Haar wavelet solutions of nonlinear oscillator equations

Applied Mathematical Modelling xxx (2014) xxx–xxx Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm Haar wavelet solutions of nonlinear oscillator equations Harpreet Kaur a,⇑, R.C. Mittal b, Vinod Mishra a a b Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148106, Punjab, India Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttrakhand, India a r t i c l e i n f o Article history: Received 7 December 2012 Received in revised form 24 January 2014 Accepted 16 March 2014 Available online xxxx Keywords: Haar wavelets Nonlinear oscillators Multiresolution analysis Operational matrix Quasilinearization process a b s t r a c t In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems. Published by Elsevier Inc. 1. Introduction Nonlinear problems are of interest to many scientists and engineers, because most of physical systems in the real world are inherently nonlinear in nature. Many nonlinear differential equations arise in physical, chemical and biological contexts. Finding innovative methods to solve and analyze these equations has been an interesting subject in the field of differential equations and dynamical systems [1]. Considerable attention has been directed toward the chaos, chaotic systems and solutions of nonlinear oscillator differential equations since they play crucial role in natural and physical simulations. These problems are important for wavelet analysis, applied mathematics, physics and engineering sciences [2,3]. The chaotic behavior can be observed in natural and man-made systems. Some of chaotic systems are represented by Duffing, Van der Pol, and Duffing–van der Pol equations which are important mathematical models for dynamical systems having a single unstable fixed point, along with a single stable limit cycle. These systems are highly sensitive to initial conditions because small differences in initial conditions yield widely diverging outcomes [4,5]. Up to now various aspects of nonlinear oscillators have been studied in the literature such as, the vibration amplitude control, synchronization dynamics and additive resonances etc. [6–10]. Recently, Duffing equation has been enhanced by Ahmad et al. [11] in the field of the prediction of diseases. A cautious measurement and analysis of a strongly chaotic voice has the potential to serve as an early warning system for more serious chaos and possible commencement of a disease. In fact, the success at analyzing and predicting the commencement of chaos in a signal and its simulation by equations lie in the problems as the Duffing, Van der Pol, and Duffing–van der Pol play very important role. In this work, we have studied these equations by Haar wavelet method. ⇑ Corresponding author. Tel.: +91 9779372138. E-mail addresses: [email protected] (H. Kaur), [email protected] (R.C. Mittal). https://dx.doi.org/10.1016/j.apm.2014.03.019 0307-904X/Published by Elsevier Inc. Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 2 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx In this work, we consider a more general nonlinear oscillator system of the form eu00 ðtÞ þ ðd þ bup ðtÞÞu0 ðtÞ  luðtÞ þ auq ðtÞ ¼ gðF; x; tÞ p; q 2 N; ð1Þ with the initial conditions uð0Þ ¼ c0 ; u0 ð0Þ ¼ c1 : ð2Þ Depending on the parameters chosen, the Eq. (1) can take a number of special forms, where differentiation is with respect to independent time variable t and all parameters e; d; b; l and a are real constants. Here x is an angular frequency and gðF; x; tÞ represents the periodic driving function of time with period T ¼ 2p=x. Among the periodically forced self-excited oscillators, one of the most extensively studied examples is the Duffing–van der Pol oscillator whose mathematical expression is assumed in the form of the second order differential equation. The Eq. (1) is referred to the Duffing–van der Pol oscillator [12] when p ¼ 2; q ¼ 3 and other parameters which are involved in Eq. (1) are