplification of a term including large number of harmonics.(ii)
How to reuse the obtained solutions for the case of harmonic terms so that there is no algebraic complexity.The outline of the presented procedure is as follows:(i)Step 1: considering the general form of oscillator by the following way:(1)x¨+fx=0.(ii)
Step 2: setting initial conditions as(2)x0=A,x˙0=0.(iii)
Step 3: making standard form by addingΩ2x on both sides, we have(3)x¨+Ω2x=Ω2x−fx≡Gx.(iv)
Step 4: formulizing the suitable iteration scheme as(4)x¨k+1+Ωk2xk+1=Gxk+Gxx0,Ωkxk−x0;k=1,2,3,….(v)
Step 5: setting initial supposition as(5)x0t=AcosΩ0t.(vi)
Step 6: setting iterated initial conditions as(6)xk+10=A,x˙k+10=0.(vii)
Step 7: executing the successive iteration fromk=1,2,3,….


## 3. Solution Procedure of Cube-Root TNL Oscillator

Consider the cube-root TNL oscillator [25](7)x¨+x1/3=0.Now, introducing the termΩ2x on equation (7), then we have the standard form:(8)x¨+Ω2x=Ω2x−x1/3=Gx,Ω,where Gx,Ω=Ω2x−x1/3.Therefore,Gx=Ω2−1/3x−2/3.Using equation (4), the iteration scheme of equation (8) is(9)x¨k+1+Ωk2xk+1=Ωk2x0−x01/3+Ωk2−13x0−2/3xk−x0.First Iteration Step. Putting k=0 in equation (9) and using initial supposition (5), we get(10)x¨1+Ω02x1=Ω02Acosθ−Acosθ1/3.Now, expanding the right hand side of equation (10), then it reduces to(11)x¨1+Ω02x1=Ω02A−1.1596A1/3cosθ+0.231919A1/3cos3θ−0.11596A1/3cos5θ.To avoid secular term from equation (11), we obtain(12)Ω0=1.076847A1/3=ΩCR0,say.Then, equation (11) is transformed into(13)x¨1+Ω02x1=0.231919A1/3cos3θ−0.11596A1/3cos5θ.The required solution of equation (13) is(14)x1t=A1.0208398−0.0249998cos3θ+0.00416cos5θ=xCR1t,say.This is known as the first iterated approximate solution of the oscillator (7).Second Iteration Step. For the second level of iteration, we have(15)x¨2+Ω12x2=Ω12x1−13x0−2/3x1−23x01/3.Now, substituting the value ofx0 and x1 from equations (5) and (14) into (15) and expanding the right side of equation (15), it reduces to(16)x¨2+Ω12x2=1.0208398AΩ12−1.169748A1/3cosθ+−0.0249998AΩ12+0.246675A1/3cos3θ+0.00416AΩ12−0.122544A1/3cos5θ.To eliminate secular term, we get(17)Ω1=1.070452A1/3=ΩCR1,say.Then, equation (16) is transferred into(18)x¨2+Ω12x2=0.2180285A1/3cos3θ−0.117777A1/3cos5θ.The required solution of equation (18) is(19)x2t=A1.019529cosθ−0.023812cos3θ+0.004283cos5θ=xCR2t,say.This is known as the second iterated approximate solution of the oscillator (7).Third Iteration Step. For the third level of iteration, we have(20)x¨3+Ω22x2=Ω22x2−13x0−2/3x2−23x01/3.Now, substituting the value ofx0 and x2 from equations (5) and (19) into (20) and expanding the right side of equation (20), it reduces to(21)x¨3+Ω22x3=1.019529AΩ22−1.169213A1/3cosθ−0.023812AΩ22−0.245996A1/3cos3θ+0.004283AΩ22−0.122395A1/3cos5θ.To eliminate secular term, we get(22)Ω2=1.070895A1/3=ΩCR2,say.Then, equation (21) is transferred into(23)x¨3+Ω22x3=0.220496A1/3cos3θ−0.117808A1/3cos5θ.The required solution of equation (23) is(24)x3t=A1.021153cosθ−0.025737cos3θ+0.004584cos5θ=xCR3t,say.This is known as the third iterated approximate solution of the oscillator (7).Fourth Iteration Step. For the fourth level of iteration, we have(25)x¨4+Ω32x4=Ω32x3−13x0−2/3x3−23x01/3.Now, substituting the value ofx0 and x3 from equations (5) and (24) into (25) and expanding the right side of equation (25), it reduces to(26)x¨4+Ω32x4=1.021153AΩ32−1.169943A1/3cosθ−0.025737AΩ32−0.247131A1/3cos3θ+0.004584AΩ32−0.122892A1/3cos5θ.To eliminate secular term, we get(27)Ω3=1.070378A1/3=ΩCR3,say.Equations (12), (17), (22), and (27) represent the first, second, third, and fourth approximate frequencies of the cube-root TNL oscillator (7), respectively.


