Analytical solutions of nonlinear oscillator with coordinate-dependent mass and Euler–Lagrange equation using the parameterized homotopy perturbation method

Abstract

This paper gives analytical solutions to a nonlinear oscillator with coordinate-dependent mass and Euler–Lagrange equation using the parameterized homotopy perturbation method. The results demonstrate the simplicity and convenience of the method to nonlinear differential equations.

Introduction

Nonlinear differential equations are fundamental aspect of mathematics that finds great interest in science and engineering. Most physical phenomena are modeled using nonlinear differential equations. Of such equations are the nonlinear oscillator with coordinate-dependent mass ¹ and the Euler–Lagrange equation.

These equations in almost all cases require valid, robust, and effective solutions. Over the years, different solution techniques have been established. These include the homotopy perturbation method (HPM),^1–6 energy balance method,^7–10 amplitude–frequency formulation,^11,12 variational iteration method,^13–18 and parameter expanding method.^19,20

The HPM given by He et al.^1–5 is applied to provide solutions to differential and integral equations—linear and nonlinear. The method is known to pair the traditional perturbation technique with the homotopy in topology. The method since introduction has seen many modifications to meet with the demand of more accurate and fast convergent results. Among the recent modification is the parameterized HPM. ³

This work utilizes the idea of the parameterized HPM to provide analytical solutions to a nonlinear oscillator with coordinate-dependent mass and Euler–Lagrange equation.

Review of parameterized homotopy perturbation method

Parameterized HPM is a powerful method developed by Adamu and his student, Ogenyi in 2017. The technique modified the classical idea of the HPM to make improvement.

For the review of the method, we consider a general nonlinear oscillator:

$$\lambda \left(u\right)-f\left(x\right)=0,\text{t}\in \mathrm{ℝ}$$

(1)

With initial conditions

$$u\left(0\right)=A\in \mathrm{ℝ},\mathrm{ }\dot{u}\left(0\right)=B\in \mathrm{ℝ}$$

(2)

where $\lambda$ , A, and B are differential operators and known initial boundary conditions respectively and $f\left(x\right)$ represents the known analytical function.

The division of $\lambda \left(u\right)$ by L and N leads to the transformation of equation to

$$L\left(u\right)+N\left(u\right)–f\left(t\right)=0$$

(3)

where L and N represent linear and nonlinear components, respectively.

The following homotopy is constructed for the nonlinear differential equation

$$H\left(u,p\right)=\left(1-\alpha p\right)[L\left(u\right)-L\left(u_0\right)+\alpha p\left[\lambda \left(u\right)-f\left(t\right)-\text{cos}\omega t\right]=p\left(\frac{1}{p^{\alpha p}}-2\alpha \right)\text{cos}\omega t$$

(4)

where $p=\frac{1}{\alpha }$ , $p\in [0,\frac{1}{\alpha }]$ is the embedding parameter, $u_0$ is the initial approximate solution of equation (1), which satisfies the boundary conditions.

Obviously

$$H\left(u,0\right)=L\left(u\right)-L\left(u_0\right)=0$$

(5)

$$H\left(u,\frac{1}{\alpha }\right)=\lambda \left(u\right)-f\left(x\right)=0$$

(6)

Equation (4) changes from linear to nonlinear equation as p changes from 0 to $\frac{1}{\alpha }$ .

Using p as an expanding parameter and the premise that the solution of equation (4) can be composed in the form

$$u_0+p{u}_1+p^2{u}_2+\dots +p^n{u}_n$$

(7)

The approximate solution of equation (3) becomes

$$\underset{p\to \frac{1}{\alpha }}{\lim }u=u_0+\frac{1}{\alpha }{u}_1+\frac{1}{{\alpha }^2}{u}_2+···+\frac{1}{{\alpha }^n}{u}_n$$

(8)

$\alpha$ is the parameter and is determined optimally. The optimal identification of the alpha is demonstrated by Adamu and Ogenyi ³ and showed in the two examples below for illustrative purposes.

