On the Multistage Differential Transformation Method for Analyzing Damping Duffing Oscillator and Its Applications to Plasma Physics

Abstract

The multistage differential transformation method (MSDTM) is used to find an approximate solution to the forced damping Duffing equation (FDDE). In this paper, we prove that the MSDTM can predict the solution in the long domain as compared to differential transformation method (DTM) and more accurately than the modified differential transformation method (MDTM). In addition, the maximum residual errors for DTM and its modification methods (MSDTM and MDTM) are estimated. As a real application to the obtained solution, we investigate the oscillations in a complex unmagnetized plasma. To do that, the fluid govern equations of plasma species is reduced to the modified Korteweg–de Vries–Burgers (mKdVB) equation. After that, by using a suitable transformation, the mKdVB equation is transformed into the forced damping Duffing equation.

Some Novel Approaches for Analyzing the Unforced and Forced Duffing–Van der Pol Oscillators

“…For instance, Du ng-type equation is one of the most famous and successful equations that has been used for modeling and interpreting many nonlinear oscillations in many different dynamical systems such as electrical circuit, optical stability, the buckled beam, and di erent oscillations in a plasma [16][17][18][19]. In plasma physics, there are many evolution equations that can be reduced to Du ng-type equation, Helmholtz-type equation, Du ng-Helmhlotz equation, and Mathieu equation in order to investigate the various oscillations that occur within complicated plasma systems [20][21][22][23]. ere is another type of equation of motion that was used for modeling the nonlinear oscillations in biology, electronics, engineering, plasma physics, and chemistry which is called Van der Pol-Du ng (VdPD) (sometimes called Du ng-Van der Pol (DVdP)) equation and its family [24,25].…”

On Reduce Differential Transformation Method for Solving Damped Kawahara Equation

“…e partial deferential equations (PDEs) describe several important applications in many branches of science such as physics, engineering, medicine, and uid dynamic [1][2][3][4]. Mathematicians put forth high e orts to develop methods that are able to nd solutions of these PDEs [5][6][7][8].…”

Novel Analytical and Numerical Approximations to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Method

“…e pendulum oscillator and some related equation have been used as a physical model to solve several natural problems related to bifurcations, oscillations, and chaos such as nonlinear plasma oscillations [1][2][3][4][5][6][7][8][9], Du ng oscillators [10][11][12][13][14], and Helmholtz oscillations [12], and many other applications can be found in [15][16][17][18][19][20][21][22][23][24]. ere are few attempts for analyzing the equation of motion of the nonlinear damped pendulum taking the friction forces into account [25].…”