A New Numerical Approximation of Fractional Differentiation: Upwind Discretization for Riemann-Liouville and Caputo Derivatives
A new numerical scheme for fractional differentiation is developed in this paper. The powerful numerical scheme known as upwind is used to establish a numerical approximation for the Riemann-Liouville and Caputo fractional operators. Using the Crank-Nicolson approach and the upwind first-order and second-order approximation, a new numerical scheme is developed. The new numerical approximation is then used to approximate for 0 < α < 1 and 1 < α < 2 the Riemann-Liouville fractional derivative. A detailed numerical analysis to prove the convergence and accuracy of the new numerical scheme is presented. The new approximation is then applied to solve numerically the advection equation. This new numerical scheme will be very useful to solving fractional differential equations.
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