## 4. Solution of Inverse Cube-Root TNL Oscillator

Consider the inverse cube-root TNL oscillator [25](28)x¨+x−1/3=0.Applying the same iteration procedure, we obtain(29)ΩICR0=1.194298A2/3ΩICR1=1.146861A2/3ΩICR2=1.158141A2/3ΩICR3=1.156090A2/3,(30)xICR1t=A1.052312cosθ−0.0249998cos3θ+0.00416cos5θxICR2t=A1.038587cosθ−0.052771cos3θ+0.014184cos5θxICR3t=A1.041074cosθ−0.054852cos3θ+0.013778cos5θ.Equation (29) represents the first, second, third, and fourth approximate frequencies of the inverse cube-root TNL oscillator (28) which are denoted by ΩICR0, ΩICR1, ΩICR2, and ΩICR3, respectively. Also, equation (30) represents the first, second, and third approximate analytic solutions of the inverse cube-root TNL oscillator (28) which are denoted by xICR0, xICR1 and xICR2, respectively.


## 5. Equation of Pendulum

Consider the following equation as a pendulum equation where friction is neglected, as shown in Figure1 [13].where K2=g/l, g is the gravitational acceleration, l is the length of the pendulum, and θ is the angle from the vertical to the pendulum.Figure 1
The simple pendulum.(31)θ¨+K2sinθ=0,θ0=A,θ˙0=0,To solve the pendulum equation, we considersinθ≈θ−1/6θ3+1/120θ5, then equation (30) reduces to(32)θ¨+K2θ−16θ3+1120θ5=0,withθ0=A,θ0=0.The iteration form of equation (32) is(33)θ¨+Ω2x=Ω2θ−K2θ−16θ3+1120θ5=Hθ,Ω,where(34)Hθ,Ω=Ω2θ−K2θ−16θ3+1120θ5,Hθ=∂H∂θ=Ω2−K21−12θ2+124θ4.Applying the approximate method (6), we get(35)θ¨k+1+Ωk2θk+1=Ωk2θ0−K2θ0−16θ03+1120θ05+Ωk2−K21+12θ02−124θ04θk−θ0.For the first iteration, we get(36)θ¨1+Ω02θ1=Ω02Acosθ−K2Acosθ−16Acosθ3+1120Acosθ5.Now, expanding the right side of equation (36) using a Fourier series, we get(37)θ¨1+Ω02θ1=Ω02A−K2A−18A3+1192A5cosθ−K2−124A3+1384A5cos3θ−K211920A5cos5θ.To avoid secular term, we obtain(38)Ω0=K1−18A2+1192A41/2.This is the first approximate frequency of the oscillator.After simplification in equation (37), we have(39)θ¨1+Ω02θ1=−K2−124A3+1384A5cos3θ−K211920A5cos5θ.Thus, the first iterated complete solution of the oscillator (31) is(40)θ1t=f1cosθ+f2cos3θ+f3cos5θ,where(41)f1=A−23/192A3+7/1440A51−1/8A2+1/192A4,f2=−1/24A3+1/384A581−1/8A2+1/192A4,f3=1/1920A5241−1/8A2+1/192A4.For the second iteration, we get(42)θ¨2+Ω12θ2=Ω12θ1−K2−12θ02θ1+124θ04θ1+13θ03−130θ05+θ1,(43)θ¨2+Ω12θ2=Ω12f1cosθ+f2cosθ+f3cosθ−K2f1cosθ+f2cos3θ+f3cos5θ+12K2A2cosθ2f1cosθ+f2cos3θ+f3cos5θ−13K2A3cosθ3−124K2A4cosθ4f1cosθ+f2cos3θ+f3cos5θ+130K2A5cosθ5.Now, expanding the right side of equation (43) using truncation principle, we get(44)θ¨2+Ω12x2=Ω12f1−K2f1−34−13A3+12A2f1−18A2f2−58130A5−124A4f1+5384A4f2+1384A4f3cosθ+Ω12f2−K2f2+14−12A2f1+13A3−14A2f2−18A2f3+516124A4f1−130A5+164A4f2+196A4f3cos3θ+Ω12f3−K2f3−18A2f2−14A2f3+116124A4f1−130A5+196A4f2+164A4f3cos5θ.To avoid secular term, we obtain(45)Ω1=K1−38A2+5192A4+−18A2+5384A4f2f1+1384A4f3f1−1f1−14A3+148A51/2.This is the second approximate frequency of the oscillator.After simplification in equation (44), we have(46)x¨2+Ω12x2=λ1cos3θ+λ2cos5θ,where(47)λ1=Ω12f2−K2f2+14−12A2f1+13A3−14A2f2−18A2f3+516124A4f1−130A5+164A4f2+196A4f3,λ2=Ω12f3−K2f3−18A2f2−14A2f3+116124A4f1−130A5+196A4f2+164A4f3.Thus, the second iterated complete solution of the oscillator (31) is(48)θ2t=24Ω12A+3λ1+λ224Ω12cosθ−λ18Ω12cos3θ−λ224Ω12cosθ.


