Abstract
In the paper, we solve two nonlinear problems related to the Duffing equations in space and in time. The first problem is the bifurcation of Duffing equation in space, wherein a critical value of the parameter initiates the bifurcation from a trivial solution to a non-trivial solution. The second problem is an unconventional periodic problem of Duffing equation in time to determine period and periodic solution. To save computational cost and even enhance the accuracy in seeking higher order analytic solutions of these two problems, a modified homotopy perturbation method is invoked after a linearization technique being exerted on the Duffing equation, whose nonlinear cubic term is decomposed at two sides via a weight
 factor, such that the Duffing equation is linearized as the Mathieu type differential equation. The constant preceding the displacement is expanded in powers of homotopy parameter and the coefficients are determined to avoid secular solutions appeared in the derived sequence of linear differential equations. Consequently, after setting the homotopy parameter equal to unity and solving the amplitude equation, the higher order bifurcated solutions can be derived explicitly. For the second problem, we can determine the period and periodic solution in closed-form, which are very accurate. For the sake of comparison the results obtained from the fourth-order Runge-Kutta numerical method are used to evaluate the presented analytic solutions.
Highlights
The first problem is the bifurcation of Duffing equation in space, wherein a critical value of the parameter initiates the bifurcation from a trivial solution to a non-trivial solution
To save computational cost and even enhance the accuracy in seeking higher order analytic solutions of these two problems, a modified homotopy perturbation method is invoked after a linearization technique being exerted on the Duffing equation, whose nonlinear cubic term is decomposed at two sides via a weight factor, such that the Duffing equation is linearized as the Mathieu type differential equation
In the last few decades, considerable attention was directed towards the analytic solutions for nonlinear oscillators, since most nonlinear phenomena are modeled by nonlinear differential equations
Summary
In the last few decades, considerable attention was directed towards the analytic solutions for nonlinear oscillators, since most nonlinear phenomena are modeled by nonlinear differential equations. The harmonic balance method approximates the periodic solutions of nonlinear oscillators expanded in terms of the Fourier series, and the terms associated with each harmonic component are balanced This process could be very complicated when the order increases. The parameter-expanding method developed by He (2001,2006) is a convenient method for nonlinear differential equations, which has been shown to effectively, and accurately solve a large class of linear and nonlinear problems with components that converge rapidly to accurate solutions. He’s homotopy perturbation method is applied to the linearized differential equation of Mathieu type to determine the period and periodic solution. We develop a linearized homotopy method to determine the period and the free vibration mode up to second-order and the accuracy of the period and the analytic solution are confirmed by comparing to the exact one and the one computed from the RK4.
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