Matrix approach to discrete fractional calculus II: Partial fractional differential equations

By Igor Podlubny, Aleksei V. Chechkin, Tomas Skovranek, YangQuan Chen, Blas M. Vinagre. Journal of Computational Physics, 2008

458 citations recorded by the source corpus.

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359-386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/d030913cad3eb9893b8f068e35516dfd405c5fd2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. YangQuan Chen
2. Changpin Li
3. Mingrong Cui
4. Dingyu Xue
5. Fanhai Zeng
6. Qiang Du
7. Max D. Gunzburger
8. Kun Zhou

In Modeling and Simulation

1. Changpin Li
2. Ivo Petrás
3. YangQuan Chen
4. Fanhai Zeng
5. Dingyu Xue
6. Qiang Du
7. Mingrong Cui
8. Max D. Gunzburger

In Numerical Analysis

1. Changpin Li
2. Ivo Petrás
3. Fanhai Zeng
4. YangQuan Chen
5. Dingyu Xue
6. Mingrong Cui
7. R. Lehoucq
8. Qiang Du

Citations in the corpus

• Compact finite difference method for the fractional diffusion equation (2009, methodology): “The fractional heat equations were discussed in [11] and matrix approach to discrete fractional calculus were given in [12,13].”
• Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints (2012, methodology): “This limitation is apparent when developing numerical methods for fractional partial differential equations; see [45] for recent work including many citations to the literature.”
• The Finite Difference Methods for Fractional Ordinary Differential Equations (2013, methodology): “in [15] proposed the matrix approach method for a wide class of FDEs.”
• Fractional order control - A tutorial (2009, methodology): “Last but not least we should mention the approach proposed by Hwang, which is based on B-splines function [29] and Podlubny’s matrix approach [78], [81].”
• High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II) (2015, methodology): “And for fractional derivatives, the Riemann–Liouville derivative and Caputo derivative are often adopted [2, 4, 12, 13, 16].”