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.
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]: where is the Caputo derivative; represents the periodic driving function of time with period , where is the angular frequency of the driving force; is the forcing strength; and is the damping parameter of the system. Duffing-Van der Pol oscillator equation can be expressed in three physical situations: (1)single-well , ;(2)double-well , ;(3)double-hump , .
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 is as follows: the Riemann-Liouville fractional order integral of order is defined as for .(2)Riemann-Liouville and Caputo fractional derivative operators of order are as follows:the Riemann-Liouville fractional order derivative of order is defined as for , where , , and .
The Caputo fractional order derivative of order is defined as for , where , , and .
3. The Haar Wavelets
The Haar functions contain just one wavelet during some subinterval of time and remain zero elsewhere and are orthogonal. The th uniform Haar wavelet , is defined as [2] where ; is dilation parameter, where and is translation parameter. is maximal level of resolution and the maximal value of is where . In particular , where is characteristic function on interval , 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 where .
3.1. Fractional Integral of the Haar Wavelets
Any function can be represented in terms of the uniform Haar series where are the Haar wavelet coefficients given as .
Any function of two variables can be approximated as where is coefficient matrix which can be determined by the inner product .
The Riemann-Liouville fractional integral of the uniform Haar wavelets is given as where , , and .
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:
Applying quasilinearization technique to (10) yields Let be some initial approximation. Each function is a solution of a linear equation (11), where is always considered to be known and is obtained from the previous iteration.
According to [6] and letting and , we have This shows that quasilinearization technique has quadratic convergence, if there is convergence at all.
4.2. Convergence of Haar Wavelet Method [15]
Let be a differentiable function and assume that have bounded first derivative on ; that is, there exist ; for all , Haar wavelet approximation for the function is given by Babolian and Shahsavaran [15] gave -error norm for Haar wavelet approximation, which is or As and 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 is increased.
Each iteration of quasilinearization technique gives linear differential equation in which is solved to get approximate value of , , by Haar wavelet method. Since solution of our problems has bounded first derivatives over , according to (16), converges fast to if we consider the higher level of resolution ; that is, we get more accurate results while increasing , and at the same time quasilinearization technique works; that is, given an initial approximation , we get solution of linear differential equation (11) by Haar wavelet method and at next iteration, we get 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 converges to exact solution when both and 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 th order fractional forced DVP oscillator equation subject to the initial conditions and .
Applying the quasilinearization technique to (17), we obtain with the initial conditions and .
Now we apply the Haar wavelet method to (18) and approximate the higher order derivative term by the Haar wavelet series as
Lower order derivatives are obtained by integrating (19) and use the initial condition Substitute (19) and (20) into (18) to get with the initial approximations and .(1)(Single-well , ). , , , , .(2)(Double-well , ). , , , , .(3)(Double-hump , ). , , , , .
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, , and level of resolution . 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.
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 of (17). Here we fixed the solution at fifth iteration and level of resolution or . Also solution by the fourth-order Runge-Kutta method (RK Solution) at 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 approaches 2.
5.2. Force-Free Duffing-Van der Pol Oscillator Equation [16]
Example 2. Consider the th order fractional force-free DVP oscillator equation subject to the initial conditions and .
The Haar wavelet-quasilinearization technique on (22) gives with the initial approximations and .
Results of fifth iteration by the Haar wavelet quasilinearization technique at fixed level of resolution and at are shown in Table 4. Here we consider , , and 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.
Results of fifth iteration by the Haar wavelet quasilinearization technique at fixed level of resolution and at different values of are shown in Figure 4, along with the RK solution at . Figure 4 showed that obtained solution converges to the RK solution when approaches 2.
5.3. Higher Order Oscillation Equation [14]
Example 3. Consider the th order fractional Duffing equation
subject to the initial conditions:
The exact solution, when , is given by
Quasilinearization technique to (24) gives with the initial conditions:
Implement the Haar wavelet method to (27) as follows: Lower order derivatives are obtained by integrating (29) and use the initial condition Substitute (29) and (30) into (27), we get with the initial approximations:
Solution by the Haar wavelet quasilinearization technique at 6th fixed level of resolution and order of (24) is shown in Table 5. It shows that obtained solution is more accurate as compared to generalized differential quadrature rule (GDQR) [14]. and represent the percentage error of generalized differential quadrature rule and the Haar wavelet quasilinearization technique, respectively.
We fix the solutions at fifth iteration, level of resolution , and plot the solution at different values of that are shown in Figure 5 along with the exact solution at and Figure 5 shows that solution by the Haar wavelet quasilinearization technique converges to the exact solution, when approaches 4.
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.
Conflict of Interests
Umer Saeed and Mujeeb ur Rehman declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the anonymous reviewers for their valuable comments which have led to the improvement of the paper.