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In this paper, we investigate the mixed generalized Laguerre-Fourier spectral method and its applications to exterior problems of partial differential equations of fourth order. Some basic results on the mixed generalized Laguerre-Fourier orthogonal approximation are established, which play important roles in designing and analyzing various spectral methods for exterior problems of fourth order. As an important application, a mixed spectral scheme is proposed for the stream function form of the Navier-Stokes equations outside a disc. The numerical solution fulfills the compressibility automatically and keeps the same conservation property as the exact solution. The stability and convergence of proposed scheme are proved. Numerical results demonstrate its spectral accuracy in space, and coincide with the analysis very well.
A spectral collocation method for solving initial value problems of first order ordinary differential equations
We propose a spectral collocation method for solving initial value problems of first order ODEs, based on the Legendre-Gauss-Lobatto interpolation. This method is easy to be implemented and possesses the spectral accuracy. We also develop a multi-step version of this process, which is very available for long-time calculation. Numerical results demonstrate the high accuracy of suggested algorithms and coincide well with the theoretical analysis.
Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains
In this paper, we develop a pseudospectral method for differential equations defined on unbounded domains. We first introduce Gauss-type interpolations using a family of generalized Laguerre functions, and establish basic approximation results. Then we propose a pseudospectral method for differential equations on unbounded domains, whose coefficients may degenerate or grow up. As examples, we consider two model problems. The proposed schemes match the underlying problems properly and exhibit spectral accuracy. Numerical results demonstrate the efficiency of this new approach.
A modified Chebyshev rational orthogonal system on the whole line is introduced. A rational spectral scheme for the Korteweg de Vries equation on the whole line is constructed. The convergence is proved. The numerical results show its efficiency.
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