Exact analytic solutions for the damped Duffing nonlinear oscillator

Abstract

We prove that the second-order damped nonlinear Duffing oscillator is reduced to an equivalent equation of the normal Abel form of the second kind. Based on a recently developed mathematical methodology for the construction of exact analytic solutions of Abel's equation, exact analytic solutions are obtained for the nonlinear damped Duffing oscillator obeying the initial conditions adapted to the physical problem. To improve the general developed methodology an application concerning the nonlinear Van der Pol free oscillator is briefly discussed.

Analytical and numerical solution to the nonlinear cubic Duffing equation: An application to electrical signal analysis of distribution lines

“…For instance, the [15][16][17][18][19][20]. Some of the above mentioned methods were employed to investigate the dynamical behaviour of a large variety of physical systems that are described by the various forms of the homogeneous Duffing equation.…”

The frequency estimation for non‐conservative nonlinear oscillation

“…The numerical illustration of the analytical solution of the cubic power will be made to the solution (38). The calculations are displayed in Figure (5). The comparison shows high agreement between the numerical solution and the analytical solution.…”

Closed-Form Solutions of the Thomas-Fermi in Heavy Atoms and the Langmuir-Blodgett in Current Flow ODEs in Mathematical Physics

“…The solution of (16), and thus the solutions of (14), (12), and (9), constitutes the intermediate integral of the LB equation (1) in the phase plane. In other words, when obtaining the solution of the transformed equation (16) in the form = ( , * ), * = first integration constant, the solution of (14) becomes ( ) = − ( ), the solution of (12) is ( ) = − ( 4/3 ), and finally the solution of (8) is ( ) = − −1/3 ( 4/3 ). Thus, the solution to the original Langmuir equation 1in the physical plane can be obtained by the integration by parts of the following equation:…”

“…Note that the reduced Abel equation (16) does not admit an exact analytic solution in terms of known (tabulated) functions [4, pages 29-45]. 3is a typical Emden-Fowler nonlinear equation with = 1, = −1/2, and = 3/2.…”

“…In this paper using series of admissible functional transformations, at first both LB and TF nonlinear ODEs are reduced to equivalent Abel's equations of the second kind of the normal form. According to a mathematical methodology recently developed [14][15][16], we extract closed-form solution of the above Abel equations in both phase and physical planes under initial data in accordance with the physical problems.…”