Pitfalls in fast numerical solvers for fractional differential equations

By Kai Diethelm, J. Ford, Neville J. Ford, Marc Weilbeer. 2006

138 citations recorded by the source corpus.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/28228034a1fafc17cc6bb44fbebc8eb8ebd9f724

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Citations in the corpus

The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation (2013, background): “1), so we omit the corresponding details of s and w (β) n,k; see [15, 18, 19, 32] for more detailed information.”
On Multistep Methods for Differential Equations of Fractional Order (2006, methodology): “Usually, near the origin, the numerical approximation is improved by using a second set of weights, the starting weights, chosen in such a way the resulting formula be correct for polynomials of some degree [ 5 ], [9].”
Explicit methods for fractional differential equations and their stability properties (2009, unknown): “, see [2,4,8,10,14]).”
Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial (2018, background): “However, round-off errors may accumulate when solving the ill-conditioned linear systems (21) [38], and hence, it is advisable to avoid very high order methods.”
Numerical solution of multi‐term fractional differential equations (2009, unknown): “Several numerical schemes have been presented to date [1, 8, 10].”
Fractional boundary value problems: Analysis and numerical methods (2011, methodology): “We believe that these results have an easy explanation: it is known that these higher numerical methods should be used with some prudence, because instability may occur due to the cancelation of digits in the linear system for the determination of the so-called starting weights ([3]).”
The Finite Difference Methods for Fractional Ordinary Differential Equations (2013, methodology): “holds for some selected nonpolynomial terms [17, 20, 21].”
Numerical analysis for distributed-order differential equations (2009, methodology): “Step 2 finally will consist of solving the system we have constructed numerically by any of the available methods; one could for example use the Adams method of [16] or the backward differentiation scheme of [9] or a fractional multistep method (see [20, 21, 22, 14]) which is likely to be efficient ”
Nonpolynomial collocation approximation of solutions to fractional differential equations (2013, background): “The theory and numerical analysis of fractional differential equations has also been widely developed in the last decades, see for example [4], [6], [7], [8], [10], [12], [14], [16], [18] and the references therein.”
A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations (2011, unknown): “[7] and Ford et al.”