Application of modified wavelet and homotopy perturbation methods to nonlinear oscillation problems

Open access

Abstract

In this paper, an accurate and efficient Chebyshev wavelet-based technique is successfully employed to solve the nonlinear oscillation problems. Numerical examples are also provided to illustrate the efficiency and performance of these methods. Homotopy perturbation methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations. In addition, the use of Chebyshev wavelet is found to be simple, flexible, accurate, efficient and less computational cost. Our analytical results are compared with simulation results and found to be satisfactory.

Abstract

In this paper, an accurate and efficient Chebyshev wavelet-based technique is successfully employed to solve the nonlinear oscillation problems. Numerical examples are also provided to illustrate the efficiency and performance of these methods. Homotopy perturbation methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations. In addition, the use of Chebyshev wavelet is found to be simple, flexible, accurate, efficient and less computational cost. Our analytical results are compared with simulation results and found to be satisfactory.

1 Introduction

Most of the oscillation problems in engineering sciences are nonlinear, and it is difficult to solve such equations analytically. Recently, nonlinear oscillator models have been widely considered in physical and chemical sciences. Due to the limitation of existing exact solutions, many approximate analytical and numerical approaches have been investigated. Many real-life problems that arise in several branches of pure and applied science can be expressed using the nonlinear differential equations. Therefore, these nonlinear equations must be solved using analytical/numerical methods. Many researchers have been working on various analytical methods for solving nonlinear oscillation systems in the last decades. Nowadays, the computational experience is significant, and several numerical methods have been suggested and analyzed under certain conditions. These numerical methods have been developed using different techniques such as Taylor series, homotopy perturbation method, quadrature formula, variation iteration method and decomposition method [1, 2, 3, 4, 5, 6, 7]. Noor et al. [8] have applied a sixth-order predictor–corrector iterative method for solving the nonlinear equations. Chun [9] proposed a class of fifth-order and sixth-order iterative methods. In this paper, we apply the wavelet transform method and homotopy perturbation method for solving the nonlinear oscillation equations. The homotopy perturbation method was developed by He [10, 11] and has been applied to a wide class of nonlinear and linear problems arising in various branches.

Recently, the homotopy perturbation method (HPM) [12, 13, 14, 15, 16] with expanding parameter is applied to solve some of the nonlinear equations. Ren and Hu [17], Yu et al. [18] and Wu and He [19] developed the nonlinear oscillation problem using HPM. Abbasbandy et al. [20] developed the homotopy perturbation method and the decomposition method [21] for solving nonlinear equations. Ganji et al. [22] applied the oscillation systems with nonlinearity terms such as the motion of a rigid rod rocking. Khudayarov and Turaev [23] developed the mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid. Nasab et al. [24] solved the nonlinear singular boundary value problems.

To our knowledge, no exact analytical expressions for oscillation problems are reported. However, it is difficult for us to obtain the exact solution for these problems. The purpose of this paper was to derive the approximate analytical expressions for some oscillation problems in engineering sciences.

2 Mathematical formulation of the problems

The general form of the differential equation describing the oscillations of single-degree-of-freedom systems can be written as [25]:

x¨+F(x)=0

with initial conditions:

x(0)=landx˙(0)=0

where F(x) represents the linear and nonlinear terms. A typical nonlinear conservative system that has been the subject of many investigations is Duffing-type oscillator. This nonlinear oscillator represents the dynamic behaviour of many engineering problems. Exact analytical solutions to the oscillatory problem in the form of Eq. (1) are generally impossible, and therefore, some numerical solution is to be reported. The main goal is to apply the discretization and linearization concepts to develop the HPM and CWM which can provide periodic solutions for oscillatory problems.

3 Some properties of shifted second kind Chebyshev polynomials [26]

In this section, we discuss some relevant properties of the function

Un(x)=sin(n+1)θsinθ,x=cosθ

These polynomials are orthogonal on [–1, 1]

ie.,111x2Um(x)Un(x)dx=0,formn,π2,form=n.

