Fractional Differential Equations

https://www.chester.ac.uk/sites/files/chester/rep377.pdf

Analysis of Fractional Differential Equations

In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf

A new Definition of Fractional Derivative without Singular Kernel

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

https://ntrs.nasa.gov/api/citations/20020024453/downloads/20020024453.pdf

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

Finite difference/spectral approximations for the time-fractional diffusion equation

Fundamentals of Fractional Calculus.- Riemann-Liouville Differential and Integral Operators.- Caputo's Approach.- Mittag-Leffler Functions.- Theory of Fractional Differential Equations.- Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations.- Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results.- Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases.- Multi-Term Caputo Fractional Differential Equations.

The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type

We investigate a method for the numerical solution of the nonlinear fractional differential equation D
*
α
y(t)=f(t,y(t)), equipped with initial conditions y
(k)(0)=y
0
(k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.

/pdf/detailed-error-analysis-for-a-fractional-adams-method-2odctuwibd.pdf

Detailed error analysis for a fractional Adams method

https://ntrs.nasa.gov/api/citations/20030112277/downloads/20030112277.pdf

Kai Diethelm

Papers

Analysis of Fractional Differential Equations

The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

Detailed error analysis for a fractional Adams method

Algorithms for the fractional calculus: A selection of numerical methods