He–Laplace method for general nonlinear periodic solitary solution of vibration equations

Abstract

This paper suggests a universal approach to solve nonlinear periodic vibration equations by He–Laplace method, a coupling of He’s perturbation method and Laplace transform. The nonlinear periodic solitary solution of vibration equation is used as an example to elucidate the effectiveness and simplicity of the method, only few iterations are needed to obtain an extremely accurate solution.

Introduction

The study of nonlinear differential equations is key in lot of areas such as physics, applied mathematics, engineering and other related fields. A challenging difficulty is to suggest precise analytical approximate solution in a computationally efficient manner of a strongly nonlinear differential equation. Particularly, the nonlinear periodic vibration equation has received considerable attention where the aim is to obtain analytical approximation of the nonlinear periodic vibration equations. The problem is even more challenging in the presence of a discontinuity. It is well known that the classical periodic solitary solutions of nonlinear vibration equations and Korteweg–de Vries converge to solitary-wave solutions as the period length grows unboundedly. This latter result is important in principle since most of the numerical simulations of solutions of these sorts of equations are actually performed with periodic boundary conditions, even though one is attempting to approximate solutions on the entire real axis, which arises from the dimensionless hydrodynamics equations describing the nonlinear propagation of the quantum ion-acoustic waves. With the aid of symbolic computation, many types of new analytical solutions of the nonlinear vibration equation are constructed in terms of some powerful ansatze, which include new doubly periodic wave, solitary wave, shock wave, rational wave, and singular wave solutions. The Euler–Bernoulli beam theory or linear string vibration results in a linear vibration equation. For a non-linear vibration, its governing equation can be written as

$$\frac{{\partial }^2\varphi }{\partial t^2}+F\left(\varphi ,\frac{\partial \varphi }{\partial x},\dots ,\frac{{\partial }^5\varphi }{\partial x^5}\right)=0$$

(1)

where F represents the nonlinear term. All nonlinear vibration equations can be converted into the following form of equation (1). For example, we consider the well-known KdV equation

$$\frac{\partial \varphi }{\partial t}-6\varphi \frac{\partial \varphi }{\partial x}+\frac{{\partial }^3\varphi }{\partial x^3}=0$$

(2)

Differentiating equation (2) with respect to time results in

$$\frac{{\partial }^2\varphi }{\partial t^2}-6\frac{\partial \varphi }{\partial t}\frac{\partial \varphi }{\partial x}-6\varphi \frac{\partial }{\partial x}\left(\frac{\partial \varphi }{\partial t}\right)+\frac{{\partial }^3}{\partial x^3}\left(\frac{\partial \varphi }{\partial t}\right)=0$$

(3)

The above equation can be converted into the equations as shown in equation (1), which reads

$$\frac{{\partial }^2\varphi }{\partial t^2}-6\frac{\partial \varphi }{\partial x}\left(6\varphi \frac{\partial \varphi }{\partial x}-\frac{{\partial }^3\varphi }{\partial x^3}\right)-6\varphi \frac{\partial }{\partial x}\left(6\varphi \frac{\partial \varphi }{\partial x}-\frac{{\partial }^3\varphi }{\partial x^3}\right)+\frac{{\partial }^2}{\partial x^2}\left(6\varphi \frac{\partial \varphi }{\partial x}-\frac{{\partial }^3\varphi }{\partial x^3}\right)=0$$

(4)

This paper focuses on nonlinear periodic solutions of vibration equations using a coupling of the He’s perturbation method and Laplace transform, which was initially been proposed to solve the nonlinear vibration equations. We consider the generalized fifth-order vibration equation of the form

$$\frac{\partial \varphi }{\partial t}+\alpha \phi \frac{{\partial }^3\varphi }{\partial x^3}+\beta \frac{\partial \varphi }{\partial x}\frac{{\partial }^2\varphi }{\partial x^2}+\gamma {\phi }^2\frac{\partial \varphi }{\partial x}+\frac{{\partial }^5\varphi }{\partial x^5}=0$$

