Composite spectral collocation method of Legendre-Legendre rational polynomials for the Fokker–Planck equation on unbounded domain

A composite scheme using Legendre-Legendre rational spectral collocation is developed for solving initial–boundary value problems of partial differential equations. This method can accommodate various boundary conditions across different subdomains by employing mixed Lagrange interpolation polynomials tailored to each subdomain. New quasi-orthogonal approximation results are introduced, combining one-dimensional quasi-orthogonal approximation with interpolation approximation. Unlike existing quasi-orthogonal projections, which only match the approximated function at the endpoints of the interval, the new two-dimensional quasi-orthogonal projection can match the approximated function accurately at all interpolation nodes within the interval. The convergence analysis is established by converting the spectral collocation scheme into a discrete inner product formulation and applying the Petrov–Galerkin method with numerical integration. Numerical experiments demonstrate the high accuracy and effectiveness of the proposed method.