Parameter-expansion Techniques for Strongly Nonlinear Oscillators

Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)

“…uðsÞ ¼ Acos us (16) Substituting (16) into the u term of (15) yields RðsÞ ¼ 1 2 u 2 A 2 sin 2 us þ FðAcos usÞ À FðAÞ ¼ 0…”

“…Nevertheless, perturbation techniques are valid only for weakly nonlinear problems. Recently, some approximate variational methods are considered to be the powerful methods capable of handling strongly nonlinear behaviors, and can converge to an accurate periodic solution for smooth nonlinear systems, such as variational iteration method [8e11] homotopy perturbation method [12e14], parameter-expansion method [15,16], maxemin approach [17e19] and energy balance method [20e22]. In the energy balance method according to basic idea of it, if q ¼ 0, it shows the whole energy is in form of kinetic energy and if q ¼ p/2, it shows the whole energy is in form of potential energy, in q ¼ p/4 there is a balance between the potential energy and kinetic energy so we can benefit from this point [23].…”

Comparing linear and nonlinear differential equations of differential transformation method by other numerical methods

“…Finally, the following series solution can be formed by applying the inverse transformation in definition 2 up to Next we consider, which is also solved by ADM [15], Noor decomposition [16] and Variational Iteration Method [17].…”

“…The author emphasized that the method presented is highly accurate solutions. Noor and Mohyuddin [17] solved fifth-order boundary value problems by using Noor decomposition method. They claimed that method had convergent series solution.…”

Periodic Solution for Strongly Nonlinear Oscillators by He’s New Amplitude–Frequency Relationship

“…It is very important in applications to have a version of the frequency (or period) to have a better understanding of the phenomena modeled through differential equations that contain terms with high nonlinearities, and a simple mathematical method is very useful for practical applications. Recently many analytical methods have appeared to obtain the approximate solutions of nonlinear systems, such as the parameter-expansion method [1], the harmonic balance method [2,3,4,6], the energy balance method [7,8], the Hamiltonian approach [10,12], the use of special functions [13,14], the max-min approach [15,16], the variational iteration method [17,18,19,20,21] and homotopy perturbation [22,23,24,25,26,27,28], and others. An excellent study, in which many of these techniques can be found in detail to solve nonlinear problems of oscillatory type can be seen in [29].…”