(5)

where $\alpha ,\beta$ and $\gamma$ are the arbitrary constants. We can assign different values to $\alpha ,\beta$ and $\gamma$ s in, which shows different variations on the properties of equation (5). For example, when $\alpha =10,\beta =20$ and $\gamma =30,$ then equation (5) becomes Lax equation and equation (5) becomes Sawada-Kotera (SK) equation if $\alpha =3,\beta =6$ and $\gamma =2$ , representing two different equations namely N-soliton solution and conservation laws with an infinite set. The nonlinear vibration equations with $\alpha =5,\beta =\frac{25}{2}$ and $\gamma =5$ its N-soliton solution are not known, but it is known to be integrable and have bilinear representation. In this article, we use the combination of homotopy perturbation method HPM^1,2 and Laplace transform introduced by He^3,4 and determined its various application.^3,4 Many researchers applied HPM and its modified forms^5,6 to solve linear and nonlinear problems arise in engineering and applied science problems. The He–Laplace method is the modified form of He’s HPM, ¹ introduced by Khan and Wu. ⁷ Khan and Wu ⁷ combined the idea of He’s homotopy perturbation method and Laplace transform for the numerical treatment of different nonlinear problems arising in various physical phenomena. The He–Laplace method is the most suitable method to solve the physical model problems because no linearization of He’s HPM method and its modified form was used by many researchers to solve nonlinear problems arising in different fields.^8,9 Assumptions and restrictions are required for He–Laplace method, which may amend the problem. He–Laplace method essentially demonstrates how we can calculate the approximate solution of nonlinear differential equations using the combination of homotopy perturbation method and Laplace transform. A solution obtained by He–Laplace method is in the form of rapidly converging series. Sometimes, it may appear in the closed form.

Many researchers studied the nonlinear vibration equations Hereman and Nuseir, ¹⁰ Hone ¹¹ scaled the similar solution of three integrable PDEs namely KK equation, fifth-order KdV equation, and SK equation. Hu et al. ¹² determined the solution of (2 + 1) dimensional KK equation. Parker^13,14 obtained the N-soliton solution of SK and KK equation. Musette and Verhoeven ¹⁵ achieved the solution of fifth-order KK equations of integrable cases of Henon–Heiles system. Tam and Hu ¹⁶ found three Soliton solutions of KK-type equations using Hirota method and Mathematica. Das and Popowicz ¹⁷ studied the nonlinear dispersive fifth-order integrable equation and its hierarchy. Inc ¹⁸ determine the Soliton solution of the KK equation numerically and convergence analysis of the decomposition method. Recently, Al-Amr^19–29 used the reduced differential transform to study the KK equation.

In the next section, the basic idea of Laplace transform and homotopy perturbation method is discussed. Implementation of the proposed method for the nonlinear problem is then explained. The solution of the problems is given in the subsequent section. Then convergence analysis of proposed method is discussed and conclusion is drawn in the last section.

He–Laplace method

To demonstrate the idea of He–Laplace method, assume partial differential equation which is nonlinear and non-homogeneous with initial conditions of the form

$$D\varphi \left(x,t\right)+R\varphi \left(x,t\right)+N\varphi \left(x,t\right)=f\left(x,t\right)$$

(6)

$$\varphi \left(x,0\right)=g\left(x\right)$$

(7)

where $g\left(x\right)$ $\in C\left(R_{+}\right),$ D represents the linear differential operator of the first order, that is $D=\frac{\partial }{\partial t}$ and R denotes the linear differential operator, and N denotes the nonlinear differential operator. $f\left(x,t\right)$ is the source term. Apply the Laplace transform on both sides of equation (6)

$$L\left[D\varphi \left(x,t\right)\right]+L\left[R\varphi \left(x,t\right)\right]+L\left[N\varphi \left(x,t\right)\right]=L\left[f\left(x,t\right)\right]$$

(8)