## 6. Results and Discussion

To obtain approximate solutions of cube-root truly nonlinear oscillator, inverse cube-root truly nonlinear oscillator, and pendulum equation, we have applied an extended iteration method. Here, we have calculated the first, second, third, and fourth approximate frequencies of both oscillators and the first and second approximate frequencies of pendulum equation. All the results are given in Tables1 and 2 and the frequency-amplitude relationships are given in Tables 3–5. To show the validity of the obtained solutions, we have compared these with the existing results determined by Mickens iteration method [25], Mickens HB method [25], He’s amplitude-frequency formulation [34], modified He’s homotopy perturbation method [15], improved harmonic balance method [6] and homotopy perturbation method [9] in Table 1, He’s energy balanced method [21] in Table 3, Mickens iteration method [25], Mickens HB method [25] in Table 2, and homotopy perturbation method [13] in Table 5. The comparison between the third-order approximate solution of equation (8) for A=10 and A=100 together with the corresponding exact solution are presented in Figures 2 and 3. The comparison between the third-order approximate solution of equation (28) for A=1 together with the corresponding exact solution are presented in Figure 4 and the comparison between the second-order approximate solution of equation (31) for A=1 and K=1 together with the corresponding exact solution are presented in Figure 5.Table 1
Comparison of the approximate frequencies with the exact frequencyΩCRe of x¨+x1/3=0.
ΩexactA=ΩCReA=1.070451/A1/3AmplitudeAΩCR0Error (%)ΩCR1Error (%)ΩCR2Error (%)ΩCR3Error (%)Mickens iteration method [25]1.0491151/A1/32.01.041424/A1/32.7——Mickens harmonic balance method [25]1.049115/A1/32.01.063410/A1/30.7——Zhang using He’s amplitude-frequency formulation [17]1.04912/A1/32.0—Beléndez et al. using modified He’s homotopy perturbation method [15]1.07685/A1/30.61.06922/A1/30.171.07078/A1/30.024—Lim and Wu using improved harmonic balance method [6]1.07685/A1/30.61.06928/A1/30.17—Beléndez using homotopy perturbation method [9]1.07685/A1/30.61.06991/A1/30.05Adopted method1.076847/A1/30.59751.070452/A1/30.001.070895/A1/30.041.070378/A1/30.00Table 2
Comparison of the approximate frequencies with exact frequencyΩICRe of x¨+x−1/3=0.
ΩexactA=ΩICReA=1.154700/A2/3AmplitudeAΩICR0Error (%)ΩICR1Error (%)ΩICR2Error (%)ΩICR3Error (%)Mickens iteration method [25]1.08148/A2/36.31.07634/A2/36.780.988591/A2/314.38—Mickens HB method [25]1.31329/A2/313.71.18824/A2/32.9——Adopted method1.194298/A2/33.431.146861/A2/30.671.158141/A2/30.291.156090/A2/30.12Table 3
Comparison of the approximate frequency-amplitude relations with exact and existing results ofx¨+x1/3=0.
AΩCR0Er (%)ΩCR1Er (%)ΩCR2Er (%)ΩCR3Er (%)Ω7HEBMEr (%)Ω7ex0.12.3199923360.59732.3062208960.00022.3071303280.03962.3070820810.03752.2699588741.5722.3062167680.51.3567397490.59731.3486861620.00021.3492180000.03961.3491897850.03751.3274800021.5721.34868374811.0768450520.59731.0704529160.00021.0708750360.03961.0708526420.03751.0536215761.5721.07045100050.6297428060.59730.6260046630.00020.6262515200.03960.6262384240.03750.61616163531.5720.626003542100.4998271970.59730.4968602300.00020.4970561610.03960.4970457660.03750.48904781411.5720.496859340500.2923007180.59730.2905656250.00020.2906802060.03960.2906741270.03750.28599689651.5720.2905651051000.2319992330.59730.2306220890.00020.2307130320.03960.2307082080.03750.22699588741.5720.2306216765000.1356739740.59730.1348686160.00020.1349218000.03960.1349189780.03750.13274800021.5720.13486837410000.1076845050.59730.1070452910.00020.1070875030.03960.1070852640.03750.10536215761.5720.107045100Note. Here, Ω7HEBM represents the approximation frequency obtained by Ganji et al. [21].Table 4
Comparison of the approximate frequency-amplitude relations with exact results ofx¨+x−1/3=0.
AΩICR0Er (%)ΩICR0Er (%)ΩICR0Er (%)ΩICR0Er (%)Ωex0.15.543443.45.323260.685.375610.295.366090.125.359640.51.895833.41.820530.681.838430.291.835180.121.8329711.194303.41.146860.681.158140.291.156090.121.1547050.408443.40.392220.680.396080.290.395380.120.39490100.257303.40.247080.680.249510.290.249070.120.24877500.087803.40.084500.680.085330.290.085180.120.085081000.055433.40.053230.680.053760.290.053660.120.053605000.018963.40.018210.680.018380.290.018350.120.0183310000.011943.40.011470.680.011580.290.011560.120.01155Table 5
