Haar Wavelet Operational Matrix Method for Fractional Oscillation Equations

Abstract

We utilized the Haar wavelet operational matrix method for fractional order nonlinear oscillation equations and find the solutions of fractional order force-free and forced Duffing-Van der Pol oscillator and higher order fractional Duffing equation on large intervals. The results are compared with the results obtained by the other technique and with exact solution.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Marianna A. Shubov

School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan

Received 25 February 2014; Accepted 29 June 2014; 15 July 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Haar wavelet is the lowest member of Daubechies family of wavelets and is convenient for computer implementations due to availability of explicit expression for the Haar scaling and wavelet functions [1]. Operational approach is pioneered by Chen and Hsiao [2] for uniform grids. The basic idea of Haar wavelet technique is to convert differential equations into a system of algebraic equations of finite variables. The Haar wavelet technique for solving linear homogeneous/inhomogeneous, constant, and variable coefficients has been discussed in [3].

The fractional order forced Duffing-Van der Pol oscillator is given by the following second order differential equation [4]: [figure omitted; refer to PDF] where ^cDα is the Caputo derivative; g(f,ω,t)=fcos...(ωt) represents the periodic driving function of time with period T=2π/ω , where ω is the angular frequency of the driving force; f is the forcing strength; and μ>0 is the damping parameter of the system. Duffing-Van der Pol oscillator equation can be expressed in three physical situations:

(1) single-well a>0 , b>0 ;

(2) double-well a<0 , b>0 ;

(3) double-hump a>0 , b<0 .

The quasilinearization approach was introduced by Bellman and Kalaba [5, 6] as a generalization of the Newton-Raphson method [7] to solve the individual or systems of nonlinear ordinary and partial differential equations. The quasilinearization approach is suitable to general nonlinear ordinary or partial differential equations of any order.

The Haar wavelets with quasilinearization technique [8-10] are applied for the approximate solution of integer order nonlinear differential equations. In [11], we extend the Haar wavelet - quasilinearization technique for fractional nonlinear differential equations.

The aim of the present work is to investigate the solution of the higher order fractional Duffing equation, fractional order force-free and forced Duffing-Van der pol (DVP) oscillator using Haar wavelet-quasilinearization technique. We have discussed the three special situations of DVP oscillator equation such as single-well, double-well, and double- hump.

2. Preliminaries

In this section, we review basic definitions of fractional differentiation and fractional integration [12].

(1) Riemann-Liouville fractional integral operator of order α is as follows: the Riemann-Liouville fractional order integral of order α∈^R+ is defined as [figure omitted; refer to PDF] for a<x...4;b .

(2) Riemann-Liouville and Caputo fractional derivative operators of order α are as follows:the Riemann-Liouville fractional order derivative of order α∈^R+ is defined as [figure omitted; refer to PDF] for a<x...4;b , where n-1<α<n , n∈N , and n=[left ceiling]α[right ceiling] .

The Caputo fractional order derivative of order α∈^R+ is defined as [figure omitted; refer to PDF] for a<x...4;b , where n-1<α<n , n∈N , and n=[left ceiling]α[right ceiling] .

3. The Haar Wavelets

The Haar functions contain just one wavelet during some subinterval of time and remain zero elsewhere and are orthogonal. The i th uniform Haar wavelet _hi (x) , x∈[a,b] is defined as [2] [figure omitted; refer to PDF] where i=^2j +k+1 ; j=0,1,2,...,J is dilation parameter, where m=^2j and k=0,1,2,...,^2j -1 is translation parameter. J is maximal level of resolution and the maximal value of i is 2M where M=^2J . In particular _h1 (x)[: =]_χ[a,b] (x) , where _χ[a,b] (x) is characteristic function on interval [a,b] , is the Haar scaling function. For the uniform Haar wavelet, the wavelet-collocation method is applied. The collocation points for the uniform Haar wavelets are usually taken as _xj =a+(b-a)((j-0.5)/2M), where j=1,2,...,2M .

