Hybrid of shifted Legendre and rational Legendre spectral methods on a semi-infinite interval

In this study, spectral and pseudospectral methods on the half-line are studied using orthogonal systems of shifted Legendre polynomials on a subinterval [0, S] and rational Legendre functions on [S, infinity). Based on this idea a hybrid orthogonal system is introduced. We establish primary results on hybrid approximations of inter-polations and some orthogonal projections. The obtained results organize the theory of developing spectral and pseudospectral methods for solving differential equations on a semi-infinite interval. Error analysis for a model problem are established. Numerical results are included to support the theoretical results and shows the effectiveness and performance of this method.