non zero with periodic deriving function gðF; x; tÞ ¼ F cosðxtÞ as eu00 ðtÞ þ ðd þ bu2 ðtÞÞu0 ðtÞ  luðtÞ þ au3 ðtÞ ¼ F cosðxtÞ: ð3Þ The choice of q ¼ 3; b; d ¼ 0 and gðF; x; tÞ ¼ F sinðxtÞ leads Eq. (1) to the Duffing oscillator [13,14] represented by eu00 ðtÞ  luðtÞ þ au3 ðtÞ ¼ FsinðxtÞ; e; l and a – 0: ð4Þ It is the true equation of a forced, damped pendulum if F – 0 and unforced damped pendulum if F ¼ 0: The Duffing equation is interesting because it allows us to examine what happens when we force on oscillator near its resonant frequency and to investigate a balance between the linear behavior and weak dissipative or nonlinear effects. In most physical systems, damping or nonlinear effects become important when the amplitude grows sufficiently large. The choice a ¼ 0 and p ¼ 2 leads Eq. (1) to the Van der Pol oscillator [15] governed by the second-order differential equation eu00 ðtÞ þ ðd þ bu2 ðtÞÞu0 ðtÞ  luðtÞ ¼ F sinðxtÞ; d; b – 0: ð5Þ This model was proposed by engineer Balthasar van der Pol (1889–1959) for an electrical circuit with a triode valve in 1920 when he was an engineer working for Philips Company (in the Netherlands) and was later extensively studied as a host of a rich class of dynamical behavior. There are various techniques to solve nonlinear oscillators. For details, one may refer to survey articles [11] which includes finite difference method, finite element method, boundary value approach, various forms of spline and wavelet methods to obtain approximate solutions of nonlinear equations of various types. Most scientific problems in solid mechanics problems are inherently nonlinear. Except a limited number of these problems, most of them do not have analytical solutions. Some of them are solved using a numerical techniques and the analytical perturbation method. In the perturbation method, the small parameter is inserted in the equation. Therefore, finding the small parameter and exerting it into the equation are deficiencies of this method [7]. In addition it requires a large amount of calculations resulting in a major computational difficulty for getting the accurate results for small damping parameter and tackling the nonlinearity. Duffing van der Pol equation is investigated recently by Ji and Zhang [8] while Duffing equation by Liu, Wu [13,14] and Van der Pol equation by Lepik [15]. But their solutions are restricted to very small parameters. In recent years the wavelet approach is becoming increasingly popular in the field of numerical approximations. Different types of wavelets and approximating functions have been used in numerical solution of differential equations. Out of these, the Haar wavelets have gained popularity among researchers due to their useful properties such as simple applicability, orthogonality and compact support. Concept of Haar wavelets for solving differential equations is used by Chen and Hsiao in [16]. Hariharan [17] has derived solutions of partial differential equations by Haar wavelets. Yousefi et al. have solved fractional optimal control problems by using Legendre multiwavelet collocation method [18]. Integro differential equations are also solved by Lakestani et al. using trigonometric wavelets [19]. Second order boundary value problems and non-linear Lane Emden equations have been successfully solved by using the Haar wavelet quasilinearization method in [20,21]. An attempt is made in this paper to use Haar wavelet quasilinearization technique to solve nonlinear oscillator differential equations. Only problem we face is that when we increase number of points, the corresponding coefficient matrix becomes illconditioned. The paper is organized as follows. In next section we have discussed the brief introduction on preliminaries of Haar wavelets. In Section 3, quasilinearization process is discussed to deal the nonlinearity in the equations and Haar wavelet method is also described here. Convergence of method is discussed in Section 4. The proposed method is implemented on different nonlinear oscillator differential equations in Section 5. These nonlinear oscillators as Duffing, Van der Pol, and Duffing– van der Pol which are encountered in structural dynamics. A comparison of Haar wavelet solutions with available ones is shown graphically and numerical results are depicted in tables for different parameters. Section 6 concludes the present work. Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 3 2. Haar wavelet preliminaries Among the different wavelet families which are defined by an analytical expression, mathematically most simple are the Haar wavelets. Due to the simplicity the Haar wavelets are very effective for solving ordinary differential and partial differential equations. In 1910, Haar [22] introduced the notion of wavelets in the form of a rectangular pulse pair function. His initial theory has been expanded recently into a wide variety of applications, but primarily it allows for the representation of various functions by a combination of step functions and wavelets over specified interval widths. The Haar wavelet is the only real valued function which is symmetric, orthogonal and have a compact support [23]. Here first Haar wavelet function is defined as 8 1 > < 1 if 0  t  2 h1 ðtÞ ¼ 1 if 0  t  12 > : 0 elsewhere: ð6Þ The following definitions illustrate the translation-dilation of Haar wavelet function hi ðtÞ: 2.1. Translation and dilation operators Let h 2 Lj 2 ðRÞ: For k 2 Z; let T k : L2 ðRÞ ! L2 ðRÞ be given by ðT k hÞðtÞ ¼ hðt  kÞ and Dh : L2 ðRÞ ! L2 ðRÞ be given by ðDj hÞðtÞ ¼ 22 hð2j tÞ operators T k and Dj are called translation and dilation operator. 2.2. Orthonormal Haar Wavelet n o n j o A function h 2 L2 ðRÞ is called an orthonormal wavelet for L2 ðRÞ if Dj T k h : j; k 2 Z ¼ 22 hð2j t  kÞ : j; k 2 Z is an orthonormal basis for L2 ðRÞ. Index j refers to dilation and k refers to translation. Thus Haar wavelet family hi ðtÞ is orthogonal square waves family which is obtained by translation and dilation operators as [16] 8 t 2 ½n1 ; n2  > < 1 hi ðtÞ ¼ 1 t 2 ½n2 ; n3  > : 0 elsewhere; Here n1 ¼ ð7Þ k k þ 0:5 kþ1 ; n ¼ and n3 ¼ : m 2 m m l12 The collocation points tl ¼ 2m ; ð8Þ l ¼ 1; 2; . . . ; 2m: For i P 2; i ¼ 2j þ k þ 1; j P 0; 0  k  2j  1 Here m is the level of the wavelet, we assume the maximum level of resolution as index J, then m ¼ 2j ; ðj ¼ 0; 1; 2; . . . ; JÞ; in case of minimal values m ¼ 1; k ¼ 0 then i ¼ 2. For any fixed level m, there are m series of i to fill the interval ½0; 1Þ corresponding to that level and for a provided J, the index number i can reach the maximum value M = 2J+1, when including all levels of wavelets. Also for the ease of implementation, we have used the same notations for Haar wavelets and their integrals as [24] and matrix form of Haar wavelet family hi ðtÞ for j ¼ 1; 2m ¼ 4 is given as 2 1 1 1 1 3 6 1 1 1 1 7 6 7 H¼6 7: 4 1 1 0 0 5 0 0 1 ð9Þ 1 We can find the required derivatives in terms of operational matrix. The operational matrix pi;n ðtÞ of order 2m  2m can be obtained by integration of Haar wavelet. Integrals can be evaluated from Eq. (7) and the first two integrals of them are given below. 8 k > < tm kþ1 pi;1 ðtÞ ¼ t > : m 0 pi;2 ðtÞ ¼ 8 > > > > > < > > > > > : 1 2 1 4m2   t 2 mk ; kþ0:5 kþ0:5 mkþ1 t2 m ; m ; elsewhere t  mk  1 kþ1  2 m 1 4m2 0 2 t 2   t 2 mk ; kþ0:5 m   t 2 kþ0:5 ; kþ1 m m :   t 2 kþ1 ;1 m ð10Þ ð11Þ elsewhere Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 4 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 3. Description of Haar wavelet technique 1 The sequence fhi gi¼0 is a complete orthonormal system in L2 ½0; 1 and by using the concept of multiresolution analysis (MRA) as an example the space V j can be defined like V j ¼ spfwj;k gj¼0;1;2;...;2j 1 ¼ W j1  V j1 ¼ W j1  W j2  V j2     ¼ Jþ1 j¼1 W j  V 0 : ð12Þ The linearly independent functions hj;k ðtÞ spanning W j are called wavelets. The Haar basis has the very important property of