A modification of He’s variational approach using the least square method to nonlinear oscillators

Abstract

In this paper, we apply the global error minimization method to nonlinear oscillators with fractional terms for determining the first- and second-order approximate frequencies and solutions. The estimation problem is transformed into a minimization problem which leads to a simple yet precise scheme for finding the frequency of oscillation and the solution of the ordinary differential equation.

Introduction

Nonlinear differential equations are fundamental in physics, applied mathematics and engineering. The last two decades have seen a surge of interest in analytical tools for studying such equations to provide a more accurate solution. In particular, the nonlinear oscillator has received considerable attention where the aim is to jointly approximate the frequency of oscillation and the solution of a nonlinear equation.

There are plenty of analytical techniques that could be used to acquire the approximate solution of a nonlinear system, for example, the parameter-expansion method, ¹ the harmonic balance method,^2–5 the energy balance method,^2,3 the Hamiltonian approach,^6,7 the use of special functions,^8,9 the amplitude–frequency formulation, ¹⁰ the max–min approach,^11,12 the variational iteration method^13–17 and homotopy perturbation.^18–24 We refer the reader to He’s survey ²⁵ for a discussion of these methods. In literature,^26–28 a modification of the homotopy perturbation method has been discussed.

Differential equations with boundary or initial conditions and variational principles are two fundamental ways to describe a physical problem.^29–34 The differential model involves strong local differentiability (smoothness) of the physical field, while variational principles involve weaker local smoothness or only local integrability. Among the analytical methods, variational methods have two benefits. First, they yield a physically meaningful interpretation of the solution and second, the possible trial functions are optimal. Some approximate variational methods, including approximate energy method^35–37 and variational iteration method ¹⁵ have received considerable attention. A modified variational approach called the global error minimization (GEM) method has been developed in Farzaneh and Akbarzadeh Tootoonchi ³⁸ which modifies He’s variational method using the least squares method. To demonstrate the method, the author developed the first-, third- and fourth-order approximate solution and frequencies of the Duffing oscillator.

In this paper, the GEM method is applied to oscillators with fractional terms with the use of Fourier series and linearization. In the proposed method, the nonlinear differential equation is converted to an equivalent minimization problem. A simple sine or cosine term is chosen as the trial solution in the first step of the GEM method. The unknown parameters are identified via the minimizing the global error using the least squares criterion. Next, more sine or cosine terms are added to increase the desired accuracy of the approximated solution. We demonstrate that by using only a few terms a solution with high accuracy is obtained.

The rest of this article is ordered as follows. In Section ‘The basic idea of GEM method’, the key idea with the essential definitions and theorems is discussed. The applicability and efficiency of our method are discussed in Section ‘Numerical experiment’ by solving a nonlinear differential equation with a fractional nonlinear term. In Section ‘Results and discussions’, the conclusions regarding the overall study are discussed.

The basic idea of GEM method

Let the general form of the nonlinear oscillator can be expressed by

$$\ddot{x}+H\left(\dot{x},x,t\right)=0$$

(1)

Such that

$$x\left(0\right)+A,\dot{x}\left(0\right)=0$$

(2)

Definition 1 ⁴ : Define the following global error functional

$$\tilde{H}\left(\dot{x},x,t\right)={\displaystyle {\int }_0^T\left|\right|\ddot{x}+H(\dot{x},x,t)\left|\right|\text{d}t}$$

(3)

where $\text{T}=\frac{2\pi }{\omega }$ .

Definition 2 ⁴ : The following minimization problem can be established from equations (1) and (2).

Minimize

$$\begin{array}{l}\text{min}\tilde{H}\left(\dot{x},x,t\right)\\ \text{subject}\text{to}x\left(0\right)+A,\dot{x}\left(0\right)=0\end{array}$$

(4)

Lemma 1 ⁴ : The necessary and sufficient conditions for $\underset{0}{\overset{T}{\int }}f\mathit{dx}=0$ is $f=0$ on $[0,T]$ where $f$ is a nonlinear continuous function over the domain $[0,T]$ .

Theorem 1 ⁴ : In the minimization problem (4) $\tilde{H}\left(\stackrel{x,\dot{x},t}{\hspace{0.17em}}\right)+0$ is the necessary and sufficient condition for $x$ to be a solution of the oscillator (1) subject to initial condition (2).

