Fractional calculus

Summary

Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D D f(x) = \frac{d}{dx} f(x),, and of the integration operator J J f(x) = \int_0^x f(s) ,ds,, and developing a calculus for such operators generalizing the classical one. In this context, the term powers refers to iterative application of a linear operator D to a function f, that is, repeatedly composing D with itself, as in D^n(f) = (\underbrace{D\circ D\circ D\circ\cdots \circ D}_n)(f) = \underbrace{D(D(D(\cdots D}_n (f)\cdots))). For example, one may ask for a meaningful interpretation of \sqrt{D} = D^{\scriptstyle{\frac12}} as an analogue of the functional square root for the differentiation operator, that is, an expres

Official source

https://en.wikipedia.org/wiki/Fractional_calculus

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (62)

Related publications

Related people (7)

Related people

Related units (6)

Related units

Related concepts (3)

Related concepts

Related courses (11)

Related courses

Related lectures (15)

Related lectures

Diffusion Process Modeling by Using Fractional-Order Models

Tomas Skovranek

This paper deals with a concept and description of a RC network as an electro-analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions. It is based on well-known analogy between heat and electrical conduction. In the paper are compared analytical solution together with numerical solution and experimentally measured data. For the first time a fractional order model of diffusion process and its modeling via lumped RC network has been used. Simple examples of simulations, measurements and their comparison are shown.

Elsevier2015

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

Daniel Henri Baffet, Jan Sickmann Hesthaven

High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel reduction method for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We present numerical results obtained with both implicit and the explicit methods applied to different problems.

Springer Verlag2017

The nonlocal nature of the fractional integral makes the numerical treatment of fractional dierential equations expensive in terms of computational eort and memory requirements. In this paper we propose a method to reduce these costs while controlling the accuracy of the scheme. This is achieved by splitting the fractional integral of a function f into a local term and a history term. Observing that the history term is a convolution of the history of f and a regular kernel, we derive a multipole approximation to the Laplace transform of the kernel. This enables the history term to be replaced by a linear combination of auxiliary variables dened as solutions to standard ordinary dierential equations. We derive a priori error estimates, uniform in f, and obtain estimates on the number of auxiliary variables required to satisfy an error tolerance. The resulting formulation is discretized to produce a time stepping method. The method is applied to some test cases to illustrate the performance of the scheme.

Society for Industrial and Applied Mathematics2017