Laguerre and composite Legendre-Laguerre dual-Petrov-Galerkin methods for third-order equations

Dual-Petrov-Galerkin approximations to linear third-order equations and the Korteweg-de Vries equation on semi-infinite intervals are considered. It is shown that by choosing appropriate trial and test basis functions the Dual-Petrov-Galerkin method using Laguerre functions leads to strongly coercive linear systems which are easily invertible and enjoy optimal convergence rates. A novel multi-domain composite Legendre-Laguerre dual-Petrov-Galerkin method is also proposed and implemented. Numerical results illustrating the superior accuracy and effectiveness of the proposed dual-Petrov-Galerkin methods are presented.