Numerical experiment

This section elucidates the application of GEM method in the following two examples.

Example 1:

Consider the following nonlinear oscillator

$$\ddot{x}+\frac{x^3}{1+x^2}=0$$

(5)

with

$$x\left(0\right)=A,\dot{x}\left(0\right)=0$$

(6)

In order to begin the solution procedure using GEM method, let the first-order approximate solution can be expressed by

$${\overline{x}}_1\left(t\right)=\mathit{Acos}(\omega t)$$

(7)

The minimization problem corresponding to equation (5) with initial condition (6) is given by

Minimize

$$\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)={\displaystyle {\int }_0^T{\left({\ddot{\overline{x}}}_1+\frac{x_1{\hspace{0.17em}}^3}{1+x_1{\hspace{0.17em}}^2}\right)}^2\text{d}t}$$

(8)

such that ${\overline{x}}_1\left(0\right)=A,{\overline{x}}_1\left(0\right)=0$ where $T=\frac{2\pi }{\omega }$ .

Present work of applying GEM method is concerned with the application of Fourier series to nonlinear term in equation (5).

Rewrite the minimization problem (8) in the following form

Minimize

$$\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)={\displaystyle {\int }_0^T{({\ddot{\overline{x}}}_1+{\displaystyle \sum }_{k=0}^{\infty }{a}_k\text{cos}((2k+1)\omega t))}^2\text{d}t}$$

(9)

such that ${\overline{x}}_1\left(0\right)=A,{\overline{x}}_1\left(0\right)=0$ where $T=\frac{2\pi }{\omega }$ and

$$a_k=\frac{4}{\pi }{\displaystyle {\int }_0^{\pi /2}\left(1+\frac{{\overline{x}}_1^3}{1+{\overline{x}}_1^2}\right)\cos \left(\left(2k+1\right)\omega t\right)\mathrm{d}\left(\omega t\right)}$$

In order to get approximate first-order frequency and solution of equations (5) and (6), the Fourier series is truncated at $k=1$ , that is the sum of first two terms of the Fourier series is used.

Truncate the Fourier series and minimization problem (9) can be expressed by

Minimize

$$\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)={\displaystyle {\int }_0^T{\left({\ddot{\overline{x}}}_1+a_0\cos \left(\omega t\right)+a_1\text{cos}(3\omega t)\right)}^2\text{d}t}$$

(10)

such that ${\overline{x}}_1\left(0\right)=A,{\overline{x}}_1\left(0\right)=0$ .

By using equation (9), the coefficients $a_0$ and $a_1$ can be founded which are given by

$$a_0=\frac{1}{A}\left(A^2-2+2\sqrt{\frac{1}{1+A^2}}\right)$$

(11)

$$a_1=\frac{1}{A^3}\left(8-8\sqrt{\frac{1}{1+A^2}}+A^2\left(2-6\sqrt{\frac{1}{1+A^2}}\right)\right)$$

(12)

Substituting the coefficients $a_{°}$ and $a_1$ in minimization problem (10) gives

$$\begin{array}{l}\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)=\frac{1}{A^6(1+A^2)\omega }[\pi (-128\left(-1+\sqrt{\frac{1}{1+A^2}}\right)-80A^4\left(-1+2\sqrt{\frac{1}{1+A^2}}\right)\\ -64A^2\left(-3+4\sqrt{\frac{1}{1+A^2}}\right)-A^8\left(-3+4\sqrt{\frac{1}{1+A^2}}-{\omega }^2\right)\left(-1+{\omega }^2\right)\\ +A^{10}{\left(-1+{\omega }^2\right)}^2-4A^6\left(-1+7\sqrt{\frac{1}{1+A^2}}\right)+\left(-1+\sqrt{\frac{1}{1+A^2}}\right){\omega }^2)]\end{array}$$

(13)

