A New Numerical Approximation of Fractional Differentiation: Upwind Discretization for Riemann-Liouville and Caputo Derivatives

  • Abdon Atangana
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 24)


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.


Caputo Fractional Derivative Crank-Nicolson Approach Upwind Numerical Scheme Riemann Liouville Derivative Second-order Upwind 
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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Abdon Atangana
    • 1
  1. 1.Faculty of Natural and Agricultural Sciences, Institute for Groundwater StudiesUniversity of the Free StateBloemfonteinSouth Africa

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