Matrix approach to discrete fractional calculus II: partial fractional differential equations (0811.1355v4)

Abstract: 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 (Fractional Calculus and Applied Analysis, vol. 3, no. 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.

• The paper introduces a novel discretization framework leveraging triangular strip matrices and Kronecker products to model fractional PDEs.
• It validates the approach with numerical examples covering classical, time-fractional, spatial, and delayed fractional diffusion equations.
• The method unifies integer and fractional derivative handling, promising enhanced accuracy and efficiency in modeling complex transport dynamics.

Overview of the Matrix Approach to Discrete Fractional Calculus for Partial Fractional Differential Equations

The paper "Matrix Approach to Discrete Fractional Calculus II: Partial Fractional Differential Equations" presents a novel methodology for the discretization of partial differential equations (PDEs) with fractional derivatives of arbitrary real order and delays. This method extends Podlubny's previous work on the matrix approach to fractional calculus and offers a framework for capturing complex transport dynamics deviating from classical models. The authors discuss the application of this method through the numerical solutions of various fractional diffusion equations, incorporating integer and fractional derivatives with respect to time and space.

Methodology

The foundation of this approach lies in utilizing triangular strip matrices and Kronecker products for the discretization of differential operators of arbitrary real order. By considering the entire time interval simultaneously, rather than incrementally progressing through time steps, this method establishes a comprehensive 2D grid of discretization nodes. The discretization process defines algebraic equations across all nodes, with known boundary values anchoring the system.

Key components include:

1. Triangular Strip Matrices: These matrices facilitate the representation of discretized fractional operators.
2. Kronecker Product: Used to extend the discretization to partial derivatives in multi-dimensional cases.
3. Eliminators and Shifters: Tools for efficiently implementing boundary conditions and handling delays.

This approach accommodates fractional PDEs' integro-differential structure, allowing a unified numerical framework applicable to integer and fractional differential equations.

Numerical Examples

The paper demonstrates the method's robustness through five illustrative examples:

1. Classical Diffusion Equation: Confirms the method's accuracy by aligning results with established analytical and numerical solutions for integer-order equations.
2. Time-Fractional Diffusion Equation: Utilizes the Caputo fractional derivative to demonstrate the approach's efficacy and validates results against alternative numeric techniques.
3. Spatial Fractional Derivative Equation: Accounts for fractional kinetics in space, showcasing the versatility afforded by the matrix methodology.
4. General Fractional Diffusion Equation: Solves equations with fractional derivatives in both time and space, exemplifying the method's comprehensive applicability.
5. Fractional Equation with Delay: Incorporates time delays into the fractional calculus framework, underscoring the approach's adaptability to complex systems.

Implications and Future Directions

This work presents a significant step towards a unified computational method for fractional differential equations, addressing the pressing need for numerical solutions in non-integer order modeling. The matrix approach simplifies handling diverse forms of fractional kinetic equations, facilitating progress in fields such as anomalous diffusion and complex transport phenomena. Future research could focus on extending these methods to nonlinear cases and multidimensional spatial variables, potentially integrating distributed and variable fractional orders.

By providing a resource-efficient and adaptable toolkit, this matrix methodology stands to impact how fractional dynamics are computed, offering improvements in accuracy and computational feasibility. This approach's flexibility in handling delays and its extension to PDEs suggest promising applications across scientific and engineering domains, particularly in scenarios involving complex physical systems exhibiting non-classical behavior.