multiresolution analysis that V jþ1 ¼ V j  W j : Original signal can be expressed as a linear combination of the box basis functions in V j . The orthogonality property puts a strong limitation on the construction of wavelets and allows us to transform any square integral function on the interval time ½0; 1Þ into Haar wavelets series as f ðtÞ ¼ a0 h0 ðtÞ þ j 1 2 1 X X j¼0 k¼0 a2j þk h2j þk ðtÞ; t 2 ½0; 1: ð13Þ P Similarly the highest derivative can be written as wavelet series 1 i¼1 ai hi ðtÞ: In applications, Haar series are always trunP2m cated to 2 m terms, that is i¼0 ai hi ðtÞ [25,26]. The presented technique is based on integral operational matrices of Haar wavelet approximated series and quasilinearization process [27]. The quasilinearization process [28] is application of the Newton Raphson Kantrovich approximation method in function space. The idea and advantage of the quasilinearization are based on the fact that linear equations can often be solved analytically or numerically while there are no useful techniques for obtaining the general solution of a nonlinear equation in terms of a finite set of particular solutions. Consider an nth order nonlinear ordinary differential equation   LðnÞ uðtÞ ¼ f uðtÞ; uð1Þ ðtÞ; uð2Þ ðtÞ; uð3Þ ðtÞ; . . . ; uðn1Þ ðtÞ; t ; ð14Þ uð0Þ ¼ k0 ; uð1Þ ð0Þ ¼ k1 ; uð2Þ ð0Þ ¼ k2 ; uð3Þ ð0Þ ¼ k3 ; . . . ; uðn1Þ ð0Þ ¼ kn1 : ð15Þ with the initial conditions ðnÞ th Here L is the linear n order ordinary differential operator, f is nonlinear function of uðtÞ and its ðn  1Þ derivatives are uðsÞ ðtÞ; s ¼ 0; 1; 2; . . . ; n  1: The quasilinearization prescription determines the ðr þ 1Þth iterative approximation to the solution of Eq. (14) with Eq. (15) and its linearized form is given by Eq. (16) ð3Þ ðn1Þ LðnÞ urþ1 ðtÞ ¼ f ður ðtÞ; urð1Þ ðtÞ; uð2Þ ðtÞ; tÞ þ r ðtÞ; ur ðtÞ::::::; ur ð3Þ ðn1Þ fuðsÞ ður ðtÞ; urð1Þ ðtÞ; uð2Þ ðtÞ; tÞ; r ðtÞ; ur ðtÞ . . . ; ur n1 X ðsÞ ðurþ1 ðtÞ  uðsÞ r ðtÞÞ; ð16Þ s¼0 ð0Þ @f th where ur ðtÞ ¼ ur ðtÞ. The functions fuðsÞ ¼ @u approximation u0 ðtÞ is s are functional derivatives of the functions. The zero chosen from mathematical or physical considerations. By using quasi-linearization process, we get the following two equations for linearization of Eq. (1) ð1Þ ð1Þ ð1Þ 2 u2 ðtÞu0 ðtÞ ¼ u2r ðtÞuð1Þ r ðtÞ þ ðurþ1 ðtÞ  ur ðtÞÞ2ur ðtÞur ðtÞ þ ðurþ1 ðtÞ  ur ðtÞÞur ðtÞ; ð17Þ u3 ðtÞ ¼ 2u3r ðtÞ þ 3u2r ðtÞurþ1 ðtÞ: ð18Þ Thus, Eq. (1) becomes after quasilinearization ð1Þ p ð1Þ p1 euð2Þ ðtÞ  ur ðtÞpur ðtÞup1 ðtÞ r rþ1 ðtÞ þ durþ1 ðtÞ þ bður ðtÞur ðtÞ þ urþ1 ðtÞpur ðtÞur ð1Þ ð1Þ p q q1 q1 þ ðurþ1  ur ðtÞÞur ðtÞÞ  lurþ1 ðtÞ þ aur ðtÞ þ aqur ðtÞurþ1 ðtÞ  aqur ðtÞur ðtÞ ¼ gðF; x; tÞ; ðnÞ urþ1 ðtÞ ¼ ðnnÞ ð19Þ 2m X i¼0 urþ1 ðtÞ ¼ ai hi ðtÞ; ð20Þ 2m X ðnnÞ ai pi;n ðtÞ þ t n uðn1Þ ð0Þ þ t n1 uðn2Þ ð0Þ þ . . . þ urþ1 ð0Þ: ð21Þ i¼0 Finally on applying Haar wavelet technique and using the Eq. (21) and (19) becomes 2m X ai i¼0 2m X e i¼0 ð1Þ ! 2m 2m X X ð1Þ q1 p1 hi ðtl Þ þ ðd þ 1Þ pi;1 ðtl Þ þ ðqur ðt l Þ þ pur ðtl Þur ðtl ÞÞ pi;2 ðt l Þ i¼0 ð1Þ i¼0 ð1Þ ð1Þ ð1Þ ð1Þ ðt l Þur ðt l Þ  ur ðt l Þpup1 ðt l Þ þ ur ð0Þupr ðtl Þ  uqþ1 ðt l Þ þdur ð0Þ þ bupr ðtl Þur ðtl Þ þ ðtl ur ð0Þ þ ur ð0ÞÞpup1 r r ð22Þ ð1Þ ðt l Þ  ur ðt l Þqurq1 ðt l Þ ¼ gðF; x; tl Þ: upr ðt l Þ  upr ðt l Þ  lur ðtl Þ þ auqr ðtl Þ þ ðt l ur ð0Þ þ ur ðt l ÞÞquq1 r Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 5 Now we will simplify Eq. (22) as the linear matrix system for getting the coefficients a0i s and finally get the Haar wavelet solution (HWS) of Eq. (1). 4. Convergence of Haar wavelet approximation A function u 2 L2 ðRÞ; MRA of L2 ðRÞ generates a sequence of subspaces V j ; V jþ1 ; V jþ2 . . . such that the projections of u onto these spaces give finer and finer approximations of the function u as J ! 