a boundary element method for the numerical solution of the variable-order time-fractional diffusion equation based on the caputo derivative

In this study, a boundary element method is employed for the numerical solution of the variable-order time-fractional diffusion equation based on the caputo fractional derivative. variable-order fractional equations are particularly effective in modeling physical and engineering processes with time-dependent memory effects. in the proposed approach, the fractional diffusion equation is reformulated in an integral form, and time discretization based on the caputo definition is applied to accurately account for the variations in the fractional order over time. the numerical algorithm is implemented in matlab, and results obtained from a numerical example demonstrate that the proposed method provides high accuracy, good stability, and fast convergence. these features make the boundary element method an efficient and reliable tool for solving variable-order time-fractional diffusion problems.