 Circuits and Systems, 2011, 2, 196-200 doi:10.4236/cs.2011.23028 Published Online July 2011 (http://www.SciRP.org/journal/cs) Copyright © 2011 SciRes. CS A Comparative Study of Analytical Solutions to the Coupled Van-der-Pol’s Non-linear Circuits Using the He’s Method (HPEM) and (BPES) Hüseyin Koçak1, Ahmet Yıldırım1, Dahong Zhang2, Karem Boubaker3, Syed Tauseef Mohyud-Din4 1Department of Mathematics, Ege University, Bornova–İzmir, Turkey 2Department of Physics, South China University, Guangzhou, China 3École Supérieure des Sciences et Techniques de Tunis, University of Tunis, Tunis, Tunisia 4Department of Basic Sciences, HITEC University, Taxila Cantt, Pakistan E-mail: hkocak.ege@gmail.com, ahmet.yildirim@ege.edu.tr, Received October 6, 2010; revised May 13, 2011; accepted May 20, 2011 Abstract In this paper, the He’s parameter-expanding method (HPEM) and the 4q-Boubaker Polynomials Expansion Scheme (BPES) are used in order to obtain analytical solutions to the non-linear modified Van der Pol’s os-cillating circuit equation. The resolution protocols are applied to the ordinary Van der Pol equation, which annexed to conjoint delayed feedback and delay-related damping terms. The results are plotted, and com-pared with exact solutions proposed elsewhere, in order to evaluate accuracy. Keywords: Van-der-Pol’s Oscillating Circuit, Delayed Feedback, Damping, BPES, HPEM, Exact Solutions, Electrical Triode-Valve Circuit 1. Introduction Originally, the Van der Pol’s equation was associated, in the 1920s, with an electrical triode-valve circuit (Figure 1). In the last decades’ literature, it was the subject of several investigations due to the panoply of dynamical oddness as relaxation oscillations, elementary bifurca-tions, quasiperiodicity, and chaos. Its application has already reached nerve pulse propagation and electric potential evolution across neural membranes. Figure 1. Van der Pol oscillator synoptic scheme. The actual study tries to give a theoretical supply to the recent attempts to yield analytical solutions to this equation, like the studies of D. D. Ganji et al. [1,2] and A. Rajabi et al.  in the heat transfer domain, the investi-gations of L. Cveticanin  and J. H. He [5-7] on non- linear mechanics, fluid dynamics and oscillating systems modelling (Figure 2). Figure 2. A prototype of Van der Pol oscillating systems modelling (The two integrators are Trapezoidal-type 112kkkkyhuu y). H. KOÇAK ET AL. 197Among the different formulations, the well-known standard boundary value-free Van der Pol oscillator problem (BVFP) is given by F. M. Atay  by the fol-lowing system (1):    2,,,, 1xt xtfxtxtxtfxt xt xtxtxtk xt   (1) where  is a positive parameter representing the delay,  > 0 and is the feedback gain. kA simpler formulation is that of W. Jiang et al. :   21xt ytytxtkf xtxtyt  (2) In this study, an attempt to give analytical solution to the nonlinear second-order Van der Pol equation annexed to conjoint delayed feedback and delay-related damping terms as presented by A. Kimiaeifar et al. :   20010000xt ytytxtxtytk xtxxHxx   (3) 2. Analytical Solutions Derivation 2.1. The Enhanced He’s Parameter-Expanding Method (HPEM) Solution The resolution protocol based on the enhanced He’s pa-rameter-expanding method (HPEM) is founded on the infinite serial expansions:   00nnnnnnxtx txtx t  (4a) Substituting these expansions in the main equation Equation (3) and processing with the standard perturba-tion method, it has been demonstrated  that a solu-tion of the kind:  0cosxt Ht (4b) where H,  and  are constant, gives: 211132( )sin()( )cos()cos( )sin( )cos() cos()sin()sin() 0xt HtxtHtHtt kH tkH t    .2. The Boubaker Polynomials Expansion y-nomials expansion scheme (BPES) [11-23]. The first step of this scheme starts by applying the (4c) with, as a final solution (Equation (4d)): 2Scheme (BPES)-Related Solution The resolution protocol is based on the Boubaker polexpressions: 401λ2kk kk01NxtBtrN (4e) where 4kB are 4k-order Boubaker polynom the normalized time (theials, is0,1t), kr are minimal po tege4kBr, and sitive roots, 0N is a prefixed in01..kkNλ are u poingConsequently, it comes that: nknownnder real coefficients.  04102dkkkkkyt xt λrNt  (5) The main advantage of these formulations (Equations (4) and (5)) is the fact of verifying the boundary conditions in d1NBtrt stage of fact, due to the properties of the Boubaker polynomials[12-18], and since Equation (3), at the earliesresolution protocol. In 01..kNkrare the roots of kr 041..kkNB, the following conditions stand : 0000104100010dd10d2 dNktkNkkkkkttxt λAxNBtrxt rtN t  (6) By introducing expressions (4) and (6) in the system (3), and by majoring and integrating along the interval 0,1 , xt is confined, through the coefficients 01..kkN, to be a weak solution of the system:  000021112142014014014000 010ddddddddNNNkk kkkkkkkkkkkkkkkkkkkkkkNkkrM rPQkRBtrMttBtrPttQBtrtRBtrtNx NH  (7) 4224 22226 336 3636coskk kkkxt Htk   (4d)Copyright © 2011 SciRes. CS 198 The set of solutions H. KOÇAK ET AL. 01, ,ˆkkNis the one which mini mizes, for given values of  and the Minimum Square function (8) (9) The condition expressed by Equation (9) ensures a non-zero solution to the system (8). The convergence of the algorithm is tested relatively to increasing values of e correspondent solutions are represented in Figure 3 for the data gathered in Table 1, solutions given by F. M. Atay  and A. Kimiaeifar et alyik,MS k: 000 221,ˆˆˆMSNNNkkkkkrM rPQkR 11kk k kk kkk under the intrinsic condition: 001ˆNkkNH 0N. Thalong with the exact . . It is noted that F. M. Atay  demonstrated that the presence of delay can change the amplitude of limit cycle oscillations, or suppress them altogether through derivative-like effects, while A. Kimiaeifar et al.  elded a highly accurate solution to the same classical Van der Pol equation with delayed feedback and a modi-fied equation where a delayed term provides the damping. The features of the proposed solutions [8-10] (namely behavior at starting phase, first derivatives at limit time, etc.) are concordant with the actually proposed results. Figure 3. Analytical solutions plots. Table 1. Solution parameters values. Parameter Value  0.1 k −1.0  1.0 0x 2.75 0x 0.0 0N Figure 4. Mean absolute error versus N0. 3. Results and Discussions The results show a good agreement between the pro-posed analytical solutions (Figure 3) and those of the recent studies published elsewhere. The mean absolute error (for) was less than 3.33% (Figure 4). The convehe BPES-related protocol has been recorded fs of superior to 30. 4. Conclusions In this paper, we have used the enhanced He’s parameter-expanding Method (HPEM) along with thBoubaker olynomr to ob-c periodical solutions. acceptable agreement into an istic nonlinear system. This simple duction is carried out through the 030Nrgence of tor the value0N -e ials Expansion Scheme (BPES) in ordePtain the Van der Pol’s characteristiThe obtained solutions were inwith those obtained from values of similarly performed methods. 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