1; then the corresponding error at Jth level may be defined as      j  X 2 þ1 1 X         eJ ðtÞ ¼ uðtÞ  uJ ðtÞ ¼ uðtÞ  ai hi ðtÞ ¼  ai hi ðtÞ:     j i¼1 i¼2 þ1 ð23Þ We can analyze the error for nonlinear oscillator Eq. (1). Convergence of the method may be discussed on the same lines as given in Saeedi et al. [29]. We can also discuss the convergence of the method for nonlinear oscillator problem if we know the exact solution. Theorem. Suppose that f ðxÞ satisfies a Lipschitz condition on [0,1], that is, 9M > 0; 8x; y 2 ½0; 1 : jf ðxÞ  f ðyÞj  Mjx  yj; M is the Lipschitz constant. The error bound for jeJ ðxÞj2 is also obtained as keJ ðxÞk2   M p ffiffiffi : 2Jþ1 3  Then the Haar wavelet method will be convergent in the sense that ej ðxÞ goes to zero as j goes to infinity and Lipschitz constant M may be small not too large. Moreover, the convergence is of order one, that is.   1 keJ ðxÞk2 ¼ O Jþ1 : 2 Proof. See Saeedi et al. [29]. 5. Applications and numerical problems In this section, the wide applicability and efficiency of the Haar wavelet method are manifested further through a set of experiments on numerical problems. We consider the following oscillators in particular cases and comparison will be made with existing available solutions in literature. All computations are carried out by programming in C++ and MATLAB R2007b at maximum level of resolution J ¼ 4 and 2m ¼ 32. 5.1. Duffing oscillator Consider the Duffing oscillator from Eq. (4) as following eu00 ðtÞ  luðtÞ þ au3 ðtÞ ¼ F sinðxtÞ: ð24Þ Where e; l; a; and F are given parameters, x is also a given constant which represents the enforcing frequency. The analytic solution is given by means of a trigonometric series [13,14]. uðtÞ ffi a1 sinðxtÞ þ a2 sinð3xtÞ þ a3 sinð5xtÞ þ . . . : ð25Þ Here we consider two cases of Duffing equation with different parameters as follows. 5.1.1. Unforced Duffing oscillator Unforced Duffing oscillator represents the free vibration of pendulum. The frequency of the oscillations depends on the initial displacement of the pendulum. By taking the parameters e ¼ 1; l ¼ 1; a ¼ 1 x ¼ 0:7 and F ¼ 0 as considered in [13] 6 and initial conditions uð0Þ ¼ 0 and u0 ð0Þ ¼ 1:6376 in Eq. (24), we get 1 u00 ðtÞ þ uðtÞ  u3 ðtÞ ¼ 0: 6 ð26Þ The trigonometric series solution up to three terms of Eq. (26) is uðtÞ ffi 2:058 sinð0:7tÞ þ 0:0816 sinð2:1tÞ þ 0:00337 sinð3:5tÞ: Using the Eqs. (18) and (21) in Eq. (26), we get Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 6 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx   2m 2m X X 1 2 3 2 ai hi ðt l Þ þ 1  0:5ð1:6376Þ2 sin ðt l Þ ai pi;2 ðtl Þ ¼ 1:6376 ð1:6376Þ2 sin ðt l Þ þ 0:5ð1:6376Þ  sin ðt l Þt l  t l ; ð27Þ 3 i¼0 i¼0 and applying the procedure mentioned in Section 3, finally Haar wavelet solution of Eq. (26) can be obtained by getting the values of coefficients which are computed after solving the Eq. (27). Computed values of u; u0 and u00 are compared with trigonometric series solution and depicted graphically in Fig. 5.1.1a and the absolute errors in the solutions for a ¼  16 are given in Table 1. For a ¼ 10, wavelet solution is shown in Fig. 5.1.1b and in Table 2. 5.1.2. Forced Duffing oscillator ; x ¼ 1 and F ¼ 2 in Eq. (24), we get the forced Duffing system with initial By taking the parameters e ¼ 1; l ¼ 1; a ¼ 1 6 conditions given as in [13], 1 u00 ðtÞ  u3 ðtÞ þ uðtÞ ¼ 2 sinðtÞ; 6 with uð0Þ ¼ 0 and u0 ð0Þ ¼ 2:7676: ð28Þ Trigonometric series solution up to three terms is given below uðtÞ ffi 2:5425 sinðtÞ  0:07139 sinð3tÞ  0:00219 sinð5tÞ: Comparison of computed u; u0 and u00 are depicted graphically in Fig. 5.1.2a and the absolute errors in the solutions are given in Table 1 for a ¼  16. Haar wavelet solution for a ¼ 10 is also shown in Fig. 5.1.2b and in Table 2. 5.2. Duffing–van der Pol oscillator 5.2.1. Unforced Duffing–van der Pol oscillator [30,31] Consider Eq. (3) with parameters e ¼ 1; d ¼ 0:1; b ¼ 0:1; l ¼ 1; a ¼ 0:01 and F ¼ 0 as below u00 ðtÞ  0:1ð1  u2 ðtÞÞu0 ðtÞ þ uðtÞ þ 0:01u3 ðtÞ ¼ 0; with uð0Þ ¼ 2 and u0 ð0Þ ¼ 0: ð29Þ Unforced Duffing Oscillator 3 Output response u,u',u'' 2 By trogonometric series u By trigonometric series u' By trogonometric series u'' Computed u Computed u' Computed u'' 1 0 -1 -2 -3 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 1 Fig. 5.1.1a. Comparison of Haar wavelet solution with trigonometric series solution for a ¼ 1 . 