3.1. Fractional Integral of the Haar Wavelets

Any function y∈_L2 [a,b] can be represented in terms of the uniform Haar series [figure omitted; refer to PDF] where _bl are the Haar wavelet coefficients given as _bl =^∫-∞∞ ...y(x)_hl (x)dx .

Any function of two variables u(x,t)∈_L2 [a,b]×[a,b] can be approximated as [figure omitted; refer to PDF] where C is 2M×2M coefficient matrix which can be determined by the inner product _cl,i =Y9;_hl (x),Y9;u(x,t),_hi (t)YA;YA; .

The Riemann-Liouville fractional integral of the uniform Haar wavelets is given as [figure omitted; refer to PDF] where a(l)=a+(b-a)(k/m) , b(l)=a+(b-a)((k+0.5)/m) , and c(l)=a+(b-a)((k+1)/m) .

4. Convergence Analysis

Our work is based on quasilinearization technique and Haar wavelet method; first, we analyze the convergence of both schemes and then describe the role of their convergence according to present work.

4.1. Convergence of Quasilinearization Technique [6]

Consider the nonlinear second order differential equation: [figure omitted; refer to PDF]

Applying quasilinearization technique to (10) yields [figure omitted; refer to PDF] Let _y0 (x) be some initial approximation. Each function _yn+1 (x) is a solution of a linear equation (11), where _yn is always considered to be known and is obtained from the previous iteration.

According to [6] and letting _max...y ...(|f(y)|,|^{f[variant prime]} (y)|)=m<∞ and k=_max...u ...|^{f[variant prime][variant prime]} (u)| , we have [figure omitted; refer to PDF] This shows that quasilinearization technique has quadratic convergence, if there is convergence at all.

4.2. Convergence of Haar Wavelet Method [15]

Let y(x) be a differentiable function and assume that y(x) have bounded first derivative on (0,1) ; that is, there exist K>0 ; for all x∈(0,1) , [figure omitted; refer to PDF] Haar wavelet approximation for the function y(x) is given by [figure omitted; refer to PDF] Babolian and Shahsavaran [15] gave _L2 -error norm for Haar wavelet approximation, which is [figure omitted; refer to PDF] or [figure omitted; refer to PDF] As M=^2J and J is the maximal level of resolution, according to (16), we conclude that error is inversely proportional to the level of resolution. Equation (16) ensures the convergence of Haar wavelet approximation at higher level of resolution, that is, when M is increased.

Each iteration of quasilinearization technique gives linear differential equation in _yn+1 (x) which is solved to get approximate value of _yn+1 (x) , _yn+1,M (x) , by Haar wavelet method. Since solution of our problems has bounded first derivatives over (0,1) , according to (16), _yn+1,M (x) converges fast to _yn+1 (x) if we consider the higher level of resolution J ; that is, we get more accurate results while increasing J , and at the same time quasilinearization technique works; that is, given an initial approximation _y0 (x) , we get solution _y1 (x) of linear differential equation (11) by Haar wavelet method and at next iteration, we get _y2 (x) and so on. Since quasilinearization technique is second order accurate so it gives rapid convergence, if there is convergence at all. We conclude that solution by Haar wavelet quasilinearization technique _yn+1,M (x) converges to exact solution y(x) when both J and n approach ∞ .

5. Applications

In this section, we solve force-free Duffing-Van der Pol oscillator of fractional order, forced Duffing-Van der Pol oscillator of fractional order, and higher order fractional Duffing equation by the Haar wavelet-quasilinearization technique and compare the results with those obtained by other methods and exact solution.

5.1. Forced Duffing-Van Der Pol Oscillator Equation [4]

Example 1.

Consider the α th order fractional forced DVP oscillator equation [figure omitted; refer to PDF] subject to the initial conditions y(0)=1 and ^{y[variant prime]} (0)=0 .

Applying the quasilinearization technique to (17), we obtain [figure omitted; refer to PDF] with the initial conditions _yn+1 (0)=1 and ^{yn+1[variant prime]} (0)=0 .