The following properties of second kind Chebyshev polynomials [26] are of fundamental importance in the sequel. They are eigenfunctions of the following singular Sturm–Liouville equation:

(1x2)D2φk(x)3xDφk(x)+k(k+2)φk(x)=0,

where Dddx, and orthogonal polynomial may be generated using the recurrence relation

Uk+1(x)=2xUk(x)Uk1(x),k=1,2,3,...,

starting from U0(x) = 1 and U1(x) = 2x, or from Rodriguez formula

Un(x)=(2)n(n+1)!(2n+1)!1x2Dn(1x2)n+12

3.1 Shifted second kind Chebyshev polynomials

The shifted second kind Chebyshev polynomials are defined on the interval [0, 1] by Un(x) = Un(2x – 1). All the results of second kind Chebyshev polynomials can be easily transformed to give the corresponding results for their shifted forms. The orthogonality relation with respect to the weight function xx2 is given by

01xx2Un(x)Um(x)dx=0,formn,π8,form=n.

The first derivative Un(x) is given in the following corollary.

Corollary 1

The first derivative of the shifted second kind Chebyshev polynomial is given by

DUn(x)=4k=0,(k+n)oddn1(k+1)Uk(x)

3.2 Shifted second kind Chebyshev operational matrix of derivatives (S2KCOM)

Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelet:

ψa,b(t)=a12ψtba,a,b,a0

Shifted second kind Chebyshev wavelets ψnm(t) = ψ(k, n, m, t) have four arguments: k, n can assume any positive integer, m is the order of second kind Chebyshev polynomials and tis the normalized time. They are defined on the interval [0, 1] by

ψnmt=2k+32πUm2ktnfortn2k,n+12km=0,1,.....M,n=0,1,.....,2k10,forotherwise

3.3 Function Approximation

A function f(t) defined on the interval [0, 1] may be expanded in terms of second kind Chebyshev wavelets as

f(t)=n=0m=0cnmψnm(t),

where

cnm=f(t),ψnm(t)ω=01ω(t)f(t)ψnm(t)dt,

and ω(t)=tt2. If the infinite series is truncated, then f(t) can be approximated as

f(t)n=02k1m=0Mcnmψnm(t)=CTψ(t),

where C and ψ(t) are 2k(M + 1) × 1 matrices defined by

C=c0,0,c0,1,...c0,M,....c2k1,M,....c2k1,1,...........c2k1,MT

ψ(t)=ψ0,0(t),ψ0,1(t),..........ψ0,M(t),.....ψ2k1,M(t),...............ψ2k1,1(t),.....ψ2k1,M(t)T

3.4 Solution pertaining to the nonlinear differential equations using shifted second kind Chebyshev wavelet method (S2KCWM) and homotopy perturbation method

In this section, we give some numerical results obtained using the algorithms presented in the previous sections. We consider the following examples.

Example 1

We consider the nonlinear differential equation [27]

x¨+x1+μx2=0

with initial conditions:

x(0)=landx˙(0)=0

Wavelet method:

Eq. (17) is solved using the wavelet scheme,

CTD2ψ(t)+CTψ(t)1+μCTψ(t)2=0

Moreover, the initial conditions are

CTψ(t)=l,CTDψ(t)=0,att=0

Using Eq. (19) and Eq. (20), the following system of algebraic equations can be obtained

64c2+2c0+8t4c1+32t232t+6c21+μ2c0+8t4c1+32t232t+6c22=0

2c04c1+6c2=l

8c132c2=0

The suggested method of solution x(t) is approximated as follows (Appendix B):

x(t)=CTψ(t)

Homotopy perturbation method:

Solving the nonlinear Eq. (17) using a new approach of homotopy perturbation method (Appendix A)

x(t)=costl+l3μ6450l3μ218432+cos3t13l4μ26144l3μ64+cos5t11l4μ218432

The velocity becomes

x˙(t)=sintll3μ64+50l3μ218432+3sin3t13l4μ26144+l3μ645sin5t11l4μ218432

Example 2

We consider equation [28]

x¨+x1+αx+βx2=0

with the initial conditions

x(0)=landx˙(0)=0

Wavelet method:

The wavelet scheme of Eq. (27) is

CTD2ψ(t)+CTψ(t)1+αCTψ(t)+βCTψ(t)2=0

The initial conditions are

CTψ(t)=l,CTDψ(t)=0,att=0

Using Eq. (29) and Eq. (30), the following system of algebraic equations can be obtained

64c2+2c0+8t4c1+32t232t+6c21+α2c0+8t4c1+32t232t+6c2+β2c0+8t4c1+32t232t+6c22=0

2c04c1+6c2=l

8c132c2=0

The solution of x(t) is approximated as follows (Appendix B):

x(t)=CTψ(t)

