A Galerkin Finite Element Method with Cubic β-Spline Basis for Multi-Term Time-Fractional Differential Equations

This paper presents a Galerkin finite element method based on cubic β-spline basis functions for the numerical solution of multi-term time-fractional differential equations. The spatial discretization is constructed using a parametric family of cubic β-splines, which provide enhanced flexibility and smoothness in the approximation space. For the temporal discretization, the Caputo fractional derivatives are approximated using the L1 scheme combined with a θ-method to improve stability and numerical accuracy for multi-term fractional models. The resulting fully discrete formulation is solved efficiently and the source terms are evaluated using Gauss–Legendre quadrature to obtain accurate load vector approximations. The convergence behavior of the proposed scheme is investigated through mesh refinement analysis and numerical accuracy is assessed using standard error norms. Several benchmark test problems are considered to validate the proposed method. The numerical results demonstrate that the cubic β-spline Galerkin approach provides accurate and stable approximations, with error norms decreasing consistently as the mesh is refined. Comparisons with available exact solutions further confirm the efficiency and reliability of the proposed numerical scheme for solving complex multi-term time-fractional differential equations.