Spectral Method For The Fourth-order Partial Differential Equations In A Semi-infinite Channel

The paper focuses on mixed Laguerre-Legendre spectral method and composite Legendre-Laguerre spectral method for the fourth-order partial differential equations(PDEs) set in a semi-infinite channel.We consider firstly the Laguerre-Legendre spectral approximation of the fourth-order equation directly on the whole channel. By constructing appropriate basis functions satisfying the boundary conditions, the coefficient matrix of the corresponding linear system is sparse, which can be solved efficiently. Convergence analysis of the method is strictly carried out. Numerical results are presented to confirm the theoretical analysis.Secondly, we consider the approximation with domain decomposition. The channel is decomposed into two parts. Then Legendre spectral method and mixed Laguerre-Legendre spectral method are presented to approximation the problem in the two domains respectively. Suitable basis functions are constructed to couple the two methods. Strict error analysis of the method is also presented. The convergence of the method is confirmed by numerical experiment.