Homotopy perturbation method:

Solving the nonlinear Eq. (27) using a new approach of homotopy perturbation method, we get

x(t)=lcost+sin2t22αl23+βl34cost+βl38cos2t+4αl23+βl38

The velocity becomes

x˙(t)=lsint+2sint2cost22αl23+βl34sintβl34sin2t

Example 3

We consider the differential equation [29]

u¨+u+ςu3=0

with initial conditions:

u(0)=landu˙(0)=0

Wavelet method:

Eq. (37) is solved using the wavelet scheme,

CTD2ψ(t)+CTψ(t)+ςCTψ(t)3=0

Moreover, the initial conditions are

CTψ(t)=l,CTDψ(t)=0,att=0

Using Eq. (39) and Eq. (40), the following system of algebraic equations can be obtained

64c2+2c0+8t4c1+32t232t+6c2+ς2c0+8t4c1+32t232t+6c23=0

2c04c1+6c2=l

8c132c2=0

The solution of u(t) is approximated as follows (Appendix B):

u(t)=CTψ(t)

Homotopy perturbation method:

Solving the nonlinear Eq. (37) using a new approach of homotopy perturbation method, we get

u(t)=lcostςl332costcos3t

The velocity becomes

u˙(t)=lsint+ςl332sint3sin3t

4 Numerical simulation and discussion

To illustrate the applicability, accuracy and effectiveness of the proposed method, we have compared the approximate analytical solution of the nonlinear differential equations with numerical data. The function ode 45 (Runge–Kutta method) in MATLAB software, which is a function of solving the initial value problems, is used. In Figure 1(a–c), we have plotted the numerical solution and the approximate solution derived by our proposed method using HPM and CWM. The figure shows the behaviour of the solution for various values of the parameter. From the figure, it is observed that the variation in the approximate solution is small, when μ ≤ 1. From Figure 1(a–c), it is also observed that the amplitude depends upon the initial conditions. Figure 2 represents the displacement and velocity versus time t. From the figure, it is noted that the amplitude of displacement and velocity are equal.

Fig. 1
Fig. 1

(a–c) Comparison of CWM (Eq.(24), HPM (Eq.(25) and numerical method (MATLAB result) for various parameter values. Fig.1(a) l = 0.5 and μ = 0.01 Fig.1(b) l = 0.5 and μ = 0.1 Fig.1(c) l = 0.5 and μ = 1.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00030

Fig. 2
Fig. 2

Plot of displacement and velocity for oscillator Eq. (26) with weak nonlinearity and small amplitude oscillations l = 1.1 and μ = 0.1.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00030

Figure 3(a–c) denotes the displacement versus time for various values of the parameters α and β. The numerical solution is compared with our analytical results in Figure 3(a–c) and found to be satisfactory. Displacement and velocity are shown in Figure 4. Here also, the amplitude of displacement and velocity are equal. Figure 5 shows the displacement versus time for the oscillation problem (Eq. (45)). Displacement and velocity versus time are shown in Figure 6. All the result also confirmed for the problem Eq. (37).

Fig. 3
Fig. 3

Comparison of CWM (Eq. (34), HPM (Eq. (35) and numerical method (MATLAB result) for various parameter values. Fig. 3(a) l = 0.1, α = 1 and β = 0.5 Fig.3(b) l = 0.1, α = 1 and β = 2 Fig. 3(c) l = 0.1, α = 2 and β = 0.5.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00030

Fig. 4
Fig. 4

Plot of displacement and velocity for oscillator Eq. (36) with weak nonlinearity and small amplitude oscillations l = 0.1, α = 1, β = 2.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00030

Fig. 5
Fig. 5

Comparison of CWM (Eq. (44), HPM (Eq. (45) and numerical method (MATLAB result) for fixed parameter values l = 0.5, ς = 0.2.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00030

Fig. 6
Fig. 6

Plot of displacement and velocity for oscillator Eq. (46) with weak nonlinearity and small amplitude oscillations l = 0.5, ς = 0.2.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00030

5 Conclusion

In this paper, a wavelet technique has been employed for the approximate solution successfully to solve the well-known nonlinear oscillator differential equations such as Duffing equation with different parameters. There is no need for iterations for achieving sufficient accuracy in numerical results. The results are also obtained via CWM, HPM and numerical solution. Moreover, the proposed method is used to compare CWM, HPM, and NM iteration with the nonlinear part. The effects of constant parameters on responses of the system for approximate solution are also shown in figures.