By imposing the optimal condition $\frac{\partial \tilde{H}}{\partial \omega }=0$ , and solving the resulting equation for $\omega$ gives the following squared of frequency

$$\begin{array}{l}{\omega }_1=\frac{1}{2\left(3A^8+3A^{10}\right)}(-4A^6-2A^8+2A^{10}+4A^6\sqrt{\frac{1}{1+A^2}}+4A^8\sqrt{\frac{1}{1+A^2}}\\ +4(96A^8+240A^{10}+205A^{12}+64A^{14}-2A^{18}+A^{20}-96A^8\sqrt{\frac{1}{1+A^2}}-288A^{10}\sqrt{\frac{1}{1+A^2}}\\ {-314A^{12}\sqrt{\frac{1}{1+A^2}}-144A^{14}\sqrt{\frac{1}{1+A^2}}-18A^{16}\sqrt{\frac{1}{1+A^2}}+4A^8\sqrt{\frac{1}{1+A^2}}+\frac{A^{12}}{1+A^2}+\frac{2A^{14}}{1+A^2}+\frac{A^{16}}{1+A^2})}^{1/2})\end{array}$$

(14)

To get second-order approximations of frequency and solution using GEM method, consider the following trivial solution

$${\overline{x}}_2\left(t\right)=c\cos \left(\omega t\right)+d\cos (3\omega t)$$

(15)

For the satisfaction of constraint in the minimization problem (10) substitute ${\overline{x}}_1\left(0\right)=A,{\overline{x}}_1\left(0\right)=0$ in (15) gives

$$A=c+d$$

(16)

Thus, the following form of a trivial solution can be achieved

$${\overline{x}}_2\left(t\right)=c\cos \left(\omega t\right)+(A-c)\cos (3\omega t)$$

(17)

By truncating the Fourier series for $k=2$ in (9), the following minimization problem can be constructed

$$\tilde{H}\left({\dot{\overline{x}}}_2,{\overline{x}}_2,t\right)={\displaystyle {\int }_0^T{(-A{\omega }^2\text{cos}(\omega t)+{\displaystyle \sum }_{k=0}^2{b}_k\text{cos}((2k+1)\omega t))}^2\text{d}t}$$

(18)

where

$$b_{°}=\frac{4}{\pi }{\displaystyle {\int }_0^{\pi /2}\frac{{\overline{x}}_2^3}{1+{\overline{x}}_2^2}\text{cos}(\omega t)d(\omega t)}$$

(19)

$$b_1=\frac{4}{\pi }{\displaystyle {\int }_0^{\pi /2}\frac{{\overline{x}}_2^3}{1+{\overline{x}}_2^2}\text{cos}(3\omega t)d(\omega t)}$$

(20)

$$b_2=\frac{4}{\pi }{\displaystyle {\int }_0^{\pi /2}\frac{{\overline{x}}_2^3}{1+{\overline{x}}_2^2}\text{cos}(5\omega t)d(\omega t)}$$

(21)

By using linearization of the term $\frac{{\overline{x}}_2^3}{1+{\overline{x}}_2^2}$ in equations (19) to (21) and applying the integration yields the coefficients $b_0,b_1$ and $b_2$ .

Imposing the optimality conditions $\frac{\partial {\tilde{H}}_2}{\partial c}=0$ and $\frac{\partial {\tilde{H}}_2}{\partial \omega }=0$ , and solving the resulting equations gives the values of the constant $c$ and second-order approximate frequency.

The exact amplitude of the problem (5) and (6) is given by ¹

$$T_{\mathit{exact}}=4{\displaystyle {\int }_0^{\pi /2}{\left(\frac{A^{2}co{s}^2(\theta )}{A^{2}co{s}^2\left(\theta \right)+\text{ln}\left(1-\frac{A^{2}co{s}^2(\theta )}{1+A^2}\right)}\right)}^{1/2}\text{d}\theta }$$

Let $A=1$ the relative error in first- and second-order approximate frequencies using GEM method with Fourier series is given by

$$\left|\frac{{\omega }_1-{\omega }_{\mathit{exact}}}{{\omega }_{\mathit{exact}}}\right|\times 100\text{\%}\approx 1.80\text{\%}$$

$$\left|\frac{{\omega }_2-{\omega }_{\mathit{exact}}}{{\omega }_{\mathit{exact}}}\right|\times 100\text{\%}\approx 0.19\text{\%}$$

Consider the following equation which can be obtained by using equation (5) with the same initial conditions as in (6)

$$\ddot{x}\left(1+x^2\right)+x^3=0$$

The relative error in approximate frequencies using first- and second-order GEM method is given by

$$\left|\frac{{\omega }_1-{\omega }_{\mathit{exact}}}{{\omega }_{\mathit{exact}}}\right|\times 100\text{\%}\approx 4.60\text{\%}$$

$$\left|\frac{{\omega }_2-{\omega }_{\mathit{exact}}}{{\omega }_{\mathit{exact}}}\right|\times 100\text{\%}\approx 0.013\text{\%}$$

The relative error using the standard GEM method for second-order approximate frequency is more accurate than the modified GEM method or variational approach using the least square. Comparisons are shown in Table 1.