Now we apply the Haar wavelet method to (18) and approximate the higher order derivative term by the Haar wavelet series as [figure omitted; refer to PDF]

Lower order derivatives are obtained by integrating (19) and use the initial condition [figure omitted; refer to PDF] Substitute (19) and (20) into (18) to get [figure omitted; refer to PDF] with the initial approximations _y0 (x)=1 and ^{y0[variant prime]} (x)=0 .

(1) (Single-well a>0 , b>0 ). a=0.5 , b=0.5 , μ=0.1 , f=0.5 , ω=0.79 .

(2) (Double-well a<0 , b>0 ). a=-0.5 , b=0.5 , μ=0.1 , f=0.5 , ω=0.79 .

(3) (Double-hump a>0 , b<0 ). a=0.5 , b=-0.5 , μ=0.1 , f=0.5 , ω=0.79 .

The results obtained using the Haar wavelet quasilinearization technique at fifth iteration for the three situations, single-well, double-well, and double-hump, are given in Tables 1, 2, and 3, respectively. Here, we fix the order of equation, α=2 , and level of resolution J=9 . We compared the obtained solution with variational iteration method [13], homotopy perturbation method [13], and numerical solution based on the fourth-order Runge-Kutta (RK) method. Also the absolute error relative to RK method is shown in Tables 1, 2, and 3. It shows that obtained results are more accurate as compared to variational iteration method and homotopy perturbation method.

Table 1: Single-well situation: comparison of solutions by the Haar wavelet-quasilinearization technique _yHAAR at 5th iteration and level of resolutions J=9 with numerical methods [13] and numerical solution based on the fourth-order Runge-Kutta.

α = 2
5th iteration				J = 9
x	y RK	y HPM [13]	y VIM [13]	y HAAR	Absolute Error
0.2	0.9900451	0.99004	0.99004	0.9900451	3.1e - 8
0.4	0.9607026	0.96075	0.9607	0.9607024	1.5e - 7
0.6	0.9134154	0.91383	0.91341	0.9134150	3.5e - 7
0.8	0.8502496	0.85216	0.85025	0.8502491	5.8e - 7
1.0	0.773523	0.77973	0.77353	0.773522	8.0e - 7

Table 2: Double-well situation: comparison of solutions by the Haar wavelet-quasilinearization technique _yHAAR at 5th iteration and level of resolutions J=9 with numerical methods [13] and numerical solution based on the fourth-order Runge-Kutta.

α = 2
5th iteration				J = 9
x	y RK	y VIM [13]	y HPM [13]	y HAAR	Absolute Error
0.2	1.009945	1.00994	1.00994	1.009945	9.8e - 9
0.4	1.039114	1.03911	1.03918	1.039114	6.7e - 8
0.6	1.085448	1.08544	1.08621	1.085448	1.9e - 7
0.8	1.145384	1.14539	1.14937	1.145384	3.9e - 7
1.0	1.213777	1.21382	1.22785	1.213778	6.4e - 7

Table 3: Double-hump situation: comparison of solutions by the Haar wavelet-quasilinearization technique _yHAAR at 5th iteration and level of resolutions J=9 with numerical methods [13] and numerical solution based on the fourth-order Runge-Kutta.

α = 2
5th iteration				J = 9
x	y RK	y VIM [13]	y HPM [13]	y HAAR	Absolute Error
0.1	1.00250	1.0025	1.0025	1.00250	2.5e - 9
0.2	1.01001	1.01001	1.01001	1.01001	4.3e - 11
0.5	1.06301	1.063	1.06296	1.06301	4.3e - 8
0.75	1.14347	1.14346	1.14209	1.14347	9.8e - 8
1.0	1.26039	1.26035	1.25055	1.26039	3.9e - 7

Figures 1, 2, and 3 showed the solution of (17) for single-well, double-well, and double-hump situations, respectively. We plot the solutions at different order α of (17). Here we fixed the solution at fifth iteration and level of resolution J=5 or J=6 . Also solution by the fourth-order Runge-Kutta method (RK Solution) at α=2 is also plotted along with the solution obtained by the Haar wavelet quasilinearization technique (HAAR Solution) and Figures 1, 2, and 3 show that Haar solution converges to the RK solution when α approaches 2.

