Higher order numerical methods for solving fractional differential equations

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

“…The L1 scheme may be obtained by the direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [25], [24], [16], [26], [37], or by the approximation of the Hadamard finite-part integral, e.g., [9], [10], [13], [14], [15], [23], [41]. Since its first appearance the L1 scheme has been extensively used in practice and currently it is one of the most popular and successful numerical methods for solving the time fractional diffusion equation.…”

Error estimates of a high order numerical method for solving linear fractional differential equations

“…We then use the similar idea to prove the error estimates of a high order numerical method for solving linear fractional differential equations proposed in Yan et al [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O(∆t 3−α ), 0 < α < 1. The numerical examples are given to show that the numerical results are consistent with the theoretical results.…”

“…Ford et al [15] used the similar numerical method as in Diethelm [6] to consider the time discretization of (6)- (7) and proved that the convergence order of the time discretization scheme is O(∆t 2−α ). In [37], Yan et al introduced a higher order numerical method for solving (1)- (2) by approximating the Hadamard finite-part integral with second-degree compound quadrature formula and proved that the error has the assympototic expansion as in [8]. However the authors in [37] can not prove the error estimates of the higher order numerical method by using the argument in Diethelm [6].…”

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…The L1 scheme first appeared in the book [41] for the approximation of the Caputo fractional derivative. The L1 scheme may be obtained by direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [28], [26], [18], [29], [47], [19], or by the approximation of the Hadamard finite-part integral, e.g., [9], [11], [15], [16], [17], [24], [51].…”