An approximation technique for solving anti-symmetric constant force and anti-symmetric quadratic nonlinear oscillators

Abstract

Earlier, an approximation technique was presented for solving strong nonlinear oscillator equations. Due to arising algebraic complexities, the method fails to determine suitable solutions of some nonlinear oscillators such as quadratic oscillators, the cubical Duffing oscillator of softening springs, and pendulum equations. Then, rearranging an algebraic equation related to amplitude and frequency, the method covers the noted problems. In this article, the latter technique is applied to handling anti-symmetric constant force oscillators and anti-symmetric quadratic nonlinear oscillators.

Introduction

Many analytical approximation methods such as the perturbation method, the homotopy perturbation method, the variational iteration method, the harmonic balance method, and the energy balance method were developed to find approximate solutions of nonlinear oscillators. The classical perturbation methods^1–4 were developed to investigate weakly nonlinear problems modeled by $\ddot{x}+{\omega }_0^{2}x+\epsilon f\left(x,\dot{x}\right)=0,$ (1)where $\epsilon \ll {\omega }_0^2$ . Herein, ${\omega }_0$ is known as unperturbed frequency, $\epsilon$ is a constant, and $f$ is a nonlinear function. Then, classical perturbation methods^5–8 were modified to overcome this limitation and applied to solve strong nonlinear problems where $\epsilon =\mathrm{{\rm O}}\left(1\right)$ . On the other hand, some analytical techniques such as the homotopy perturbation method (HPM),^9–15 variational iteration method (VIM),^16–19 harmonic balance method (HBM),^20–24 energy balance method (EBM),^25–28 and frequency–amplitude formula^29–31 were presented to solve equation (1) for $\epsilon =\mathrm{{\rm O}}\left(1\right)$ . Most of the above methods were developed based on trigonometric functions.

The differential transform method (DTM) is a semi-analytical method, in which the solution is chosen as a truncated Maclaurin series. The method was first introduced by Zhou ³² and applied to analyze linear and nonlinear electric circuit problems. Then, some authors applied the DTM^33–37 to solve linear and nonlinear differential equations. It provides satisfactory results within a small region only and diverges for large regions. In order to improve the accuracy of the DTM, an alternative technique was used which modifies the series solution for nonlinear oscillatory systems. A modification of the DTM, based on the use of Pade’ approximants, was proposed by El-Shahed ³⁸ for solving different types of nonlinear problems. Further, several authors^39–43 modified the DTM by applying the Laplace–Pade’ resummation method. Though the solution obtained by the MDTM is better than the DTM, the utilization of MDTM is a laborious task.

Recently, Alam et al. ⁴⁴ presented an analytical technique for solving equation (1). But the method ⁴⁴ is useless for handling some nonlinear oscillators such as quadratic oscillator (in region $x<0$ ), cubical oscillators of softening springs (i.e., Duffing equation, $\ddot{x}+{\omega }_0^{2}x+\epsilon x^3=0$ , $\epsilon <0$ ), and pendulum equation. The approximate solutions of these oscillators were obtained by Huq et al. ⁴⁵ by rearranging only an algebraic equation related to amplitude and frequency. The aim of this article is to find the approximate solutions for anti-symmetric constant force, anti-symmetric quadratic nonlinear, and Duffing (cubical) oscillators on the basis of the latter approach. ⁴⁵

The method

By variable transformation, $\tau =\omega t$ , equation (1) with initial conditions $[x\left(0\right)=0,\dot{x}\left(0\right)=b$ ] becomes: ${\omega }^2{x}^{″}+{\omega }_0^{2}x+\epsilon f\left(x,\omega x^{\prime }\right)=0;\hspace{0.17em}x\left(0\right)=0,x^{\prime }\left(0\right)=\frac{b}{\omega }=\beta ,$ (2)where primes denote differentiation with respect to $\tau$ . In general, initial conditions are chosen as $x\left(0\right)=A,\dot{x}\left(0\right)=0$ , but in Ref. 44 the former conditions were considered.

