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

Differential equations involving derivatives of non-integer order have shown to be adequate models for various physical phenomena in areas like damping laws, diffusion processes, etc. A small number of algorithms for the numerical solution of these equations has been suggested, but mainly without any error estimates. In this paper, we propose an implicit algorithm for the approximate solution of an important class of these equations. The algorithm is based on a quadrature formula approach. Error estimates and numerical examples are given.

/pdf/an-algorithm-for-the-numerical-solution-of-differential-4u6ey2q2k5.pdf

An algorithm for the numerical solution of differential equations of fractional order

The authors have recently developed a mathematical model for the description of the behavior of viscoplastic materials. The model is based on a nonlinear differential equation of order β, where β is a material constant typically in the range 0 < β < 1. This equation is coupled with a first-order differential equation. In the present paper, we introduce and discuss a numerical scheme for the numerical solution of these equations. The algorithm is based on a PECE-type approach.

On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity

Multi-order fractional differential equations and their numerical solution

This chapter is devoted to a brief summary of the most important properties of Mittag-Leffler functions. These functions play a fundamental role in many questions related to fractional differential equations, and they will be used frequently in the later chapters.

Mittag-Leffler Functions

We propose a new mathematical model for the simulation of the dynamics of a dengue fever outbreak. Our model differs from the classical model in that it involves nonlinear differential equations of fractional, not integer, order. Using statistics from the 2009 outbreak of the disease in the Cape Verde islands, we demonstrate that our model is capable of providing numerical results that agree very well with the real data.

Kai Diethelm

Papers

An algorithm for the numerical solution of differential equations of fractional order

On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity

Multi-order fractional differential equations and their numerical solution

Mittag-Leffler Functions

A fractional calculus based model for the simulation of an outbreak of dengue fever