On spectral approximations using modified Legendre rational functions: Application to the Korteweg-de Vries equation on the half line (Q2767801)

A new set of modified orthogonal Legendre rational functions is introduced, with uniform weight equal to \(1\). The main motivation is the use of spectral methods for solving partial differential equations in unbounded domains when some type of conservation properties of the solution must be satisfied by the numerical approximation as well; this last point is not true when orthogonal Legendre rational functions with non-uniform weight, as defined by \textit{B.-Y. Guo, J. Shen} and \textit{Z.-Q. Wang} [J. Sci. Comput. 15, No. 2, 117-147 (2000; Zbl 0984.65104)], are employed. As an application, the authors propose a spectral conservative method for the initial boundary value problem of the Korteweg-de Vries equation on the half line. In the meanwhile, the authors study several projection operators and interpolation operators associated with the modified Legendre rational functions, which are used to prove the convergence of the spectral method.