A polynomial type of solution of $\phi =\tau \left(\pi -\tau \right)$ of equation (2) was found in a form ⁴⁴ : $x\left(\tau \right)=B_1\left(\beta ,{\omega }_0,\epsilon ,\omega \right)\phi \left(\tau \right)+B_2\left(\beta ,{\omega }_0,\epsilon ,\omega \right){\phi }^2\left(\tau \right)+B_3\left(\beta ,{\omega }_0,\epsilon ,\omega \right){\phi }^3\left(\tau \right)+\dots ,$ (3)when $f\left(-x,-\dot{x}\right)=-f\left(x,\dot{x}\right)$ . Solution equation (3) is also valid when $f\left(-x,-\dot{x}\right)=f\left(x,\dot{x}\right)$ . It is obvious that equation (3) satisfies the first initial condition given in equation (2). Usually, solution equation (3) is used for a semi-period of oscillation, $0<\tau \le \pi$ . For the next semi-period, $\phi$ and $\beta$ are changed, respectively, by $\phi =\left(\tau -\pi \right)\left(2\pi -\tau \right)$ , $\pi \le \tau \le 2\pi$ , and $–\beta$ . Substituting equation (3) into equation (2) and then equating the coefficients of ${{\phi }^0,\phi }^1,{\phi }^2,{\phi }^3,\cdots$ , a set of linear algebraic equations of $B_i,i=1,2,3,\cdots$ are obtained whose solution provides the noted unknown coefficients. The value of $x\left(\tau \right)$ has the maximum value (known as amplitude of oscillation, say $a>0$ ) at $\tau =\frac{\pi }{2}$ . Therefore, a relation between amplitude and frequency becomes: $a=\frac{B_1\left(\beta ,{\omega }_0,\epsilon ,\omega \right){\pi }^2}{4}+\frac{B_2\left(\beta ,{\omega }_0,\epsilon ,\omega \right){\pi }^4}{16}+\frac{B_3\left(\beta ,{\omega }_0,\epsilon ,\omega \right){\pi }^6}{64}+\cdots .$ (4)

It is noted that equation (4) contains two unknown constants $a$ and $\omega$ . So, it requires another equation involving $a$ and $\omega$ to determine them. A lot of important nonlinear oscillators follow the relation: ${{\omega }^2{x}^{\prime }}^2+G\left(x\right)={\omega }^2{\beta }^2,$ (5)

It has already been mentioned that $x\left(\tau \right)$ has a maximum at $\tau =\frac{\pi }{2}$ . So, $x^{\prime }$ vanishes at this limit and equation (5) becomes: $G\left(a\right)={\omega }^2{\beta }^2.$ (6)Thus, $a$ and $\omega$ can be found solving equations (4) and (6).

Examples

Anti-symmetric constant force oscillator

Let us consider the anti-symmetric constant force oscillator $\ddot{x}+sgn\left(x\right)=0,x\left(0\right)=0,\dot{x}\left(0\right)=b,$ (7)where $sgn\left(x\right)$ is defined as $sgn\left(x\right)=\{\begin{array}{l}+1,\\ -1,\end{array}\begin{array}{l}x>0\\ x<0\end{array}$ .

When $sgn\left(x\right)=+1$ , substituting solution equation (3) into equation (7) and then equating the coefficients of ${{\phi }^0,\phi }^1,{\phi }^2,{\phi }^3,\cdots$ , the following algebraic equations are derived: ${1-(2B}_1-2B_2{\pi }^2){\omega }^2=0,$ ${6(-2B}_2+B_3{\pi }^2){\omega }^2=0,$ ${6(-5B}_3+{2B}_4{\pi }^2){\omega }^2=0,$ ${\hspace{0.17em}4(-14B}_4+{5B}_5{\pi }^2){\omega }^2=0,\dots$ (8)

Solving equation (8), the unknown coefficients are derived as $B_2=\frac{-{1+2B}_1{\omega }^2}{2{\pi }^2{\omega }^2},B_3=\frac{-{1+2B}_1{\omega }^2}{{\pi }^4{\omega }^2},B_4=\frac{5\left(-{1+2B}_1{\omega }^2\right)}{2{\pi }^6{\omega }^2},B_5=\frac{7\left(-{1+2B}_1{\omega }^2\right)}{{\pi }^8{\omega }^2},\cdots$ (9)Now, differentiating equation (3) with respect to $\tau$ and then substituting $\tau =0$ (i.e., by utilization of the second initial condition), it becomes: $x^{\prime }\left(0\right)=\beta =\pi B_1$ (10)Now from equations (9) and (10), all unknown coefficients can be written in terms of $\beta$ as ${B_1=\frac{\beta }{\pi },B}_2=\frac{2\beta {\omega }^2-\pi }{2{\pi }^3{\omega }^2},{\hspace{0.17em}B}_3=\frac{2\beta {\omega }^2-\pi }{{\pi }^5{\omega }^2},B_4=\frac{5\left(2\beta {\omega }^2-\pi \right)}{2{\pi }^7{\omega }^2},{\hspace{0.17em}B}_5=\frac{7\left(2\beta {\omega }^2-\pi \right)}{{\pi }^9{\omega }^2},\cdots$ (11)

