A Comparative B-Spline Collocation Framework for the Caputo Time-Fractional Mobile–Immobile Transport Equation

In this study, we present a novel and efficient computational framework for obtaining approximate solutions of the time-fractional mobile–immobile transport equation, where the temporal derivative is considered in the Caputo sense with order 0<γ≤1. The spatial derivatives are discretized using three different B-spline basis functions, namely cubic B-splines (CuBS), trigonometric cubic B-splines (TCuBS), and extended cubic B-splines (ECuBS), while the time-fractional derivative is approximated through an appropriate finite difference scheme. The theoretical reliability of the proposed schemes is established by conducting stability and convergence analyses. To evaluate the accuracy and computational efficiency of the proposed methods, several numerical experiments are performed. The obtained results demonstrate a strong agreement between the numerical and exact solutions and confirm the effectiveness of the developed schemes. Furthermore, a comparative study of the three B-spline approaches is carried out, highlighting their advantages under different conditions. The proposed schemes provide reliable, accurate, and efficient numerical tools for solving time-fractional mobile–immobile transport equations.