A Comparative Study of Analytical Solutions to the Coupled Van-der-pol's Non-linear Circuits Using the He's Method (hpem) and (bpes)

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 oscillating 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 compared with exact solutions proposed elsewhere, in order to evaluate accuracy.


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 bifurcations, quasiperiodicity, and chaos.Its application has already reached nerve pulse propagation and electric potential evolution across neural membranes.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. [3] in the heat transfer domain, the investigations of L. Cveticanin [4] and J. H.He [5][6][7] on nonlinear mechanics, fluid dynamics and oscillating systems modelling (Figure 2).Among the different formulations, the well-known standard boundary value-free Van der Pol oscillator problem (BVFP) is given by F. M. Atay [8] by the following system (1): where  is a positive parameter representing the delay,  > 0 and is the feedback gain.k A simpler formulation is that of W. Jiang et al. [9]: x t y t y t x t kf x t x t y t 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. [10]:

The Enhanced He's Parameter-Expanding Method (HPEM) Solution
The resolution protocol based on the enhanced He's parameter-expanding method (HPEM) is founded on the infinite serial expansions: Substituting these expansions in the main equation Equation ( 3) and processing with the standard perturbation method, it has been demonstrated [10] that a solution of the kind: where H ,  and  are constant, gives: .

Scheme (BPES)-Related Solution
The resolution protocol is based on the Boubaker pol expressions: The main advantage of these formulations (Equations ( 4) and ( 5)) is the fact of verifying the boundary conditions in d 1 N B t r  t stage of fact, due to the properties of the Boubaker polynomials [12][13][14][15][16][17][18], and since Equation (3), at the earlies resolution protocol.In the following conditions stand : By introducing expressions (4) and (6) in the system (3), and by majoring and integrating along the interval , to be a weak solution of the system: The set of solutions H. KOÇAK ET AL.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 [8] and A. Kimiaeifar et al under the intrinsic condition: Th along with the exact .[10].It is noted that F. M. Atay [8] 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. [10] elded a highly accurate solution to the same classical Van der Pol equation with delayed feedback and a modified equation where a delayed term provides the damping.The features of the proposed solutions [8][9][10] (namely behavior at starting phase, first derivatives at limit time, etc.) are concordant with the actually proposed results.

Results and Discussions
The results show a good agreement between the proposed 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 conve he BPES-related protocol has been recorded f s of superior to 30.

Conclusions
In this paper, we have used the enhanced He's parame ter-expanding Method (HPEM) along with th Boubaker olynom r to obc periodical solutions.acceptable agreement into an istic nonlinear system.This simple duction is carried out through the The obtained solutions were in with those obtained from values of similarly performed methods.The typical periodical aspect of the oscillations, already yielded [2,10,[24][25][26][27] by the enhanced He's parameter-expanding method (HPEM) could be reproduced using a simple and convergent polynomial approximation.This method was based on an original protocol hich reduces the stochastic nonlinear system w equivalent determin nd controllable re a verification of the initial conditions, in the solution basic expression, prime to launching the resolution process.

Figure 2 .
Figure 2. A prototype of Van der Pol oscillating systems modelling (The two integrators are Trapezoidal-type     1 1
(BPES) in orde P tain the Van der Pol's characteristi and A. Rajabi, "Assessment of Homotopy Perturbation and Perturbation Methods in Heat Transfer Radiation Equations," International Communications in 31 0.0