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.