Substituting these values of unknown coefficients (up to ${\mathrm{B}}_{10}$ ) in equation (4), it becomes: $a=\frac{215955\mathrm{\beta }\pi }{524288}-\frac{84883{\pi }^2}{1048576{\omega }^2}$ (12)

For this, oscillator equation (6) becomes: $2a={{\omega }^2\beta }^2$ (13)and it can be written as $\beta =\frac{\eta }{\mathrm{\omega }},\text{where}\hspace{0.17em}\eta =\sqrt{\left(2a\right)}$ (14)

By substitution of $\omega =\eta /\beta$ in equation (12), it becomes: $\beta =a/\left(\frac{215955\pi }{524288}-\frac{84883{\beta \pi }^2}{1048576{\eta }^2}\right)$ (15)

When $a$ is given, the value of $\beta$ is obtained from equation (15) via an iterative procedure, starting with initial ${\beta }_0=a$ . Therefore, the solution of anti-symmetric constant force oscillator equation (7) is obtained from equation (3) by substituting the values of $B_1,B_2,\cdots ,B_{10}$ from equation (11).

When $sgn\left(x\right)=-1$ , applying a similar procedure, the corresponding equation (15) becomes: $\beta =a/\left(\frac{215955\pi }{524288}+\frac{84883{\beta \pi }^2}{1048576{\eta }^2}\right)$ (16)

Anti-symmetric quadratic oscillator

Let us consider the anti-symmetric quadratic oscillator $\ddot{x}+sgn\left(x\right)x^2=0,x\left(0\right)=0,\dot{x}\left(0\right)=b,$ (17)where $sgn\left(x\right)$ is defined as $sgn\left(x\right)=\{\begin{array}{l}+1,\\ -1,\end{array}\begin{array}{l}x>0\\ x<0\end{array}$ .

When $sgn\left(x\right)=+1$ , the unknown coefficients for this oscillator have been obtained: ${B_1=\frac{\beta }{\pi },B}_2=\frac{\beta }{{\pi }^3},{\hspace{0.17em}B}_3=\frac{2\beta }{{\pi }^5},B_4=\frac{5\beta }{{\pi }^7}-\frac{{\beta }^2}{12{\pi }^4{\omega }^2},{\hspace{0.17em}B}_5=\frac{14\beta }{{\pi }^9}-\frac{{\beta }^2}{3{\pi }^6{\omega }^2},\cdots$ (18)Substituting these values of unknown coefficients (up to ${\mathrm{B}}_{10}$ ) in equation (4), it becomes: $a=\frac{215955\mathrm{\beta }\pi }{524288}-\frac{{\mathrm{\beta }}^4{\pi }^{10}}{6341787648{\omega }^6}+\frac{235{\mathrm{\beta }}^8{\pi }^7}{132120576{\omega }^4}-\frac{10889{\mathrm{\beta }}^2{\pi }^4}{6291456{\omega }^2}$ (19)

For this, oscillator equation (6) becomes: ${\frac{2a^3}{3}=\omega }^2{\beta }^2$ (20)and it can be written as $\beta =\frac{a\eta }{\mathrm{\omega }},\text{where}\hspace{0.17em}\eta =\sqrt{\left(2a/3\right)}$ (21)

By substitution of $\omega =a\hspace{0.17em}\eta /\beta$ in equation (19), it becomes: $\beta =a/\left(\frac{215955\pi }{524288}-\frac{{\beta }^9{\pi }^{10}}{6341787648{\hspace{0.17em}a}^6{\eta }^6}+\frac{235{\beta }^6{\pi }^7}{132120576{\hspace{0.17em}a}^4{\eta }^4}-\frac{10889{\beta }^3{\pi }^4}{6291456\hspace{0.17em}{a}^2{\eta }^2}\right)$ (22)

When $a$ is given, the value of $\beta$ is obtained from equation (22) via an iterative procedure, starting with initial ${\beta }_0=a$ .Therefore, the solution of anti-symmetric quadratic oscillator (equation (17)) is obtained from equation (3) by substituting the values of $B_1,B_2,\cdots ,B_{10}$ from equation (18).