6 Table 1 Computed absolute errors at different points for problems 5.1.1 and 5.1.2. t Error for Prob. 5.1.1 Error for Prob. 5.1.2 Error for Prob. 5.2.1 0.0156 0.1094 0.2344 0.3594 0.4844 0.6094 0.7344 0.8594 0.9844 0.180e005 0.600e005 0.900e005 0.300e004 0.690e004 0.131e003 0.230e003 0.320e003 0.400e003 0.100e005 0.200e004 0.100e003 0.300e003 0.400e003 0.600e003 0.700e003 0.900e003 0.150e002 0.100e005 0.100e002 0.240e002 0.390e002 0.600e002 0.910e002 0.143e001 0.226e001 0.356e001 Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 7 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx Unforced Duffing Oscillator for α = -10 1200 Computed u Computed u' Computed u'' Output Response u,u',u'' 1000 800 600 400 200 0 -200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Fig. 5.1.1b. Haar wavelet solution for a = 10. Table 2 Computed Haar wavelet solution for different values of a for Problems 5.1.1 and 5.1.2. t Prob. 5.1.1 for a ¼ 10 uðtÞ Prob. 5.1.1 for a ¼ 10 u0 ðtÞ Prob. 5.1.2 for a ¼ 10 uðtÞ Prob. 5.1.2 for a ¼ 10 u0 ðtÞ 0.0156 0.0781 0.1406 0.2031 0.2656 0.3281 0.3906 0.4531 0.5156 0.5781 0.6406 0.7031 0.7656 0.8281 0.8906 0.9531 0.9844 0.025368 0.126753 0.228181 0.331277 0.440353 0.563629 0.714659 0.914337 1.19409 1.60125 2.20813 3.12759 4.53940 6.73545 10.1974 15.7316 19.6709 1.62338 1.62091 1.62934 1.68232 1.83239 2.15257 2.74221 3.74033 5.35134 7.89044 11.8613 18.0885 27.9441 43.7384 69.4009 111.676 142.409 0.043234 –0.215634 –0.383921 –0.537104 –0.651968 –0.690770 –0.604461 –0.344828 0.112827 0.740349 1.433721 2.017480 2.296010 2.149951 1.637560 1.026532 0.802753 2.76637 –2.74109 –2.61254 –2.22046 –1.34120 0.250025 2.66299 5.72811 8.83971 10.9285 10.7186 7.34634 1.17692 –5.68788 –9.87672 –8.48304 –5.49208 Forced Duffing Oscillator Output response u,u' and u'' 3 2 By trigonometric series u Computed u By trigonometric series u' Computed u' By trigonometric series u'' Computed u'' 1 0 -1 -2 -3 0 0.2 0.4 0.6 0.8 1 t Fig. 5.1.2a. Comparison of Haar wavelet solution with trigonometric series solution for a ¼ 1 . 6 Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 8 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 150 Computed u Computed u' Computed u'' Output response u,u',u'' for α =10 100 50 0 -50 -100 -150 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 1 Fig. 5.1.2b. Haar wavelet solution for a = 10. By quasilinearization, we get the form ð2Þ ð1Þ urþ1 ðtÞ  0:1urþ1 ðtÞ þ urþ1 ðtÞ ¼ 0:1u2r ðtÞu0rþ1 ðtÞ þ 0:2u2r ðtÞu0r ðtÞ  0:2u0r ðtÞur ðtÞurþ1 ðtÞ þ 0:02u3r ðtÞ  0:03u2r ðtÞurþ1 ðtÞ: ð30Þ By using the Eqs. (19) in (27),we obtain " # 2m 2m 2m 2m X X X X ai hi ðtl Þ þ pi;1 ðtl Þð0:1 þ 0:4 cos2 ðt l ÞÞ þ pi;2 ðtl Þð1 þ 0:4 cos2 ðt l Þð4 cosðt l Þ þ 0:3ÞÞ i¼0 i¼0 i¼0 i¼0 ¼ 2  8 sinð2t l Þ cosðt l Þ  cos2 ðt l Þð15:4 cosðtl Þ þ 0:24Þ: ð31Þ 0 After solving system of Eq. (31), we have computed solutions u and u . Comparison of obtained solutions with those solutions which have been obtained by Adomian decomposition method [29] is shown in Fig. 5.2.1. The absolute errors in the solutions are also reported in Table 1. Unforced Duffing-van der Pol 2.5 2 ADM[24] u Computed u u' Computed u' Output response u, u' 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Fig. 5.2.1. Comparison of Haar wavelet solution with Adomain decomposition method solution. Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 9 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx Forced Duffing -van der Pol 1.6 α=-0.5 α=0.5 α=1.5 α=2.5 Output response u 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 1 0.9 1 Fig. 5.2.2a. The effect of different values of l keeping a = 0.5.2a. Forced Duffing-van der Pol 1.4 μ=-2.5 μ=-1.5 μ=.5 1.2 Output response u 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 Fig. 5.2.2b. The effect of different values of a keeping l = 0.5. Table 3 Computed Haar wavelet Solution uðtÞ for problem 5.2.2 at different values of l and a. t l ¼ 0:5 a ¼ 0:5 l ¼ 0:5 a ¼ 1:5 l ¼ 0:5 a ¼ 2:5 l ¼ 0:5 a ¼ 0:5 a ¼ 0:5 l ¼ 1:5 a ¼ 0:5 l ¼ 2:5 a ¼ 0:5 l ¼ 0:5 0.0156 0.1094 0.2031 0.2969 0.3906 0.5156 0.6094 0.7031 0.7969 0.8906 0.9844 0.999939 0.997014 0.989735 0.978191 0.962516 0.935526 0.911024 0.883181 0.852307 0.818731 0.782791 0.999817 0.991072 0.969560 0.936134 0.892052 0.819362 0.756565 0.688489 0.616619 0.542236 0.466374 0.999695 0.985167 0.949797 0.895886 0.826694 0.717307 0.627435 0.534554 0.441452 0.350018 0.261395 1.00006 1.00299 1.01033 1.02212 1.03834 1.06693 1.09352 1.12435 1.15913 1.19739 1.23849 0.999817 0.991054 0.96935 0.935213 0.889407 0.811893 0.742905 0.665995 0.582432 0.493472 0.400332 0.999695 0.985106 0.949106 0.892876 0.818179 0.693792 0.585343 0.467045 0.341798 0.212456 0.081763 1.00006 1.00299 1.01026 1.02181 1.03753 1.06486 1.09013 1.11957 1.15343 1.19207 1.23698 Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 10 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 4 Comparison of Haar wavelet solution with those solutions available in [15] by different methods for problem 5.2.2 with parameters d ¼ 0:1; b ¼ 0:1; l ¼ 0:5; a ¼ 0:5 and F ¼ 0:5; x ¼ 0:79. t uðtÞ uðtÞ uðtÞ uðtÞ 0.2 0.4 0.6 0.8 1.0 HWM 0.9900 0.9609 0.9128 0.8502 0.7766 VIM[15] 0.9900 0.9607 0.9134 0.8502 0.7735 HPM[15] 0.9900 0.9607 0.9138 0.8521 0.7797 NUMERICAL[15] 0.9900 0.9607 0.9134 0.8502 0.7735 Table 5 Comparison of Haar wavelet solution with those solutions available in [15] by different methods for problem 5.2.2 with parameters for following parameters: d ¼ 0:1; b ¼ 0:1; l ¼ 0:5; a ¼ 0:5 and F ¼ 0:5; x ¼ 0:79. t uðtÞ uðtÞ uðtÞ uðtÞ 0.2 0.4 0.6 0.8 1.0 HWM 1.0099 1.0392 1.0834 1.1440 1.2037 VIM[15] 1.0099 1.0391 1.0854 1.1453 1.2138 HPM[15] 1.0099 1.0391 1.0862 1.1493 1.2278 NUMERICAL[15] 1.0099 1.0391 1.0854 1.1453 1.2137 Table 6 Comparison of Haar wavelet solution with those solutions available in [15] by different methods for problem 5.2.2 with parameters d ¼ 0:1; b ¼ 0:1; l ¼ 0:5; a ¼ 0:5 and F ¼ 0:5; x ¼ 0:79. t uðtÞ uðtÞ uðtÞ uðtÞ 0.1 0.2 0.5 .75 1.0 HWM[15] 1.0029 1.0101 1.0631 1.1458 1.2609 VIM[15] 1.0025 1.0100 1.0629 1.1420 1.2603 HPM[15] 1.0025 1.0100 1.0629 1.1420 1.2505 NUMERICAL[15] 1.0025 1.0100 1.0629 1.1420 1.2604 0.45 Haar Wavelet Solution 0.4 0.35 0.3 u 0.25 0.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Fig.5.3.1. Haar wavelet solution. 5.2.2. Forced Duffing -van der Pol oscillator [31,32] Consider Eq. (3) with initial conditions uð0Þ ¼ 1; u0 ð0Þ ¼ 0 and parameters are given as below e ¼ 1; d ¼ 0:1; b ¼ 0:1; l ¼ 0:5; a ¼ 0:5 and F ¼ 0:5; x ¼ 0:79: Haar wavelet solutions are shown graphically in Figs. 5.2.2a and 5.2.2b and reported in Table 3 for different values of parameters l and a: Comparison of solutions with those available in [32] are given in Tables 4, 5 and 6 for different values of parameters. Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 11 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 7 Comparison of Haar wavelet solution uðtÞ with those obtained by different numerical methods [33] for Prob. 5.3.1. t Runge kutta method Gear method Chebyshev series solution Haar wavelet solution 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05004 0.09983 0.14886 0.19665 0.24270 0.28653 0.32770 0.36577 0.40034 0.43105 0 0.05004 0.09983 0.14886 0.19665 0.24270 0.28653 0.32770 0.36577 0.40034 0.43105 0 0.05004 0.09982 0.14885 0.19664 0.24275 0.28653 0.32771 0.36576 0.40040 0.43104 0 0.05004 0.09983 0.14886 0.19665 0.24270 0.28653 0.32770 0.36577 0.40034 0.43105 80 HWS for ε = 0.0001 60 40 u 20 0 -20 -40 -60 -80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 following initial t Fig. 5.3.2a. Haar wavelet solution. 2000 HWS for ε = 0.001 1500 1000 500 u 0 -500 -1000 -1500 -2000 -2500 0 0.1 0.2 0.3 0.4 0.5 0.6 t Fig. 5.3.2b. Haar wavelet solution. 