Fractional Spectral Collocation with Generalized Rational Jacobi Functions on the Half-Line

We develop a fractional pseudo-spectral collocation method for steady and time-dependent fractional differential equations on the half-line, which are based on previously introduced generalized rational Jacobi functions (GJRFs). Rational cardinal functions are constructed at mapped Jacobi–Gauss points, yielding explicit nodal differentiation matrices for left-sided Riemann–Liouville and right-sided Weyl operators, including higher-order derivatives. A boundary-adapted second-kind construction enforces the homogeneous condition at the finite endpoint. For the representative left-sided order-μ matrix with 0<μ<1, we make explicit its diagonal equivalence to a Jacobi-weighted fractional Lagrange matrix on [−1,1]; in GJRF modal coordinates, this gives a finite-N factorization, determinant identity, inverse bound, and a conditional consistency-to-solution nodal error estimate. These transferred finite-dimensional structural results do not constitute a uniform-in-N convergence theorem. The method is applied to fractional advection and diffusion equations, multi-term problems, Burgers-type equations, and Schrödinger-type equations with unilateral fractional spatial operators. A Jacobi-parameter sensitivity study shows that matching the basis parameter to the endpoint behavior of the solution is essential for rapid convergence; with this choice, the linear and nonlinear tests exhibit rapid error reduction across the tested fractional orders. Temporal refinement recovers the expected L1 convergence rates, and matched-degree-of-freedom comparisons with GJRF Petrov–Galerkin discretizations are also reported. The results demonstrate the effectiveness of the GJRF collocation framework for the half-line problems considered.