Using the differential property of Laplace transform, we have

$$L\left[\varphi \left(x,t\right)\right]=\frac{g\left(x\right)}{s}+\frac{1}{s}L\left[f\left(x,t\right)\right]-\frac{1}{s}L\left[R\varphi \left(x,t\right)\right]-\frac{1}{s}L\left[N\varphi \left(x,t\right)\right]$$

(9)

Operating Laplace inverse transforms on both sides of equation (9) gives

$$\varphi \left(x,t\right)=G\left(x,t\right)-L^{-1}\left[\frac{1}{s}L\left[R\varphi \left(x,t\right)+N\varphi \left(x,t\right)\right]\right]$$

(10)

where $G\left(x,t\right)$ represents the terms obtained from the source term and prescribed the initial conditions. He–Laplace method admits the solution in the form

$$\varphi \left(x,t\right)={\displaystyle \sum _{n=0}^{\infty }{p}^n}{\varphi }_n\left(x,t\right)$$

(11)

and nonlinear terms can decompose as

$$N\varphi \left(x,t\right)={\displaystyle \sum _{n=0}^{\infty }{p}^n}{H}_n\left(\varphi \right)$$

(12)

Using He’s polynomial

$$H\left({\varphi }_0,{\varphi }_1,\mathrm{\dots ..}{\varphi }_n\right)n=\frac{1}{n!}\frac{{\partial }^n}{\partial p^n}\left[N\left({\displaystyle \sum _{i=0}^{\infty }{p}^i}{\varphi }_i\right)\right]$$

(13)

substitute equations (11) and (12) in equation (10), according to He–Laplace method, a homotopy is constructed of the form

$${\displaystyle \sum _{n=0}^{\infty }{p}^n}{\varphi }_n\left(x,t\right)=G\left(x,t\right)-p\left(L^{-1}\left[\frac{1}{s}L\left[R{\displaystyle \sum _{n=0}^{\infty }{p}^n}{\varphi }_n\left(x,t\right)+{\displaystyle \sum _{n=0}^{\infty }{p}^n}{H}_n\left(\varphi \right)\right]\right]\right)$$

(14)

which is the coupling of Laplace transform and homotopy perturbation method using He’s polynomials. Equating the coefficients of like powers of p, we have

$$p^0:{\varphi }_0\left(x,t\right)=G\left(x,t\right)$$

(15)

$$p^1:{\varphi }_1\left(x,t\right)=L^{-1}\left[\frac{1}{s}L\left[R{\varphi }_0\left(x,t\right)+H_0\left(\varphi \right)\right]\right]$$

(16)

$$p^2:{\varphi }_2\left(x,t\right)=L^{-1}\left[\frac{1}{s}L\left[R{\varphi }_1\left(x,t\right)+H_1\left(\varphi \right)\right]\right]$$

(17)

$$p^3:{\varphi }_3\left(x,t\right)=L^{-1}\left[\frac{1}{s}L\left[R{\varphi }_2\left(x,t\right)+H_2\left(\varphi \right)\right]\right]$$

(18)

The general recursive relation is given by

$$p^{n+1}:{\varphi }_{n+1}\left(x,t\right)=L^{-1}\left[\frac{1}{s}L\left[R{\varphi }_n\left(x,t\right)+H_n\left(\varphi \right)\right]\right]$$

(19)

Implementation of the method

Consider the generalized fifth-order nonlinear evolution equation given in equation (5) known as KK equation of the form

$$\varphi \left(x,0\right)=g\left(x\right)$$

(20)

