Algorithms for the fractional calculus: A selection of numerical methods

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Luchko, Yury

doi:10.1016/j.cma.2004.06.006

Cited by 573 publications

(331 citation statements)

References 31 publications

Supporting

Mentioning

314

Contrasting

Unclassified

Order By: Relevance

“…Many numerical algorithms for initial value problems of the form (1.1) have been proposed in recent years; see, e.g., [1,4,6,7,8,21,22] and the references cited therein. We have chosen one of these methods as a prototype for our parallelization project, namely the fractional AdamsBashforth-Moulton scheme of [6].…”

Section: The Basic Numerical Methodsmentioning

confidence: 99%

An efficient parallel algorithm for the numerical solution of fractional differential equations

Diethelm

2011

fcaa

242

300

View full text Add to dashboard Cite

The numerical solution of differential equations of fractional order is known to be a computationally very expensive problem due to the nonlocal nature of the fractional differential operators. We demonstrate that parallelization may be used to overcome these difficulties. To this end we propose to implement the fractional version of the second-order AdamsBashforth-Moulton method on a parallel computer. According to many recent publications, this algorithm has been successfully applied to a large number of fractional differential equations arising from a variety of application areas. The precise nature of the parallelization concept is discussed in detail and some examples are given to show the viability of our approach.MSC 2010 : Primary 65Y05; Secondary 65L05, 65R20

show abstract

Section: The Basic Numerical Methodsmentioning

confidence: 99%

An efficient parallel algorithm for the numerical solution of fractional differential equations

Diethelm

2011

fcaa

242

300

View full text Add to dashboard Cite

show abstract

“…The old and ubiquitous requirement for physical interpretation of such initial conditions was most clearly formulated recently by Diethelm et al (2005): "A typical feature of differential equations (both classical and fractional) is the need to specify additional conditions in order to produce a unique solution. For the case of Caputo FDEs, these additional conditions are just the static initial conditions .…”

Section: Introductionmentioning

confidence: 99%

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

2005

View full text Add to dashboard Cite

show abstract

“…ds n ds (t − s) α−n+1 , assuming that it is convergent (see [19,30,50,69,78] for more details). Based on the discussion in [72], the Caputo derivative is frequently used for the derivative with respect to time.…”

Section: The Fractional Derivativementioning

confidence: 99%

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Dolgov

Pearson

Savostyanov

et al. 2016

Applied Mathematics and Computation

View full text Add to dashboard Cite

Fractional differential equations have recently received much attention within computational mathematics and applied science, and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms. An optimization problem with constraints given by fractional differential equations is considered, which in its discretized form leads to a high-dimensional tensor equation. To reduce the computation time and storage, the solution is sought in the tensor-train format. We compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned Krylov solvers or tailored alternating schemes. The competitiveness of these approaches is presented using several examples with constant and variable coefficients.

show abstract

Algorithms for the fractional calculus: A selection of numerical methods

Cited by 573 publications

References 31 publications

An efficient parallel algorithm for the numerical solution of fractional differential equations

An efficient parallel algorithm for the numerical solution of fractional differential equations

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Contact Info

Product

Resources

About