differint 1.0.0

Collection of algorithms for numerically calculating fractional derivatives.

differint

This package is used for numerically calculating fractional derivatives and integrals (differintegrals). Options for varying definitions of the differintegral are available, including the Grunwald-Letnikov (GL), the 'improved' Grunwald-Letnikov (GLI), the Riemann-Liouville (RL), and the Caputo (L1, L2, and L2C). Through the API, you can compute differintegrals at a point or over an array of function values.

See below for an example of how to use this package, or check out the wiki for references, signatures, and examples for each function.

Motivation

There is little in the way of readily available, easy-to-use code for numerical fractional calculus. What is currently available are functions that are generally either smart parts of a much larger package, or only offer one numerical algorithm. The differint package offers a variety of algorithms for computing differintegrals and several auxiliary functions relating to generalized binomial coefficients.

Installation

This project requires Python 3+ and NumPy to run.

Installation from the Python Packaging index (https://pypi.python.org/pypi) is simple using pip.

pip install differint

Included Files

Core File	Description
differint/differint.py	Contains algorithms for fractional differentiation and integration.
tests/test.py	Testing suite containing all unit tests.

Both of the above files have corresponding __init__.py files.

Setup File	Description
.gitignore	List of files to ignore during git push/pull requests.
CONTRIBUTING.md	Instructions for potential contributors to the differint project.
LICENSE	MIT license agreement.
MANIFEST.in	Selects the README file for uploading to PyPi.
README.md	This README file.
README.rst	This README file in ReStructuredText format.
init.py	__init__ file for overall package.
changelog.txt	List of updates to package.
setup.py	Script for downloading package from pip.

Example Usage

Taking a fractional derivative is easy with the differint package. Let's take the 1/2 derivative of the square root function on the interval [0,1], using the Riemann-Liouville definition of the fractional derivative.

import numpy as np
import differint.differint as df

def f(x):
   return x**0.5

DF = df.RL(0.5, f)
print(DF)

You can also specify the endpoints of the domain and the number of points used as follows.

DF = df.RL(0.5, f, 0, 1, 128)

For a description of all functions, their signatures, and more usage examples, see the project's wiki.

Tests

All tests can be run with nose from the command line. Setup will automatically install nose if it is not present on your machine.

python setup.py tests

Alternatively, you can run the test script directly.

cd <file_path>/differint/tests/
python test.py

API Reference

In this section we cover the usage of the various functions within the differint package.

Main Function	Usage
GLpoint	Computes the GL differintegral at a point
GL	Computes the GL differintegral over an entire array of function values using the Fast Fourier Transform
GLI	Computes the improved GL differintegral over an entire array of function values
CRONE	Calculates the GL derivative approximation using the CRONE operator.
RLpoint	Computes the RL differintegral at a point
RL	Computes the RL differintegral over an entire array of function values using matrix methods
CaputoL1point	Computes the Caputo differintegral at a point using the L1 algorithm
CaputoL2point	Computes the Caputo differintegral at a point using the L2 algorithm
CaputoL2Cpoint	Computes the Caputo differintegral at a point using the L2C algorithm
PCsolver	Solves IVPs for fractional ODEs of the form ${}^CD^\alpha[y(x)]=f(x,y(x))$ using the predictor-corrector method

Auxiliary Function	Usage
isInteger	Determine if a number is an integer
isPositiveInteger	Determine if a number is an integer, and if it is greater than 0
checkValues	Used to check for valid algorithm input types
GLIinterpolat	Define interpolating coefficients for the improved GL algorithm
functionCheck	Determines if algorithm function input is callable or an array of numbers
poch	Computes the Pochhammer symbol
Gamma	Computes the gamma function, an extension of the factorial to complex numbers
Beta	Computes the beta function, a function related to the binomial coefficient
MittagLeffler	Computes the two parameter Mittag-Leffler function, which is important in the solution of fractional ODEs
GLcoeffs	Determines the convolution filter composed of generalized binomial coefficients used in the GL algorithm
RLcoeffs	Calculates the coefficients used in the RLpoint and RL algorithms
RLmatrix	Determines the matrix used in the RL algorithm
PCcoeffs	Determines the coefficients used in the PC algorithm

Contribute

To contribute to this project, see the contributing guidelines.

Credits

Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods. World Scientific.

Oldham, K.B. & Spanier, J. (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press Inc.

Karniadakis, G.E.. (2019). Handbook of Fractional Calculus with Applications Volume 3: Numerical Methods. De Gruyter.

License

MIT © Matthew Adams