Abstract
The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obtained using the fourth-order Runge-Kutta numerical solution method. A comparison of the result by the numerical Runge-Kutta fourth-order accuracy method is compared with the result by the MDTM and plotted in a long time domain.
Highlights
We are improving the accuracy of the differential transform solution using the modified differential transform method (MDTM) [22]
It is clear that the MDTM result obtained by the real part of Padé approximate gives an excellent agreement with the result obtained using the fourth-order Runge-Kutta numerical method
The main goal of researchers who are interested in solving nonlinear differential equations is to obtain analytical solutions along with numerical solutions
Summary
(1b) where x is the position coordinate which is a function of the time t, ω is the system’s natural frequency, η is a scalar parameter indicating the damping factor in Duffing equation, e the nonlinearity and strength of the damping in van der Pol equation, respectively. α is a nonlinear parameter factor, A and. With Lorenz, Thompson, and Appleton, van der Pol experimented with oscillations in a vacuum tube triode circuit and concluded that all initial conditions converged to the same orbit of a finite amplitude. Since this behavior is different from the behavior of solutions of linear. This work is the derivation to obtain approximate analytical oscillatory solutions for the nonlinear oscillator Equations (1a) and (1b) with initial conditions x (0) = a and ẋ (0) = b using the modified differential transform method This is a powerful method for solving linear and nonlinear differential equations. Padé approximant [31] which can successfully predict the solution of differential equations with finite numbers of terms [32,33]
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