An accurate finite-difference scheme for the numerical solution of a fractional differential equation

In this article, a steady-state fractional-order boundary-value problem is considered with a fractional convection term. The highest-order derivative term involves a mixed-fractional derivative, which appears as a combination of a first-order classical derivative and a Caputo fractional derivative. We propose an L۱ scheme over a uniform mesh for the numerical solution of the fractional differential equation. With the help of a properly chosen barrier function, we discuss error analysis and prove that the proposed method converges with almost first-order. The proposed scheme is also applied to a semilinear fractional differential equation. Numerical experiments are presented to validate the proposed method.

کلیدواژه ها:

Mixed-fractional derivative ، fractional differential equation ، Fractional-convection term ، L۱ method ، Discrete Comparison Principle ، Stability ، Convergence