5.3. Van der Pol oscillator [15] 5.3.1. Unforced Van der Pol oscillator Consider the Van der Pol oscillator from e ¼ 1; d ¼ 0:05; b ¼ 0:05; l ¼ 1; a ¼ 0 and F ¼ 0: u00 ðtÞ  0:05ð1  u2 ðtÞÞu0 ðtÞ þ uðtÞ ¼ 0; Eq. (5) with with uð0Þ ¼ 0 and u0 ð0Þ ¼ 0:5: conditions and parameters ð32Þ Computed Haar wavelet solution is depicted in Fig. 5.3.1. Comparison of results with available solutions are also reported in Table 7. Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 12 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 2 x 10 7 HWS for ε = 0.01 1.5 1 u 0.5 0 -0.5 -1 -1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 t Fig. 5.3.2c. Haar wavelet solution. 2 x 10 4 HWS for ε = 0.1 0 -2 u -4 -6 -8 -10 -12 0 0.1 0.2 0.3 0.4 0.5 0.6 t Fig. 5.3.2d. Haar wavelet solution. 1.5 HWS for ε = 1000 1.4 1.3 1.2 u 1.1 1 0.9 0.8 0.7 0.6 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Fig. 5.3.2e. Haar wavelet solution. Please cite this article in press as: H. Kaur et al., Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Modell. (2014), https://dx.doi.org/10.1016/j.apm.2014.03.019 13 H. Kaur et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 8 Computed Haar wavelet solution uðtÞ for different values of e for prob. 5.3.2. t e ¼ 0:0001 e ¼ 0:001 e ¼ 0:1 e ¼ 1000 0.0156 0.1094 0.2031 0.2969 0.3906 0.5156 0.6094 0.7031 0.7969 0.8906 0.9844 0.503418 0.426271 0.346682 –0.650879 12.9768 24.4202 –28.1094 42.7498 –46.5388 66.4663 –71.4282 0.515625 0.609375 0.703125 –0.301753 53.2588 331.704 –610.707 881.696 –1272.02 1588.06 –2117.17 0.515625 0.609375 0.703125 0.796875 0.890625 1.01563 1.10938 1.20313 –108.566 –1744.21 –106017 0.515625 0.609367 0.703098 0.796818 0.890525 1.01544 1.10911 1.20271 1.29623 1.38947 1.48229 5.3.2. Unforced Van der Pol oscillator From Eq. (5), consider the Van der Pol oscillator with the following initial conditions and parameters d ¼ 1; b ¼ 1; l ¼ 1; a ¼ 0 and F ¼ 0: eu00 ðtÞ þ ð1  u2 ðtÞÞu0 ðtÞ þ uðtÞ ¼ 0; with uð0Þ ¼ 0:5 and u0 ð0Þ ¼ 1: ð33Þ Haar wavelet solutions of Eq. (33) are shown in Figs. 5.3.2a, 5.3.2b, 5.3.2c, 5.3.2d, 5.3.2e for different values of e. Also see Table 8 for solutions. 6. Conclusion The aim of this paper is to represent a Haar wavelet method to solve well known nonlinear oscillator differential equations such as Duffing, Van der Pol, and Duffing–van der Pol with different parameters. To overcome the nonlinearities, quasilinearization is used. It is observed that the quasilinearization makes easier procedure for the Haar wavelet method to handle nonlinearity in a shorter time of computations. There is no need of iterations for achieving sufficient accuracy in numerical results. Therefore, it is suggested that quasilinearization can effectively be used to solve the nonlinear oscillator differential equations. In our method, when we increase number of points m ¼ 2j ; then coefficient matrix becomes ill-conditioned and it becomes difficult to find direct solutions. The Haar wavelet collocation method computes the solutions only at odd points. However, results can be obtained at any point of the domain. The obtained numerical solutions are in very good coincidence with those solutions which are available in literature computed by other methods and indicate that the proposed method is feasible and convergent. The effects of constant parameters on responses of system for Haar wavelet method are also shown in figures. Therefore, it is recommended to use Haar wavelets to compute solutions of nonlinear vibration problems. Acknowledgements The authors thankfully acknowledge the comments of anonymous referees which improve the manuscript and indebted to editor for his illuminating advice and valuable discussion. Author Harpreet Kaur is thankful to Sant Longowal Institute of Engineering and Technology(SLIET), Longowal, India for providing financial support as a senior research fellowship. References [1] D.W. Jordon, P. 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