Table 1.

Percentage relative errors in first- and second-order GEM frequencies and values of the parameter ‘ $d$ ’ with some chosen values of the amplitudes for Example 1.

$A$	% relative error
	Present modified method			Standard method
	${\omega }_1$	${\omega }_2$	$d$	${\omega }_1$	${\omega }_2$	$d$
0.1	0.24	9.199	–0.00004	3.64	0.3482	0.0042
1	1.15	0.1900	0.0323	4.60	0.0132	0.0310
5	0.059	0.0057	0.0294	1.01	0.4583	0.0139
10	0.0066	0.0015	0.0189	0.316	0.1742	0.0071
100	0.000003	0.000002	0.0024	0.0039	0.0025	0.0007

GEM: global error minimization.

Example 2:

Consider the following oscillator,

$$\ddot{x}+\frac{x}{1+ϵx^2}=0$$

(22)

With the following initial conditions

$$x\left(0\right)=A,\dot{x}\left(0\right)=0$$

(23)

In order to apply GEM method or variational approach using the least square, construct the following minimization problem corresponding to equations (22) and (23),

$$\text{Minimize}\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)={\displaystyle {\int }_0^T{\left({\ddot{\overline{x}}}_1+\frac{x_1}{1+ϵx_1{\hspace{0.17em}}^2}\right)}^2\text{d}t}$$

(24)

such that ${\overline{x}}_1\left(0\right)=A,{\dot{\overline{x}}}_1\left(0\right)=0.$

Assume that the following first-order approximate solution

$${\overline{x}}_1\left(t\right)=A\cos (\omega t)$$

(25)

The minimization problem corresponds to equations (22) and (23) can be constructed as;

Since the present work is dealt with the modified form of GEM method, so the rational term in minimization problem (24) can be expressed by the Fourier series and the new minimization problem corresponding to equations (22) and (23) is given by

$$\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)={\displaystyle {\int }_0^T{\left({\ddot{\overline{x}}}_1+{\displaystyle \sum }_{n=0}^{\infty }{a}_n\text{cos}((2n+1)\omega t)\right)}^2\text{d}t}$$

(26)

such that ${\overline{x}}_1\left(0\right)=A,{\stackrel{\dot{\overline{x}}}{\hspace{0.17em}}}_1\left(0\right)=0$ , where;

For $n=0$ ,

$$=\frac{1}{ϵA}\left(2-2\sqrt{\frac{1}{1+ϵA^2}}\right)$$

(27)

In this example, only the first term of the Fourier series is used for finding the first-order approximate solution and frequency of equation (22) with initial condition (23).

Since the first-order trial solution already satisfies the constraint in minimization problem (26), so the following minimization problem can be constructed;

Minimize

$$\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)={\displaystyle {\int }_0^T{\left(-A{\omega }^2\cos \left(\omega t\right)+a_0\text{cos}(\omega t)\right)}^2\text{d}t}$$

(28)

By evaluating the integral, the following equation can be obtained

$$\tilde{H}\left({\dot{\overline{x}}}_1,{\overline{x}}_1,t\right)=\frac{\pi }{A^2{ϵ}^2\omega }{\left(-2+A^2ϵ{\omega }^2+2\sqrt{\frac{1}{1+ϵA^2}}\right)}^2$$

(29)

By applying the optimality condition $\partial \tilde{H}/\partial \omega =0$ and solving the resulting equation for $\omega$ gives the following expression for first-order approximate frequency

$${\omega }_1=\frac{\sqrt{2}}{A\sqrt{ϵ}}\sqrt{1-\sqrt{\frac{1}{1+ϵA^2}}}$$

(30)

In order to get second-order approximate frequency using modified GEM method, assume the following trial solution of equation (22) with initial conditions (23)

$${\overline{x}}_2\left(t\right)=(A-d)\cos \left(\omega t\right)+d\cos (3\omega t)$$

(31)

The solution procedure for second-order will be preceded by considering the first two terms of the Fourier series in the minimization problem (26). To get the first two components of the Fourier series, linearize the rational term in equation (22) in the following manner.

