Abstract
The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) propounded by Liao [1]. The formulation has been kept quite general to keep open the possibility of obtaining the solution of the GL equation for different continua as limiting cases of the present study. New ideas have been added and clear explanations are provided in the paper to the existing concepts in HAM. The method can easily be extended to solve complex GL equation, system of GL equations or even the GL equations with a diffusion term, each having a time-periodic coefficient. The necessary code in Mathematica that implements the HAM for the current problem is appended to the paper for use by the readers.
Highlights
The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) propounded by Liao [1]
GL equations arise as a solvability condition in a wide variety of problems in continuum mechanics while dealing with a weakly nonlinear stability of systems, e.g., one comes across the GL equation with constant and real coefficients in the case of Rayleigh-Bénard convection in fluids wherein instability sets in as a direct mode
In certain other problems one may come across a system of GL equations with constant coefficients
Summary
GL equations arise as a solvability condition in a wide variety of problems in continuum mechanics while dealing with a weakly nonlinear stability of systems, e.g., one comes across the GL equation with constant and real coefficients in the case of Rayleigh-Bénard convection in fluids wherein instability sets in as a direct mode ( called stationary mode). When the Hopf mode ( called oscillatory mode) is the preferred one, like in viscoelastic liquids or as in constrained systems, the GL equation has complex yet constant coefficients. When one considers problems in which gravity experienced in a fluid-based system is perturbed by a time-periodic vibration of the system, the GL equation turns out to be an equation like the one considered in the paper and the same has been solved here using the HAM, as propounded by Liao[1,2,3,4,5]. The great advantage in using the method is that it gives the solution of non-linear equations in the form of a series whose convergence is assured. The method is illustrated here in unabridged form using the example of the GL equation with a time-periodic coefficient. The readers may refer to Liao [1,2,3,4,5] and others [6,7,8,9,10,11,12,13] for many other versatile applications of the method
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