The Exact And Numerical Solution Of A Kind Of Landau-lifshitz And Ginzburg-landau Equation

In this paper, we study the Landau-Lifshitz equation of ferromagnetisms on R with a damping term. We disperse this equation of this kind by using the Euler-forward finite difference method. The proposed scheme is explicit so that the algorithm can be used to simulate numerically on computer. The convergence and stability of the proposed scheme are proved by the finite extensive method of the nonlinear function on R . Moreover, the numerical experiments are provided to check the theoretical results. The convergence of the semi-discrete Fourier spectral schemes of the above equation is proved. Numerical experiment of pseudo-spectral scheme is given.In another part of this article, we construct the exact solution of the Ginzburg-Landau equation. New exact periodic wave solutions for this 2D Ginzburg-Landau equation are obtained using the homogeneous balance principle and general Jacobi elliptic-function method. Furthermore, a blow up solution is provided. At the end, some properties about these solutions are showed by the graphs.