Let

$$f\left(x\right)=\frac{x}{1+ϵx^2}$$

Then using Taylor series with first two terms gives

$$f\left({\overline{x}}_2\right)\approx \frac{{\overline{x}}_1}{1+ϵ{\overline{x}}_1{\hspace{0.17em}}^2}+f^{\text{′}}({\overline{x}}_1)({\overline{x}}_2-{\overline{x}}_1)$$

where ${\overline{x}}_1$ and ${\overline{x}}_2$ are the first- and second-order trial solutions, respectively.

The following minimization problem can be constructed by using first-two components of the Fourier series

$$\tilde{H}\left({\dot{\overline{x}}}_2,{\overline{x}}_2,t\right)={\displaystyle {\int }_0^T{\left({\ddot{\overline{x}}}_2+{\displaystyle \sum }_{n=0}^1{b}_n\text{cos}((2n+1)\omega t)\right)}^2\text{d}t}$$

(32)

such that ${\overline{x}}_2\left(0\right)=A,{\overline{x}}_2\left(0\right)=0$ .

The coefficients $b_0$ and $b_1$ in minimization problem (32) can be computed as

$$=-\frac{2}{{ϵ}^2{A}^4}\left(ϵA^3\left(-1+\sqrt{\frac{1}{1+ϵA^2}}\right)+2d\left(6\left(-1+\sqrt{\frac{1}{1+ϵA^2}}\right)+ϵA^2(-1+4\sqrt{\frac{1}{1+ϵA^2}})\right)\right)$$

(33)

and

$$\begin{array}{l}=-\frac{2}{{ϵ}^3{A}^6}(ϵA^3\left(4\left(-1+\sqrt{\frac{1}{1+ϵA^2}}\right)+ϵA^2(-1+3\sqrt{\frac{1}{1+ϵA^2}})\right)+4d(20\left(-1+\sqrt{\frac{1}{1+ϵA^2}}\right)\\ +5ϵA^2\left(-3+5\sqrt{\frac{1}{1+ϵA^2}}\right)+{ϵ}^2{A}^4\left(-1+6\sqrt{\frac{1}{1+ϵA^2}}\right)))\end{array}$$

(34)

Now minimization problem (32) can be expressed by

$$\tilde{H}\left({\dot{\overline{x}}}_2,{\overline{x}}_2,t\right)={\displaystyle {\int }_0^T{\left(\left(d-A\right){\omega }^2\cos \left(\omega t\right)+9d{\omega }^2\cos \left(3\omega t\right)+b_{°}\cos \left(\omega t\right)+b_1\cos \left(3\omega t\right)\right)}^2\text{d}(\omega t)}$$

By applying the optimality conditions $\frac{\partial {\tilde{H}}_2}{\partial d}=0$ and $\frac{\partial {\tilde{H}}_2}{\partial \omega }=0$ and solve the resulting equations numerically for $\omega$ and $d$ for a fixed value of $ϵ$ and $A$ gives the numerical values of $ϵ,d$ and $\omega$ .

The problem in Example 2 can be solved by GEM method in a different way if the equation (22) can be written as

$$\ddot{x}\left(1+ϵx^2\right)+\text{x}=0$$

The minimization problem of the above equation can be constructed and its first- and second-order approximate frequencies can be found and some of the approximate frequencies are given in Table 2 with fixed values of the $ϵ$ and $A$ .

Table 2.

Percentage relative errors in first- and second-order GEM frequencies and values of the parameter ‘ $d$ ’ with some chosen values of the amplitudes and $ϵ$ for Example 2.