First, we apply the Laplace transform on both sides of equation (5)

$$\begin{array}{l}L\left[\frac{\partial \varphi }{\partial t}\right]=-\alpha L\left[\varphi \frac{{\partial }^3\varphi }{\partial x^3}\right]-\beta L\left[\frac{\partial \varphi }{\partial x}\frac{{\partial }^2\varphi }{\partial x^2}\right]-\gamma L\left[{\varphi }^2\frac{\partial \varphi }{\partial x}\right]-L\left[\frac{{\partial }^5\varphi }{\partial x^5}\right]\\ sL\left[\varphi \left(x,t\right)\right]-\varphi \left(x,0\right)=-\alpha L\left[\varphi \frac{{\partial }^3\varphi }{\partial x^3}\right]-\beta L\left[\frac{\partial \varphi }{\partial x}\frac{{\partial }^2\varphi }{\partial x^2}\right]-\gamma L\left[{\varphi }^2\frac{\partial \varphi }{\partial x}\right]-L\left[\frac{{\partial }^5\varphi }{\partial x^5}\right],\\ L\left[\varphi \left(x,t\right)\right]=\frac{1}{s}g\left(x\right)-\frac{1}{s}\alpha L\left[\varphi \frac{{\partial }^3\varphi }{\partial x^3}\right]-\frac{1}{s}\beta L\left[\frac{\partial \varphi }{\partial x}\frac{{\partial }^2\varphi }{\partial x^2}\right]-\frac{1}{s}\gamma L\left[{\varphi }^2\frac{\partial \varphi }{\partial x}\right]-\frac{1}{s}L\left[\frac{{\partial }^5\varphi }{\partial x^5}\right],\\ \hspace{0.17em}=\frac{1}{s}g\left(x\right)-\frac{1}{s}L\left[\alpha L\varphi \frac{{\partial }^3\varphi }{\partial x^3}+\beta \frac{\partial \varphi }{\partial x}\frac{{\partial }^2\varphi }{\partial x^2}+\gamma {\varphi }^2\frac{\partial \varphi }{\partial x}+\frac{{\partial }^5\varphi }{\partial x^5}\right]\end{array}$$

(21)

Apply the inverse Laplace transform to both sides of equation (21) and using equation (20), we get

$$\varphi \left(x,t\right)=g\left(x\right)-L^{-1}\left[\frac{1}{s}L\left[\alpha L\varphi \frac{{\partial }^3\varphi }{\partial x^3}+\beta \frac{\partial \varphi }{\partial x}\frac{{\partial }^2\varphi }{\partial x^2}+\gamma {\varphi }^2\frac{\partial \varphi }{\partial x}+\frac{{\partial }^5\varphi }{\partial x^5}\right]\right]$$

(22)

Now according to the homotopy perturbation method, we construct a homotopy of the form

$$\begin{array}{c}{\displaystyle \sum _{n=0}^{\infty }{p}^n}{\varphi }_n\left(x,t\right)=g\left(x\right)-p\left(L^{-1}\left[\frac{1}{s}L\left[{\displaystyle \sum _{n=0}^{\infty }{p}^n}{H}_{1n}\left(\varphi \right)+{\displaystyle \sum _{n=0}^{\infty }{p}^n}{H}_{2n}\left(\varphi \right)+{\displaystyle \sum _{n=0}^{\infty }{p}^n}{H}_{3n}\left(\varphi \right)+{\displaystyle \sum _{n=0}^{\infty }{p}^n}{\varphi }_n\left(x,t\right)\right]\right]\right)\end{array}$$

(23)

where $H_{1n}\left(\varphi \right)$ , $H_{2n}\left(\varphi \right)$ and $H_{3n}\left(\varphi \right)$ are the He’s polynomials representing the nonlinear terms in the solution.

Comparing the coefficients of like powers of p on both sides of equation (23), we get

$$p^0:{\varphi }_0\left(x,t\right)=g(x)$$

(24)

$$p^1:{\varphi }_1\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left[H_{10}\left(\varphi \right)+H_{20}\left(\varphi \right)+H_{30}\left(\varphi \right)+{\varphi }_0\left(x,t\right)\right]\right]$$

(25)

$$p^2:{\varphi }_2\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left[H_{11}\left(\varphi \right)+H_{21}\left(\varphi \right)+H_{31}\left(\varphi \right)+{\varphi }_1\left(x,t\right)\right]\right]$$