Applications/Results

Example 1

Consider a Lev–Tymehyshgn–Zagorodny nonlinear oscillator with coordinate-dependent mass ¹

$$\left(1+\beta u^2\right)\ddot{u}+\beta u{\dot{u}}^2-u\left(1-u^2\right)=0$$

(9)

With initial condition

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

(10)

Rewriting equation (9) we have

$$\dot{u}-u+\beta u^2\ddot{u}+\beta u{\dot{u}}^2+u^3-\text{cos}\omega t=-\text{cos}\omega t$$

(11)

Using the constructed homotopy ³

$$\ddot{u}+(-)u+\alpha p\left[\beta u^2\ddot{u}+\beta u{\dot{u}}^2+u^3-cos\omega t\right]=p\left(\frac{1}{p^{\alpha p}}-2\alpha \right)cos\omega t,p\in \left[0,\frac{1}{\alpha }\right]$$

(12)

Equation (12) transits from linear to nonlinear oscillator as p changes from p to $\frac{1}{\alpha }$

Expanding the linear term (-1), ¹ we have

$$-1={\omega }^2+{\zeta }_{1}p+{\zeta }_2{p}^2+···+{\zeta }_n{p}^n$$

(13)

Studies by various authors^1–5 give

$$u=u_0+u_{1}p+u_2{p}^2+···+$$

(14)

Substituting equation (13) and equation (14) into equation (12), and solving the resulting equation leads to

$$p^0:{\ddot{u}}_0+{\omega }^2{u}_0=0,u_0=0,{\dot{u}}_o\left(0\right)=0$$

(15)

$$p^1:{\ddot{u}}_1+{\omega }^2{u}_1+{\zeta }_1{u}_0+\alpha p\left[\beta u_0{\hspace{0.17em}}^2{\ddot{u}}_0+\beta u_0{\dot{u}}_0^2+u^3-\text{cos}\omega t\right]=p\left(\frac{1}{p^{\alpha p}}-2\alpha \right)\text{cos}\omega t$$

(16)

Equation (15) gives the following solution

$$u_0=A\text{cos}\omega t$$

(17)

Substituting equation (17) into equation (16) results

$$\begin{array}{l}{\ddot{u}}_1+{\omega }^2{u}_1+{\zeta }_{1}A\text{cos}\omega t+\alpha \left[\beta A^3{\omega }^2{\text{cos}}^3\omega t+\beta A^3{\omega }^2\text{cos}\omega t{\text{sin}}^2\omega t+A^3{\text{cos}}^3\omega t-\text{cos}\omega t\right]\\ \hspace{1em}=\left(\frac{1}{p^{\alpha p}}-2\alpha \right)\text{cos}\omega t\end{array}$$

(18)

Further evaluation of equation (18) provides

$$\begin{array}{l}{\ddot{u}}_1+{\omega }^2{u}_1+\left[{\zeta }_{1}A+\alpha \left(\beta A^3{\omega }^2+\frac{3}{4}{A}^2\left(1-2\beta {\omega }^2\right)-1\right)\right]cos\omega t+\frac{1}{4}\alpha A^3\left(-2\beta {\omega }^2+1\right)cos3\omega t\\ \hspace{1em}=\left(\frac{1}{p^{\alpha p}}-2\alpha \right)cos\omega t\end{array}$$

(19)

The absence of secular term in equation (19) implies

$${\zeta }_{1}A+\alpha \left(\beta A^3{\omega }^2+\frac{3}{4}{A}^2\left(1-2\beta {\omega }^2\right)-1\right)=\left(\frac{1}{p^{\alpha p}}-2\alpha \right)$$

(20)

First approximation of equation (13) gives ¹

$$-1={\omega }^2{\zeta }_1$$

(21)

The following solution results after substituting equation (21) into equation (20)

$${\omega }_p=\sqrt{\frac{\left(\frac{1}{\alpha p^{\alpha p}}-1\right)+\left(\frac{A}{\alpha }-\frac{3}{4}{A}^2\right)}{A\left(\beta A^2-\frac{3}{2}\beta A-\frac{1}{\alpha }\right)}}$$

(22)

Taking ³

$$p^{\alpha p}=1$$

(23)