Comparison of the approximate frequency-amplitude relations with exact and existing results ofθ¨+K2sinθ=0.
AKΩ28exΩ0Er (%)Ω1Er (%)Ω28HPMEr (%)0.121.998751.998750.00001.998750.00001.998750.00000.543.9375793.9376650.00223.9375830.00013.9376650.0022110.9377920.9381940.04290.9378530.00650.9381940.04291.510.8606080.8632020.30130.8617810.1362990.8632020.3013210.75250.7637631.49680.7474820.66680.7637631.4968Note. Here, Ω28HPM represents the approximation frequency obtained by Bayat et al. [13].Figure 2
A comparison between the third-order approximate solutions ofx¨+x1/3=0 for A=10 together with the corresponding exact solution.Figure 3
A comparison between the third-order approximate solutions ofx¨+x1/3=0 for A=100 together with the corresponding exact solution.Figure 4
A comparison between the third-order approximate solutions ofx¨+x−1/3=0 for A=1 together with the corresponding exact solution.Figure 5
A comparison between the second-order approximate solutions ofθ¨+K2sinθ=0 for A=1.5 and K=1 together with the corresponding exact solution.Here, the method of Mickens’ iteration [25] is divergent, as error pattern shows an increasing phenomenon for corresponding iteration steps. The process of Mickens’ HB method [13] is too much complicated for higher order approximations for its algebraic complexity, that is why the process was terminated up to second approximation and He could not determine the better solutions. Not only that, there is no flexibility to take any ordered harmonic terms in each iterated step. The improvement of solutions depends on only starting consideration. It is a disadvantage of this method.For both of the oscillators, we can say, the adopted method is convergent because the solution yields lesser error in each forward iterative step:(1)
For cube-root TNL oscillatorx¨+x1/3=0,(49)ΩCR3−ΩCRe=1.070378−1.070451<ε.(2)
For inverse cube-root TNL oscillatorx¨+x−1/3=0,(50)ΩIC3−ΩICRe=1.156090−1.154700<ε.(3)
For pendulum equationθ¨+Ω2sinθ=0,(51)ω1−ωex=0.937853−0.937792<ε,fora=1,Ω=1,(i)
whereε is taken to be of very small positive quantity.(ii)
Also, the consistency of the adopted method is acquired because(4)
For cube-root TNL oscillatorx¨+x1/3=0,(52)ΩCR3−ΩCRe=1.070378−1.070451≈0.(5)
For inverse cube-root TNL oscillatorx¨+x−1/3=0,(53)ΩICR3−ΩICRe=1.156090−1.154700≈0.(6)
For pendulum equationθ¨+Ω2sinθ=0,(54)ω1−ωex=0.937853−0.937792≈0,fora=1,Ω=1.


## 7. Conclusion

After reflecting on each point of view about all the methods compared in Tables1–3 and 5, we conclude that the adopted method is significantly better than each comparable stages which have been shown by other methods. Finally, we summarized:(i)
The proposed method is a powerful technique for analyzing random oscillations. The technique is also easy and useful to obtain approximate frequencies and the corresponding periodic solutions of strongly nonlinear oscillator.(ii)
The proposed method gives high validity for both small and large initial amplitudes of oscillations and better results of the approximate frequencies and corresponding periodic solutions than those obtained by other existing results.(iii)
The percentage errors for the fourth approximate frequencies of cube-root truly nonlinear oscillator and inverse cube-root truly nonlinear oscillator are 0.006 and 0.12, respectively. Also, the percentage error for the second approximate frequency (for amplitude = 1) of equation of pendulum is 0.0065.(iv)
It is investigated that most of the researchers have used the procedure to modify the method to improve the solutions in the iteration method, but we have paid attention to rearranging the leading oscillators with their own merit and choosing suitable harmonic terms from trigonometric expansion. It has been concluded that these two are also imperative matter for obtaining improved solutions.(v)
The proposed method is also convergent for the considered equations.



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