Figure 1: Solution by RK method (RK Solution) at α=2 and solution by Haar wavelet-quasilinearization technique (HAAR Solution) at J=6 and different values of α for a=0.5 , b=0.5 , μ=0.1 , f=0.5 , and ω=0.79 .

[figure omitted; refer to PDF]

Figure 2: Solution by RK method (RK Solution) at α=2 and solution by Haar wavelet-quasilinearization technique (HAAR Solution) at J=5 and different values of α for a=-0.5 , b=0.5 , μ=0.1 , f=0.5 , and ω=0.79 .

[figure omitted; refer to PDF]

Figure 3: Solution by RK method (RK Solution) at α=2 and solution by Haar wavelet-quasilinearization technique (HAAR Solution) at J=5 and different values of α for a=0.5 , b=-0.5 , μ=0.1 , f=0.5 , and ω=0.79 .

[figure omitted; refer to PDF]

5.2. Force-Free Duffing-Van der Pol Oscillator Equation [16]

Example 2.

Consider the α th order fractional force-free DVP oscillator equation [figure omitted; refer to PDF] subject to the initial conditions y(0)=1 and ^{y[variant prime]} (0)=0 .

The Haar wavelet-quasilinearization technique on (22) gives [figure omitted; refer to PDF] with the initial approximations _y0 (x)=1 and ^{y0[variant prime]} (x)=0 .

Results of fifth iteration by the Haar wavelet quasilinearization technique at fixed level of resolution J=9 and at α=2 are shown in Table 4. Here we consider μ=0.1 , a=1 , and b=0.01 and compare the obtained solution with Adomian decomposition method [16]. Equation (22) is also solved by the fourth-order Runge-Kutta method to show the applicability of the Haar wavelet quasilinearization technique. Table 4 shows that solution by the Haar wavelet quasilinearization technique gives more accurate results as compared to Adomian decomposition method.

Table 4: Force-free Duffing-Van der Pol Oscillator Equation: comparison of solutions by the Haar wavelet-quasilinearization technique _yHAAR at 5th iteration and level of resolutions J=9 with decomposition method _yADM [13] and numerical solution based on the fourth-order Runge-Kutta.

α = 2
5th iteration			J = 9
x	y RK	y ADM [13]	y HAAR	Absolute Error
0.0	2.00000	1.99750	2.00000	2.1e - 12
0.1	1.98971	1.98724	1.98971	1.7e - 7
0.2	1.95936	1.95697	1.95936	3.5e - 7
0.3	1.90980	1.90758	1.90980	5.4e - 7
0.4	1.84202	1.84008	1.84202	7.3e - 7
0.5	1.75702	1.75552	1.75702	9.2e - 7
0.6	1.65586	1.65493	1.65586	1.1e - 6
0.7	1.53958	1.53937	1.53958	1.3e - 6
0.8	1.40923	1.53937	1.40923	1.4e - 6
0.9	1.26586	1.26726	1.26586	1.6e - 6
1.0	1.11054	1.11267	1.11054	1.7e - 6
1.1	0.94435	0.94704	0.94435	1.9e - 6
1.2	0.76846	0.77147	0.76846	2.0e - 6
1.3	0.58411	0.58715	0.58410	2.1e - 6
1.4	0.39267	0.39545	0.39267	2.3e - 6
1.5	0.19567	0.19795	0.19566	2.4e - 6

Results of fifth iteration by the Haar wavelet quasilinearization technique at fixed level of resolution J=5 and at different values of α are shown in Figure 4, along with the RK solution at α=2 . Figure 4 showed that obtained solution converges to the RK solution when α approaches 2.

Figure 4: Solution by RK method (RK Solution) at α=2 and solution by Haar wavelet-quasilinearization technique (HAAR Solution) at J=5 and different values of α for a=1 , μ=0.1 , b=0.01 , and f=0 .

[figure omitted; refer to PDF]

5.3. Higher Order Oscillation Equation [14]

Example 3.