(26)

$$p^3:{\varphi }_3\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left[H_{12}\left(\varphi \right)+H_{22}\left(\varphi \right)+H_{32}\left(\varphi \right)+{\varphi }_2\left(x,t\right)\right]\right]$$

(27)

$$p^4:{\varphi }_4\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left[H_{13}\left(\varphi \right)+H_{23}\left(\varphi \right)+H_{33}\left(\varphi \right)+{\varphi }_3\left(x,t\right)\right]\right]$$

(28)

$$p^{n+1}:{\varphi }_{n+1}\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left[H_{1n}\left(\varphi \right)+H_{2n}\left(\varphi \right)+H_{3n}\left(\varphi \right)+{\varphi }_n\left(x,t\right)\right]\right]$$

(29)

Where some of the He’s polynomial used are as follows

$$H_{10}\left(\varphi \right)={\varphi }_0\frac{{\partial }^3{\varphi }_0}{\partial x^3}$$

(30)

$$H_{11}\left(\varphi \right)={\varphi }_0\frac{{\partial }^3{\varphi }_1}{\partial x^3}+{\varphi }_1\frac{{\partial }^3{\varphi }_0}{\partial x^3}$$

(31)

$$H_{12}\left(\varphi \right)={\varphi }_0\frac{{\partial }^3{\varphi }_2}{\partial x^3}+{\varphi }_1\frac{{\partial }^3{\varphi }_1}{\partial x^3}+{\varphi }_2\frac{{\partial }^3{\varphi }_0}{\partial x^3}$$

(32)

$$H_{13}\left(\varphi \right)={\varphi }_0\frac{{\partial }^3{\varphi }_3}{\partial x^3}+{\varphi }_1\frac{{\partial }^3{\varphi }_2}{\partial x^3}+{\varphi }_2\frac{{\partial }^3{\varphi }_1}{\partial x^3}+{\varphi }_3\frac{{\partial }^3{\varphi }_0}{\partial x^3}$$

(33)

$$H_{20}\left(\varphi \right)=\frac{\partial {\varphi }_0}{\partial x}\frac{{\partial }^2{\varphi }_0}{\partial x^2}$$

(34)

$$H_{21}\left(\varphi \right)=\frac{\partial {\varphi }_0}{\partial x}\frac{{\partial }^2{\varphi }_1}{\partial x^2}+\frac{\partial {\varphi }_1}{\partial x}\frac{{\partial }^2{\varphi }_0}{\partial x^2}$$

(35)

$$H_{22}\left(\varphi \right)=\frac{\partial {\varphi }_0}{\partial x}\frac{{\partial }^2{\varphi }_2}{\partial x^2}+\frac{\partial {\varphi }_1}{\partial x}\frac{{\partial }^2{\varphi }_1}{\partial x^2}+\frac{\partial {\varphi }_2}{\partial x}\frac{{\partial }^2{\varphi }_0}{\partial x^2}$$

(36)

$$H_{23}\left(\varphi \right)=\frac{\partial {\varphi }_0}{\partial x}\frac{{\partial }^2{\varphi }_3}{\partial x^2}+\frac{\partial {\varphi }_1}{\partial x}\frac{{\partial }^2{\varphi }_2}{\partial x^2}+\frac{\partial {\varphi }_2}{\partial x}\frac{{\partial }^2{\varphi }_1}{\partial x^2}+\frac{\partial {\varphi }_3}{\partial x}\frac{{\partial }^2{\varphi }_0}{\partial x^2}$$

(37)

$$H_{30}\left(\varphi \right)={\varphi }_0^2\frac{\partial {\varphi }_0}{\partial x}$$

(38)

$$H_{31}\left(\varphi \right)=2{\varphi }_0{\varphi }_1\frac{\partial {\varphi }_0}{\partial x}+{\varphi }_0^2\frac{\partial {\varphi }_1}{\partial x}$$

