On the Generalized Method of Lines Applied to Ginzburg–Landau Type Equations

Abstract

The main focus of this work is the presentation of some new developments concerning the generalized method of lines (GMOL). We develop some examples about the method related to Ginzburg–Landau type equations. It is worth emphasizing that, as a typical parameter \(\varepsilon >0\) is too small, to obtain the relation between two adjacent lines through the contraction mapping theorem is not viable. To overcome such a problem, we suggest the procedure of using the same line expressions generated by GMOL, but calculating the real function coefficients by numerically minimizing the \(L^2\) norm equation error.