Consider the α th order fractional Duffing equation [figure omitted; refer to PDF] subject to the initial conditions: [figure omitted; refer to PDF]

The exact solution, when α=4 , is given by [figure omitted; refer to PDF]

Quasilinearization technique to (24) gives [figure omitted; refer to PDF] with the initial conditions: [figure omitted; refer to PDF]

Implement the Haar wavelet method to (27) as follows: [figure omitted; refer to PDF] Lower order derivatives are obtained by integrating (29) and use the initial condition [figure omitted; refer to PDF] Substitute (29) and (30) into (27), we get [figure omitted; refer to PDF] with the initial approximations: [figure omitted; refer to PDF]

Solution by the Haar wavelet quasilinearization technique at 6th fixed level of resolution J=10 and order of (24) α=4 is shown in Table 5. It shows that obtained solution is more accurate as compared to generalized differential quadrature rule (GDQR) [14]. _EGDQE and _EHAAR represent the percentage error of generalized differential quadrature rule and the Haar wavelet quasilinearization technique, respectively.

Table 5: Higher order oscillation equation: comparison of solutions by the Haar wavelet-quasilinearization technique at 6th iteration and level of resolutions J=10 with generalized differential quadrature rule (GDQR) method [14] and exact solution.

α = 4
6th iteration				J = 10
x	y Exact [14]	y GDQR [14]	E GDQR [14]	y HAAR	E HAAR
0.0	0	0	0	0	0
0.7	1.2692	1.2693	-0.002	1.2692	0.0025
1.4	2.0990	2.0993	-0.010	2.0990	0.0037
2.1	2.0929	2.0933	-0.019	2.0928	0.0048
2.8	1.2541	1.2545	-0.027	1.2541	0.0059
3.5	-0.0179	-0.0177	0.813	-0.0179	-0.1679
4.2	-1.2843	-1.2842	0.003	-1.2843	-0.0010
4.9	1.0880	-2.1051	-0.004	1.0879	0.0063
5.6	-2.0866	-2.0868	-0.014	-2.0865	-0.0046
6.3	-1.2390	-1.2395	-0.039	-1.2389	-0.0083
7.0	0.0357	0.0352	1.276	0.0358	0.2095
7.7	1.2992	1.2990	0.013	1.2992	0.0010
8.4	2.1109	2.1111	-0.009	2.1108	0.0031
9.1	2.0801	2.0805	-0.021	2.0800	0.0056
9.8	1.2237	1.2243	-0.044	1.2236	0.0099
10.5	-0.0536	-0.0529	1.146	-0.0537	-0.1965
11.2	-1.3141	-1.3136	0.037	-1.3141	-0.0042
11.9	-2.1166	-2.1166	-0.002	-2.1166	-0.0022
12.6	-2.0734	-2.0741	-0.030	-2.0733	-0.0068
13.3	-1.2084	-1.2093	-0.071	-1.2082	-0.0136
14.0	0.0714	0.0706	1.057	0.0715	0.1888

We fix the solutions at fifth iteration, level of resolution J=5 , and plot the solution at different values of α that are shown in Figure 5 along with the exact solution at α=4 and Figure 5 shows that solution by the Haar wavelet quasilinearization technique converges to the exact solution, when α approaches 4.

Figure 5: Higher order oscillation equation. Exact solution at α=4 and solution by Haar wavelet-quasilinearization technique at J=5 and different values of α .

[figure omitted; refer to PDF]

6. Conclusion

It is shown that Haar wavelet method with quasilinearization technique gives excellent results when applied to fractional order nonlinear oscillation equations. The results obtained from Haar wavelet quasilinearization technique are better than the results obtained by other methods and are in good agreement with exact solutions or solution by the fourth-order Runge-Kutta method, as shown in Tables and Figures. The solution of the fractional order nonlinear oscillation equation converges to the solution of integer order nonlinear oscillation differential equation as shown in Figures 1, 2, 3, 4, and 5.

Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable comments which have led to the improvement of the paper.

Conflict of Interests

Umer Saeed and Mujeeb ur Rehman declare that there is no conflict of interests regarding the publication of this paper.

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