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.
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.
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| 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. |
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.
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 testsAlternatively, you can run the test script directly.
cd <file_path>/differint/tests/
python test.pyIn 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 |
| 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 |
To contribute to this project, see the contributing guidelines.
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.
MIT © Matthew Adams
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