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Matrix approach to discretization of ODEs and PDEs of arbitrary real order

version (1.04 MB) by Igor Podlubny
Functions illustrating matrix approach to discretization of ODEs / PDEs with fractional derivatives.


Updated 04 Mar 2016

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This submission contains the basic functions that are necessary for using the matrix approach to discretization of fractional differential equations, and demos.

The method is described in the following articles:
[1] I. Podlubny, "Matrix approach to discrete fractional calculus", Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359-386 ( ).
[2] I. Podlubny, A. Chechkin, T. Skovranek, YQ. Chen, B. M. Vinagre Jara, "Matrix approach to discrete fractional calculus II: partial fractional differential equations", Journal of Computational Physics, vol. 228, no. 8, 1 May 2009, pp. 3137-3153, (preprint: ).
For more information about fractional differential equations (i.e., differential equations containing derivatives of arbitrary real order) see, for example,
[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999, ISBN 0125588402.
Update notes 2008-11-27:

(1) Added a tutorial article (in the form of a "published m-file") with examples. The examples include: evaluation of integer-order derivatives; evaluation of left-sided and right-sided Riemann-Liouville fractional derivatives; evaluation of symmetric fractional derivatives (symmetric Riesz derivatives); solution of a fractional integral equation with Riesz kernel; solution of an ordinary fractional differential equation (the Bagley-Torvik equation); solution of a partial fractional differential equation (fractional diffusion equation); solution of a partial fractional differential equation with delayed fractional derivatives (fractional diffusion equation with delayed fractional derivative).

(2) Added two demo functions (bagleytorvikequation.m and rieszpotential.m)

(3) Updated the title of this submission by adding the words "of arbitrary real order".

Update notes 2008-12-04:

(1) Corrected typos in the description.
(2) Deleted unused files in 'html' directory.
(3) Low quality PNG images of equations in the tutorial, that were generated by Matlab when "publishing to HTML", are replaced with PNG images of good quality obtained using TeX.

Update notes 2009-01-07:

Spelling corrections in the "published m-file" and in descriptions inside the functions.

Update notes 2009-02-05:
Added journal reference.

Update notes 2009-04-24:
Updated function fracdiffdemou.m
(thanks to Dr. Mridula Garg, University of Rajasthan)

Cite As

Igor Podlubny (2021). Matrix approach to discretization of ODEs and PDEs of arbitrary real order (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (11)

Ahmed raafat

thnks for sharing the code
is that fo two dimesnsions too??

kazim atman



ibrahim Avci


Very nice & simple to use. I was wondering if it is possible/how you could apply this tool to non-linear ODEs? Thanks

Peter Kocúr

sk damarla

Yi Sui



Gunvant Birajdar

Verry Nice


Excellent tool for anyone interested in working with the solution of fractional differential equations. The programs are well-documented and easy to follow. The toolkit is accompanied by detailed instructions on how to get started with the tool. It provides numerous examples of its use in solving a variety of problems. Thank you.

MATLAB Release Compatibility
Created with R2016a
Compatible with any release
Platform Compatibility
Windows macOS Linux

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