Communicated by Juan Luis García Guirao
Acknowledgment

The authors thank Shri J. Ramachandran, Chancellor, and Col. Dr. G. Thiruvasagam, Vice-Chancellor, Academy of Maritime Education and Training (AMET), Deemed to be University, Chennai, India, for their constant encouragement.

Appendix A

Approximate analytical solution of oscillations [27] using HPM

Eq. (17) can be rewritten as follows:

x¨(t)+x(t)+μx(t)2x¨(t)=0

This equation is in the form

x¨(t)+x(t)=μfx,x˙,x(i)

where

fx,x˙,x(i)=x(t)2x¨(t)

The solution of Eq. (A.2) is

x(t)=x0+k1μkx0(k)(t)k!

Using HPM, we get the following Eq. (A.1)

x¨0(0)(t)+x0(0)(t)=0

x¨0(1)(t)+x0(1)(t)=fx0,x˙0,x0(i)Λ0(1)x0(t)

x¨0(2)(t)+x0(2)(t)=2f(1)x0,x˙0,x0(i)2Λ0(1)_Λ0(2)x0(t)

The solution of Eq. (A.5) with initial conditions and is x0(0)(0)=lx˙0(0)(0)=0,

x0(0)(t)=lcost

From Eq. (A.5), Eq. (A.8) and Eq. (A.6), we yield,

x¨0(1)(t)+x0(1)(t)=μl3cos3tΛ0(1)lcost=lcost3l24Λ0(1)+l3μ4cos3t

Neglect the presence of a secular term in Eq. (A.9),

Λ0(1)3l24=0

From Eq. (A.9) using initial conditions x0(1)(0)=0andx˙0(1)(0)=0,

x0(1)(t)=l3μ32costcos3t

Using Eqns. (A.8), (A.10), and (A.11) in Eq. (A.7), we yields,

x¨0(2)(t)+x0(2)(t)=3l4128Λ0(2)lcost13l4μ2128cos3t11l4μ2128cos5t

Neglect the presence of a secular term in Eq. (A.12),

Λ0(2)3l4128=0

From Eq. (A.12) using initial conditions: x0(2)(0)=0andx˙0(2)(0)=0, we get

x0(2)(t)=13l4μ21024cos3tcost+11l4μ23072cos5tcost

Substituting x0(0)(t),x0(1)(t),x0(2)(t)) in Eq. (A.4), we get the solution of Eq. (25) in the text.

Approximate analytical solution of oscillations [28] using HPM

Eq. (27) can be rewritten as follows:

x¨(t)+x(t)+αx(t)+βx(t)2x¨(t)=0

This equation is in the form

x¨(t)+x(t)=fx,x˙,x(i)

where fx,x˙,x(i)=αx(t)+βx(t)2x¨(t)

The above equation can be rewritten as follows:

x¨0(0)(t)+x0(0)(t)=0

x¨0(1)(t)+x0(1)(t)=fx,x˙,x(i)Λ0(1)x0(t)

The solution of Eq. (A.17) with initial conditions and x0(0)(0)=l is x˙0(0)(0)=0,

x0(0)(t)=lcost

From Eq. (A.17), Eq. (A.19) and Eq. (A.18), we yield,

x¨0(1)(t)+x0(1)(t)=αl2cos2t+βl3cos3tΛ0(1)lcost

=3βl34Λ0(1)lcost+αl221+cos2t+βl34cos3t

Avoid the presence of a secular term in Eq. (A.21), i.e.,

Λ0(1)3βl34=0

The solution of Eq. (A.22) using initial conditions x0(1)=0andx˙0(1)=0, can be obtained as follows:

x0(1)(t)=sin2t22αl23+βl34cost+βl38cos2t+4αl23+βl38

Substituting x0(0)(t),x0(1)(t) in Eq. (A.4), we get a solution of Eq. (35) in the text.