When $sgn\left(x\right)=-1$ , applying a similar procedure, the corresponding equation (22) becomes: $\beta =a/\left(\frac{215955\pi }{524288}+\frac{{\beta }^9{\pi }^{10}}{6341787648{\hspace{0.17em}a}^6{\eta }^6}+\frac{235{\beta }^6{\pi }^7}{132120576{\hspace{0.17em}a}^4{\eta }^4}+\frac{10889{\beta }^3{\pi }^4}{6291456\hspace{0.17em}{a}^2{\eta }^2}\right)$ (23)

Duffing (cubical) oscillator

Let us consider the Duffing oscillator $\ddot{x}+x+\epsilon x^3=0,x\left(0\right)=0,\dot{x}\left(0\right)=b.$ (24)

For this oscillator, unknown coefficients have been obtained: ${B_1=\frac{\beta }{\pi },B}_2=\frac{\beta }{{\pi }^3},{\hspace{0.17em}B}_3=-\frac{\beta }{6{\pi }^3{\omega }^2}+\frac{2\beta }{{\pi }^5},B_4=\frac{5\beta }{{\pi }^7}-\frac{\beta }{2{\pi }^5{\omega }^2},\cdots$ (25)

Substituting these values of unknown coefficients (up to ${\mathrm{B}}_{10}$ ) in equation (4), it becomes: $a=\frac{215955\mathrm{\beta }\pi }{524288}-\frac{15751\mathrm{\beta }{\pi }^3}{1572864{\omega }^2}+\frac{459\mathrm{\beta }{\pi }^5}{8388608{\omega }^4}-\frac{1377{\mathrm{\beta }}^3{\pi }^5\epsilon }{4194304{\omega }^2}+\frac{517{\mathrm{\beta }}^3{\pi }^7\epsilon }{88080384{\omega }^4}-\frac{47\mathrm{\beta }{\pi }^7}{528482304{\omega }^6}+\frac{13\mathrm{\beta }{\pi }^9}{380507258880{\omega }^8}-\frac{221{\mathrm{\beta }}^3{\pi }^9\epsilon }{10569646080{\omega }^6}+\frac{13{\mathrm{\beta }}^5{\pi }^9{\epsilon }^2}{503316480{\omega }^4}$ (26)

For this oscillator, equation (6) becomes: ${\omega }^2{\beta }^2=a^2+\frac{\epsilon a^4}{2}$ (27)and it can be written as $\beta =\frac{a\eta }{\mathrm{\omega }},\text{where}\hspace{0.17em}\eta =\sqrt{\left(1+\frac{\epsilon a^2}{2}\right)}$ (28)

By substitution of $\omega =a\hspace{0.17em}\eta /\beta$ in equation (26), it becomes: $\beta =a/\left(\frac{215955\pi }{524288}+\frac{13{\beta }^4{\pi }^9{\epsilon }^2}{503316480a^4{\eta }^4}-\frac{1377{\beta }^2{\pi }^5\epsilon }{4194304a^2{\eta }^2}+\frac{517{\beta }^2{\pi }^7\epsilon }{88080384a^4{\eta }^4}-\frac{15751{\pi }^3}{1572864a^2{\eta }^2}-\frac{221{\beta }^2{\pi }^9\epsilon }{10569646080a^6{\eta }^6}+\frac{459{\pi }^5}{8388608a^4{\eta }^4}-\frac{47{\pi }^7}{528482304a^6{\eta }^6}+\frac{13{\pi }^9}{380507258880a^8{\eta }^8}\right)$ (29)

When $a$ is given, the value of $\beta$ is obtained from equation (29) via an iterative procedure, starting with initial ${\beta }_0=a$ . Therefore, the solution of the Duffing oscillator (equation (24)) is obtained from equation (3) by substituting the values of $B_1,B_2,\cdots ,B_{10}$ from equation (25).

Results and discussion

In this section, the approximate periods and the solutions of anti-symmetric constant force oscillators, anti-symmetric quadratic nonlinear oscillators, and the Duffing (cubical) oscillator obtained by use of the present method are compared with exact and an effective improvement of the homotopy perturbation method. ¹⁵ First, the periods of the anti-symmetric constant force oscillator and anti-symmetric quadratic nonlinear oscillator for different initial conditions are presented, respectively, in Tables 1 and 2. From Table 1, it is clear that the present method provides better results than that of an effective improvement of the homotopy perturbation method. ¹⁵ From Table 2, it is also clear that the present method provides better results than that of an effective improvement of the homotopy perturbation method ¹⁵ when more terms are used.

Table 1.

Comparison of the present approximate periods with exact and other existing periods of $\ddot{x}+sgn\left(x\right)=0$ , $x\left(0\right)=A,\dot{x}\left(0\right)=0$ .

A	Exact	He et al. 15	Present (using up to $B_2$ )
	$T_e$	Er (%)	Er (%)
1.0	5.65685424	5.674400920.310184	5.656854250.00000016
5.0	12.64911056	12.688346190.310185	12.649110640.00000063
10.0	17.88854371	17.944031260.310185	17.888543820.00000063
20.0	25.29822112	25.376692370.310185	25.298221280.00000063

where Er (%) represents the absolute percentage error.

