The energy balance to nonlinear oscillations via Jacobi collocation method

A Modified Solution of the Nonlinear Singular Oscillator by Extended Iteration Procedure

“…Ordinarily, the nonlinear problems are solved by converting it into linear equations attributing some terms; but such linearization is not possible or feasible at all times. In these circumstances, there are a few analytical approaches to find approximate solutions to nonlinear problems, for instance; perturbation [1][2][3][4][5], standard as well as modified Linstedt-Poincare [6], Harmonic balance [7][8][9][10][11], Homotopy [12], Iterative [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] methods, He's new amplitudefrequency relationship [28], Jacobi collocation [29] etc. Among them, the perturbation method is the most widely utilized method in which the nonlinear term is small.…”

The Solution of the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries by Analytical Techniques

“…Therefore, considerable attention has been given to the solutions of differential equations of fractional order, of fluid mechanics and physics interest. Most nonlinear differential equations of fractional order do not have exact analytic solutions, so numerical and approximation techniques, such as Jacobi collocation method [19], Galerkin method [20], He's frequency-amplitude formulation and energy balance methods [21], max-min and Hamiltonian methods [22,23], variational iteration method [24][25][26][27][28] and Adomian decomposition method [29][30][31][32][33] must be used. The variational iteration and Adomian decomposition methods, which are special techniques of the homotopy analysis method [34], are relatively new techniques to provide analytical approximation to nonlinear problems and they are particularly valuable as tools for researchers, because they provide immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to nonlinear differential equations without linearization or discretization.…”

An accurate relationship between frequency and amplitude to nonlinear oscillations

“…Surveys of the literature express there are different approximate analytical techniques for dealing with the nonlinear problems. Among them, one may attract attention to the weighted linearization technique [1], the energy balance method [2][3][4], the optimal homotopy asymptotic method [5], the linearized harmonic balance method [6], the global residue harmonic balance method [7], the homotopy analysis method [8,9], Max-Min approach [10,11], Hamiltonian approach [12,13], the variational approach [14][15][16], the variational iteration method [17], the rational harmonic balance method [18], the Chebyshev polynomial approximation [19] and so on [20][21][22].…”