A semi-analytical meshless collocation technique for the distributed-order time-fractional cable equation

This paper proposes a semi-analytical meshless collocation method for solving the distributed-order time-fractional cable equation (DO-TFCE). The distributed-order terms are first approximated using Gauss–Legendre quadrature, which reduces them to finite weighted sums of fractional derivatives. The resulting Riemann–Liouville derivatives are then discretized in time by a second-order weighted and shifted Grünwald scheme. The stability and convergence of the semi-discrete formulation are established through an energy-based analysis. For the spatial discretization, the backward substitution method is employed as a semi-analytical meshless collocation technique in which the boundary conditions are treated separately from the governing equation. The method begins by constructing an approximation directly from the prescribed boundary data. A corrective function is then introduced to enforce the governing equation while preserving homogeneous boundary conditions. The final numerical solution is obtained by combining the boundary approximation with the corrective component. Numerical results are presented to validate the theoretical findings and demonstrate the accuracy of the proposed method.