Equation (22) becomes

$${\omega }_p=\sqrt{\frac{\left(\frac{1}{\alpha }-1\right)+\left(\frac{A}{\alpha }-\frac{3}{4}{A}^2\right)}{\beta A\left(A-\frac{1}{\alpha }-\frac{3}{2}\right)}}$$

(24)

To identify α we consider

$${\omega }_c=\sqrt{\frac{\frac{3A^2}{4}-1}{\left(1+\frac{A^2}{2}\beta \right)}}$$

(25)

Equation (25) is the approximate solution of equation (9) provided in He and Yue. ¹

Equating equations (24) and (25)

$$\sqrt{\frac{\left(\frac{1}{\alpha p^{\alpha p}}-1\right)+\left(\frac{A}{\alpha }-\frac{3}{4}{A}^2\right)}{A\left(\beta A^2-\frac{3}{2}\beta A-\frac{1}{\alpha }\right)}}=\sqrt{\frac{\frac{3A^2}{4}-1}{\left(1+\frac{A^2}{2}\beta \right)}}$$

(26)

The choice of equation (25) in the identification of the optimal alpha is based on the fact that there is no known exact solution for equation (9).

Solving for the $\alpha$ in equation (26) gives the following equations

$$\frac{1}{\alpha }=\frac{\left(1+\frac{3}{4}{A}^2\right)\left(1+\frac{A^2\beta }{2}\right)+\beta A\left(A-\frac{3}{2}\right)\left(\frac{3A^2}{4}-1\right)}{\left(1+A\right)\left(1+\frac{A^2\beta }{2}\right)+\beta A\left(\frac{3A^2}{4}-1\right)}$$

(27)

$$\alpha =\frac{\left(1+A\right)\left(1+\frac{A^2\beta }{2}\right)+\beta A\left(\frac{3A^2}{4}-1\right)}{\left(1+\frac{3}{4}{A}^2\right)\left(1+\frac{A^2\beta }{2}\right)+\beta A\left(A-\frac{3}{2}\right)\left(\frac{3A^2}{4}-1\right)}$$

(28)

Since for periodic solution, $A>\frac{2}{\sqrt{3}}$ , ¹ we can choose $A=\frac{2}{\sqrt{2}}>\frac{2}{\sqrt{3}}$ Now taking, $A=\frac{2}{\sqrt{2}}$ equation (28) becomes

$$\alpha =\frac{2\left(1+\sqrt{2}\right)\left(1+\beta \right)+\beta \sqrt{2}}{5\left(1+\beta \right)+\beta \left(4-3\sqrt{2}\right)}$$

(29)

Table 1 gives the numerical values of $\alpha$ , comparison of the results of equation (9) obtained using the parameterized and the classic HPMs and its relative errors.

Table 1.

Numerical results of equation (9).

$\beta$	$\alpha$	${\omega }_p$	${\omega }_c$	$E=\left|{\omega }_c{\omega }_c\right|$
1.0	1.134638	0.521705	0.5	2.2 × 10−1
1.1	1.142893	0.509269	0.487950	2.1 × 10−2
1.2	1.150415	0.497683	0.476731	2.1 × 10−2
1.3	1.157296	0.486851	0.466252	2.1 × 10−2
1.4	1.163617	0.476700	0.456435	2.0 × 10−2
1.5	1.169442	0.467157	0.447214	2.0 × 10−2

Example 2

Consider an Euler–Lagrange equations

$$\ddot{u}+3ku\dot{u}+k^2{u}^3=0$$

(30)

With the initial condition

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

(31)

Rewriting equation (30)

$$\ddot{u}+{\omega }^{2}u+3ku\dot{u}+k^2{u}^3-{\omega }^{2}u-\text{cos}\omega t=-\text{cos}\omega t$$

(32)

Homotopy is constructed for equation (32)

$$\ddot{u}+{\omega }^{2}u+\alpha p\left[3ku\dot{u}+k^2{u}^3-{\omega }^{2}u-\text{cos}\omega t\right]=p\left(\frac{1}{p^{\alpha p}}-2\alpha \right)\text{cos}\omega t$$

(33)

With reference to various studies^1–5 we can write

$$u=u_0+p{u}_1+p^2{u}_2$$

(34)