Table 2.

Comparison of the present approximate periods with exact and other existing periods of $\ddot{x}+sgn\left(x\right)x^2=0$ , $x\left(0\right)=A,\dot{x}\left(0\right)=0$ .

A	Exact	He et al. 15	Present (using up to $B_6$ )	Present (using up to $B_8$ )
	$T_e$	Er (%)	Er (%)	Er (%)
1.0	6.86926136	6.871257150.0290539	6.878157440.129506	6.870086330.0120097
5.0	3.07202707	3.072919610.0290539	3.076005520.129506	3.072396010.0120097
10.0	2.17225117	2.172882300.0290539	2.175064360.129506	2.172512050.0120097
20.0	1.53601354	1.536459810.0290539	1.538002760.129506	1.536198010.0120097

where Er (%) represents the absolute percentage error.

Second, the solutions of the anti-symmetric constant force oscillator, anti-symmetric quadratic nonlinear oscillator, and the Duffing (cubical) oscillator for different initial conditions are presented, respectively, in Figures 1–6 together with an effective improvement of the homotopy perturbation method ¹⁵ and numerical solutions (4th-order Runge–Kutta method). From these figures, it is clear that the present solutions agree well with the corresponding numerical solutions and better than that of an effective improvement of the homotopy perturbation method ¹⁵ for different values of A.

Figure 1.

Comparison of the present solutions with numerical and He et al. ¹⁵ solutions of $\ddot{x}+sgn\left(x\right)=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0$ .

Figure 2.

Comparison of the present solutions with numerical and He et al. ¹⁵ solutions of $\ddot{x}+sgn\left(x\right)=0$ for $x\left(0\right)=5,\dot{x}\left(0\right)=0$ .

Figure 3.

Comparison of the present solutions with numerical and He et al. ¹⁵ solutions of $\ddot{x}+sgn\left(x\right)x^2=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0.$

Figure 4.

Comparison of the present solutions with numerical and He et al. ¹⁵ solutions of $\ddot{x}+sgn\left(x\right)x^2=0$ for $x\left(0\right)=5,\dot{x}\left(0\right)=0.$

Figure 5.

Comparison of the present solutions with numerical and He et al. ¹⁵ solutions of $\ddot{x}+x+\epsilon \hspace{0.17em}{x}^3=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0,\epsilon =1.$

Figure 6.

Comparison of the present solutions with numerical and He et al. ¹⁵ solutions of $\ddot{x}+x+\epsilon \hspace{0.17em}{x}^3=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0,\epsilon =100.$

Third, the absolute errors of the anti-symmetric constant force oscillator, anti-symmetric quadratic nonlinear oscillator, and the Duffing (cubical) oscillator for different initial conditions are presented, respectively, in Figures 7–12 together with an effective improvement of the homotopy perturbation method. ¹⁵ From these figures, it is clear that the present method provides less absolute error than that of an effective improvement of the homotopy perturbation method ¹⁵ for different values of A.

Figure 7.

Comparison of the absolute error between the present method and He et al. ¹⁵ method of $\ddot{x}+sgn\left(x\right)=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0$ .

Figure 8.

Comparison of the absolute error between the present method and He et al. ¹⁵ method of $\ddot{x}+sgn\left(x\right)=0$ for $x\left(0\right)=5,\dot{x}\left(0\right)=0.$

Figure 9.

Comparison of the absolute error between the present method and He et al. ¹⁵ method of $\ddot{x}+sgn\left(x\right)x^2=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0.$

Figure 10.

Comparison of the absolute error between the present method and He et al. ¹⁵ method of $\ddot{x}+sgn\left(x\right)x^2=0$ for $x\left(0\right)=5,\dot{x}\left(0\right)=0.$

Figure 11.

Comparison of the absolute error between the present method and He et al. ¹⁵ method of $\ddot{x}+x+\epsilon \hspace{0.17em}{x}^3=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0,\epsilon =1.$

Figure 12.

Comparison of the absolute error between the present method and He et al. ¹⁵ method of $\ddot{x}+x+\epsilon \hspace{0.17em}{x}^3=0$ for $x\left(0\right)=1,\dot{x}\left(0\right)=0,\epsilon =100$ .

Conclusion

An approximation technique is presented for solving anti-symmetric constant force oscillators, anti-symmetric quadratic nonlinear oscillators, and the Duffing (cubical) oscillator. The trial solution is completely algebraic, and the coefficients of unknown variables are determined by solving a set of simple linear equations. But similar algebraic equations become nonlinear when trigonometric functions are used for an approximate solution (i.e., the harmonic balanced method).