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Chebyshev wavelet collocation method for Ginzburg-Landau equation

Secer Aydin (Yildiz Technical University, Faculty of Chemistry-Metallurgical, Department of Mathematical Engineering, Davutpaşa Campus, İstanbul, Туркеy)
Bakir Yasemin (Yildiz Technical University, Faculty of Chemistry-Metallurgical, Department of Mathematical Engineering, Davutpaşa Campus, İstanbul, Туркеy)

The main aim of this paper is to investigate the efficient Chebyshev wavelet collocation method for Ginzburg-Landau equation. The basic idea of this method is to have the approximation of Chebyshev wavelet series of a non-linear PDE. We demonstrate how to use the method for the numerical solution of the Ginzburg-Landau equation with initial and boundary conditions. For this purpose, we have obtained operational matrix for Chebyshev wavelets. By applying this technique in Ginzburg-Landau equation, the PDE is converted into an algebraic system of non-linear equations and this system has been solved using MAPLE computer algebra system. We demonstrate the validity and applicability of this technique which has been clarified by using an example. Exact solution is compared with an approximate solution. Moreover, Chebyshev wavelet collocation method is found to be acceptable, efficient, accurate and computational for the non-linear or PDE.

Keywords: Chebyshev wavelet collocation method, Ginzburg-Landau equation, operational matrices, non-linear PDE