This follows that

$$\dot{u}={\dot{u}}_0+p{\dot{u}}_1+p^2{\dot{u}}_2$$

(35)

$$\ddot{u}={\ddot{u}}_0+p{\ddot{u}}_1+p^2{\ddot{u}}_2$$

(36)

Equations (34), (35), and (36) are substituted into equation (33) and simplified to give

$$p^0:{\ddot{u}}_0+{\omega }^2{u}_0=0,u_0=0,{\dot{u}}_o\left(0\right)=0$$

(37)

$$p^1:{\ddot{u}}_1+{\omega }^2{u}_1+\alpha \left[3k{u}_0{\dot{u}}_0+k^2{u}_0{\hspace{0.17em}}^3-{\omega }^2{u}_0-\text{cos}\omega t\right]=\left(\frac{1}{p^{\alpha p}}-2\alpha \right)\text{cos}\omega t$$

(38)

Consequently

$$u_0=A\text{cos}\omega t$$

(39)

Combining equations (39) and (38)

$${\ddot{u}}_1+{\omega }^2{u}_1+\alpha \left[-3k\omega A^2\text{cos}\omega t\text{sin}\omega t+k^2{A}^3{\text{cos}}^3\omega t-{\omega }^{2}A\text{cos}\omega t-\text{cos}\omega t\right]=\left(\frac{1}{p^{\alpha p}}-2\alpha \right)\text{cos}\omega t$$

(40)

Further simplification leads to

$$\begin{array}{c}{\ddot{u}}_1+{\omega }^2{u}_1+\alpha \left[\frac{3k^2{A}^3}{4}-{\omega }^{2}A-1\right]\text{cos}\omega t+\frac{\alpha k^2{A}^3}{4}\text{cos}3\omega t\\ -\frac{3\alpha k\omega A^2}{2}{\omega }^{2}A\text{cos}\omega t-\text{cos}\omega t=\left(\frac{1}{p^{\alpha p}}-2\alpha \right)\text{cos}\omega t\end{array}$$

(41)

The absence of secular term in $u_1$ requires:

$$\alpha \left[\frac{3k^2{A}^3}{4}-{\omega }^{2}A-1\right]=\left(\frac{1}{p^{\alpha p}}-2\alpha \right)$$

(42)

This implies

$$\frac{3k^2{A}^3}{4}-{\omega }^{2}A-1=\left(\frac{1}{\alpha p^{\alpha p}}-2\right)$$

(43)

Taking $p^{\alpha p}=1$ gives ³

$$\frac{3k^2{A}^3}{4}-{\omega }^{2}A-1=\left(\frac{1}{\alpha }-2\right)$$

(44)

$${\omega }_p=\sqrt{\frac{3k^2{A}^3}{4}-\left(\frac{1}{\alpha }-1\right)}$$

(45)

Considering an approximate solution of equation (30) given by the classic HPM

$${\omega }_c=\sqrt{\frac{3k^2{A}^3}{4}-1}$$

(46)

Equating equations (44) and (45)

$$\sqrt{\frac{3k^2{A}^3}{4}-\left(\frac{1}{\alpha }-1\right)}=\sqrt{\frac{3k^2{A}^3}{4}-1}$$

(47)

Solving for α equation (47) gives

$$\alpha =\frac{1}{2}$$

(48)

The numerical solution of equation (30) is given in Table 2.

Table 2.

Numerical results of equation (30).

k	$\alpha$	${\omega }_p$	${\omega }_c$	$E=\left|{\omega }_c{\omega }_c\right|$
1.0	0.5	1.058924	1.058924	0.00
1.1	0.5	1.251718	1.251718	0.00
1.2	0.5	1.433423	1.433423	0.00
1.3	0.5	1.607803	1.607803	0.00
1.4	0.5	1.777017	1.777017	0.00
1.5	0.5	1.942414	1.942414	0.00

Conclusion

The parameterized HPM is successfully used to obtain analytical solutions to a nonlinear oscillator with coordinate-dependent mass and Euler–Lagrange equations. The method is simple, convenient, and able to handle wide varieties of nonlinear differential equations with ease.