A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation - Maria Carmela De Bonis; Donatella Occorsio (Axioms)
In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order <semantics>α</semantics><semantics>(Dαf)(y)=<mstyle>1Γ(m−α)</mstyle><msubsup>∫0y</msubsup>(y−x)m−α−1f(m)(x)dx,<mspace width="1.em"></mspace>y>0</semantics>, with <semantics>m−1<α≤m,<mspace width="0.166667em"></mspace>m∈N.</semantics> The numerical procedure is based on approximating <semantics>f(m)</semantics> by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order <semantics>α</semantics>. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.