(39)

$$H_{32}\left(\varphi \right)=2{\varphi }_0{\varphi }_2\frac{\partial {\varphi }_0}{\partial x}+{\varphi }_1^2\frac{\partial {\varphi }_0}{\partial x}+2{\varphi }_0{\varphi }_1\frac{\partial {\varphi }_2}{\partial x}+{\varphi }_0^2\frac{\partial {\varphi }_2}{\partial x}$$

(40)

$$H_{33}\left(\varphi \right)=\left(2{\varphi }_0{\varphi }_3+2{\varphi }_1{\varphi }_2\right)\frac{\partial {\varphi }_0}{\partial x}+{\varphi }_1^2\frac{\partial {\varphi }_1}{\partial x}+2{\varphi }_0{\varphi }_2\frac{\partial {\varphi }_1}{\partial x}+2{\varphi }_0{\varphi }_1\frac{\partial {\varphi }_2}{\partial x}+{\varphi }_0^2\frac{\partial {\varphi }_3}{\partial x}$$

(41)

And so on.

Illustration of the method by numerical examples

Example

Consider

$${\phi }_t+6\phi {\phi }_x+{\phi }_{xxx}=0$$

(42)

$$\phi \left(x,0\right)=2x$$

(43)

Applying the He–Laplace method to equation (42) and using equation (43), we get

$${\displaystyle \sum }_{n=0}^{\infty }{P}^n{\phi }_n\left(x,t\right)=2x-L^{-1}\left[\frac{1}{s}L\left({\left({\displaystyle \sum }_{n=0}^{\infty }{P}^n{\phi }_n\left(x,t\right)\right)}_{xxx}+6\left({\displaystyle \sum }_{n=0}^{\infty }{P}^n{H}_n\left(\phi \right)\right)\right)\right]$$

(44)

Now, equating the like powers of p and using the values of He’s polynomial from equation (34) to equation (41) to obtain values ${\varphi }_1\left(x,t\right)\left(i=0,1,\mathrm{2\dots \dots .}\right)$ .

$$P^0:{\phi }_0\left(x,t\right)=2x$$

(45)

$$P^1:{\phi }_1\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left({\left({\phi }_0\right)}_{xxx}+6\left(H_0\left(\phi \right)\right)\right)\right]=-24tx$$

(46)

$$P^2:{\phi }_2\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left({\left({\phi }_1\right)}_{xxx}+6\left(H_1\left(\phi \right)\right)\right)\right]=288t^{2}x$$

(47)

$$P^3:{\phi }_3\left(x,t\right)=-L^{-1}\left[\frac{1}{s}L\left({\left({\phi }_2\right)}_{xxx}+6\left(H_2\left(\phi \right)\right)\right)\right]=-3456t^{3}x$$

(48)

and so on $\phi \left(x,t\right)$ is the approximate solution of (42) obtained by the He–Laplace method

$$\begin{array}{l}\phi \left(x,t\right)=2x-24tx+288t^{2}x-3456t^{3}x+\dots \\ \phi \left(x,t\right)=2x\left[1-\left(12t\right)+{\left(12t\right)}^2-{\left(12t\right)}^3+\dots \right]\\ \phi \left(x,t\right)=\frac{2x}{1+12t}\end{array}$$

(49)

which is the closed form solution of equation (42).

Numerical results obtained by the proposed method show that a reasonable approximation has been achieved with exact solution as shown in Figure 1. By the addition of more approximation terms, absolute error can be minimized. Numerical results proved the accuracy and efficiency of the proposed method. Furthermore, discretization of variables is not necessarily required for the He–Laplace method.

Figure 1.

Representing the numerical solution of vibration equation.

Conclusion

In this manuscript, we presented the periodic solitary solutions of nonlinear vibration equations. The periodic soliton solution obtained by our method captures all the characteristics of vibrational properties that are associated with periodic solitons. We also studied the bifurcation and chaotic solutions of nonlinear wave equations, and the influence of integration constant on the periodicity of the solution of nonlinear vibration equations, which is named as bifurcation