Approximate analytical solution of oscillations [29] using HPM

Eq. (37) can be rewritten as follows:

u¨(t)+u(t)=ςu3

This equation is in the form

u¨(t)+u(t)=ςfu,u˙,u(i)

where fu,u˙,u(i)=u(t)3

The above equation can be rewritten as follows:

u¨0(0)(t)+u0(0)(t)=0

u¨0(1)(t)+u0(1)(t)=ςfu,u˙,u(i)Λ0(1)lcost

We obtain the solution of Eq. (A.26) with initial conditions u0(0)(0)=landu˙0(0)(0)=0, and we get

u0(0)(t)=lcost

From Eq. (A.26), Eq. (A.28) and Eq. (A.27), we yield,

u¨0(1)(t)+u0(1)(t)=ςl3cos3tΛ0(1)lcost

=3ςl34Λ0(1)lcostςl34cos3t

Avoid the presence of a secular term in Eq. (A.30), i.e.,

Λ0(1)3ςl34=0

We obtain the solution of Eq. (A.30) using initial conditions u0(1)(0)=landu˙0(1)(0)=0, and we get

u0(1)(t)=ςl332cos(t)cos(3t)

Substituting u0(0)(t),u0(1)(t) in Eq. (A.4), we get a solution of Eq. (45) in the text.

Appendix B

Shifted second Chebyshev kind wavelets operational matrix of derivatives:

We solve the nonlinear equation using the algorithm described in shifted second kind Chebyshev wavelet method for the case corresponding to m = 2, k = 0. To obtain the approximate solution of x(t) and u(t). Using Eq. (17), Eq. (27), and Eq. (37), the two operational matrices D and D2 can be obtained

D=000400080,D2=0000003200

ψ(t)=2π28t432t232t+6,CT=π2c0c1c2

The second kind Chebyshev wavelet expansion of a function, to define the residual, ℜ (t) of this equation can be written as follows:

(t)=CTD2ψ(t)+F1Tψ(t)(ψ(t))TDTC+F2Tψ(t)(ψ(t))TCG1TTψ(t)

The approximate solution is

x(t)=CTψ(t)

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    C.Chun (2005) Iterative methods improving Newton’s method by the decomposition method. Computers and Mathematics with Applications 50 (10-12) 1559-1568.

    • Crossref
    • Export Citation
  • [2]

    C. Chun (2006) A new iterative method for solving nonlinear equations. Applied Mathematics and Computation 178 (2006) 415-422.

    • Crossref
    • Export Citation
  • [3]

    M. Javidi (2009) Fourth-order and fifth-order iterative methods for nonlinear algebraic equations. Mathematical and Computer Modelling. 50 66-71.

    • Crossref
    • Export Citation
  • [4]

    J.F. Traub (1964) Iterative methods for the solution of equations Prentice-Hall Englewood Cliffs NJ USA.

  • [5]

    M.A. Noor (2008) Extended general variational inequalities Applied Mathematics Letters. 22 182-186.

    • Crossref
    • Export Citation
  • [6]

    M.A. Noor (2010) Some iterative methods for solving nonlinear equations using homotopy perturbation method International Journal of Computer Mathematics 87 141-149.

    • Crossref
    • Export Citation
  • [7]

    M.A. Noor (2010) Iterative methods for nonlinear equations using homotopy perturbation technique. Applied Mathematics and Information Sciences. 4 227–235.

  • [8]

    M.A. Noor W.A. Khan A. Hussain (2006) A new modified Halley method without second derivatives for the nonlinear equation.. Applied Mathematics and Computation. 189 1268-1273.

    • Crossref
    • Export Citation
  • [9]

    C. Chun (2007) Some improvements of Jarrat’s methods with sixth order convergences. Applied Mathematics and Computation 190 1432–1437.

    • Crossref
    • Export Citation
  • [10]

    J.H. He (2006) Some asymptotic methods for strongly nonlinear mappings. International Journal of Modern Physics. 20 1144-1199.

    • Crossref
    • Export Citation
  • [11]

    J.H. He (1999) Homotopy perturbation technique Computer. Methods in Applied Mechanics and Engineering 178 257-262.

    • Crossref
    • Export Citation
  • [12]

    J.H. He (2012) Homotopy perturbation method with an auxiliary term Abstract and Applied Analysis Article ID 857612 .

    • Crossref
    • Export Citation
  • [13]

    J.H. He (2013) Homotopy perturbation method with two expanding parameters Indian Journal of Physics 88 193-196.

    • Crossref
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