Numerical Solution of Time-Fractional Sub-Diffusion Equation via Laguerre-Galerkin Method

This article presents a Laguerre-Galerkin approximation for the numerical solution of time
fractional sub-diffusion equations involving the Caputo fractional derivative. The proposed
methodology employs the separation of variables technique to decompose the governing equa
tion into a spatial ordinary differential equation and a fractional temporal equation into spatial
and temporal components. The resulting spatial boundary value problem is approximated using
Laguerre polynomial basis functions, while the temporal component is represented through the
Mittag-Leffler function. Both Neumann and Dirichlet boundary conditions are investigated.
For problems with Dirichlet boundary conditions, a shooting technique is utilized to obtain an
equivalent Neumann formulation suitable for the proposed approach. Numerical experiments
are carried out for different values of the fractional order and approximation parameters. The
obtained numerical solutions are compared with available exact solutions through graphical and
error analysis. The results demonstrate that the proposed method provides accurate approxi
mations with good convergence characteristics and serves as an efficient Computationally tool
for the study of time-fractional sub-diffusion models.