$ϵ$		$A$	Numerical values of frequencies
			Present modified method			Standard method
			${\omega }_1$	${\omega }_2$	$d$	${\omega }_1$	${\omega }_2$	$d$
1	1		0.7654	0.7612	–0.0213	0.7503	0.7604	–0.0195
	5		0.2536	0.2352	–0.3735	0.2172	0.2287	–0.1978
	10		0.1342	0.1206	–0.8515	0.1104	0.1166	–0.4057
	100		0.0141	0.0122	–9.2865	0.0111	0.0117	–4.0905
5	1		0.4865	0.4716	–0.0504	0.4475	0.4655	–0.0345
	5		0.1207	0.1080	–0.4309	0.0989	0.1044	–0.2032
	10		0.0618	0.0544	–0.9020	0.0496	0.0524	–0.4084
	100		0.0063	0.0055	–9.3245	0.0050	0.0052	–4.0908
100	1		0.1342	0.1206	–0.0851	0.1104	0.1166	–0.0406
	5		0.0280	0.0244	–0.4607	0.0222	0.0235	–0.2045
	10		0.0141	0.0122	–0.9287	0.0111	0.0117	–0.4091
	100		0.0014	0.0012	–9.3476	0.0011	0.0011	–4.0908

GEM: global error minimization.

Results and discussions

Tables 1 and 2 show the percentage of relative errors in first- and second-order GEM frequencies in Examples 1 and 2, which shows that the percentage relative error in first-order GEM frequencies has dual behavior with the increase of the amplitude. But after one increase it decreases when the amplitude is enhanced. The percentage relative error in second-order approximate frequencies decreases with the increase in amplitude. Table 2 shows that first-order GEM frequencies also decreases with the increasing values of the amplitudes at fixed values of the parameter ‘ $ϵ$ ’ used in the problem of Example 2 and the same decreasing variation of the percentage relative error can be seen for second-order GEM frequencies.

Figures 1 and 2 show the comparison of first- and second-order GEM solutions with the numerical solution obtained from the Matlab solver ‘ode45’ for different amplitudes of Example 1. In these figures, it can be concluded that the first-order GEM solution is improved by increasing the value of the amplitude. So the error of the first-order GEM solution can be reduced by choosing the larger value of the amplitude but second-order is more accurate for small and large amplitudes. Figures 3 and 4 show the phase portraits of first- and second-order GEM solutions and numerical solutions. These figures can be used to observe the difference between the phase plots of numerical solutions and GEM solutions. The phase portraits show that whether the limit cycle is present for particular chosen values of the amplitudes. So limit cycles are present for the mentioned values of the amplitudes. Similarly, phase portraits can be drawn for different finite values of the amplitudes and comparison of the GEM first-order and second-order solutions with the numerical solutions can be seen for those values of the amplitudes.

Figure 1.

Comparison of the numerical solution with first- and second-order GEM solutions of Example 1.

Figure 2.

Comparison of the numerical solution with first- and second-order GEM solutions of Example 1.

Figure 3.

Comparison of the phase portraits from numerical solution with phase portraits obtained from first- and second-order GEM solutions of Example 1.

Figure 4.

Comparison of the phase portraits from numerical solution with phase portraits obtained from first- and second-order GEM solutions of Example 1.

Figures 5 and 6 show the comparison of numerical solutions and comparison of phase portraits obtained from Matlab solver ‘ode45’ with GEM first and second-order solutions and phase portraits for the mentioned values of the amplitudes of Example 2.

Figure 5.

Comparison of the numerical solution with first- and second-order GEM solutions of Example 2.

Figure 6.

Comparison of the phase portraits from numerical solution with phase portraits obtained from first- and second-order GEM solutions of Example 2.

It can also be noted from the solution procedures of both solved problems above that linearization of the nonlinear term is performed instead of adopting the whole nonlinear term to carry out the integration which is performed during the procedures of determining the coefficients of Fourier series. So, the first step of the present solution procedure is to approximate any rational, fractional and discontinuous term of the oscillator to Fourier series and coefficients of the Fourier series are found by performing the integration for the same rationale, fractional and discontinuous term and linearized form of the rational, fractional and discontinuous term which is evaluated at the trial solution chosen, i.e. if the first-order solution is needed to find then no need to linearize the rational term and in order to get second order or high order of the solutions then linearization of the rational term will be performed and linearized form of the rational term can be used to carry out the integration which would be needed for determining the coefficients of the Fourier series.

Conclusion

The GEM method is extended using Fourier series for oscillators with fractional terms. In our proposed extension of the GEM method, Fourier series has been adopted to approximate the fractional term. Also, in order to get the second-order approximate frequencies and approximate solutions, linearization of the fractional term in the variational principle is performed. The linearization leads to a computationally efficient integration. Two examples have been presented to demonstrate our proposed method. The proposed approach generalizes to nonlinear oscillators with discontinuity and fractional power.