Efficient numerical schemes for linear fractional differential systems with applications to diffusion problems

This study introduces an efficient numerical method based on the fractional backward differentiation formula of order l (FBDFl[jls-end-space/]) for solving linear systems of fractional differential equations (LSFDEs) within the Caputo derivative framework. The proposed approach reformulates the original LSFDEs into systems of algebraic equations at each time step, thereby streamlining the computational process. A detailed error analysis confirms the convergence of the method, with a global error of order (Formula presented), where l indicates the scheme’s order. The performance of the method is assessed through four representative numerical examples, including three cases involving time-fractional diffusion equations. Numerical results demonstrate the method’s high accuracy, computational efficiency, and adaptability to various fractional-order models. Particular attention is given to the FBDF2 and FBDF3 schemes, which exhibit excellent agreement with exact solutions and highlight the potential of the method for practical applications in fractional diffusion problems.