DeepAI

# Stability and Complexity Analyses of Finite Difference Algorithms for the Time-Fractional Diffusion Equation

Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With the advent of increasing computational capacity along with advances in numerical methods, there has been increased interest in using FDEs to represent complex physical processes, where dynamics may not be as accurately captured with classical differential equations. The time-fractional diffusion equation is an FDE that represents the underlying physical mechanism of anomalous diffusion. But finding tractable analytical solutions to FDEs is often much more involved than solving for the solutions of integer order differential equations, and in many cases it is not possible to frame solutions in a closed form expression that can be easily simulated or visually represented. Therefore the development of numerical methods is vital. In previous work we implemented the full 2D time-fractional diffusion equation as a Forward Time Central Space finite difference equation by using the Grünwald-Letnikov definition of the fractional derivative. In addition, we derived an adaptive time step version that improves on calculation speed, with some tradeoff in accuracy. Here, we explore and characterize stability of these algorithms, in order to define bounds on computational parameters that are crucial for performing accurate simulations. We also analyze the time complexity of the algorithms, and describe an alternate adaptive time step approach that utilizes a linked list implementation, which yields better algorithmic efficiency.

01/25/2022

### Substitution Method for Fractional Differential Equations

Numerical solving differential equations with fractional derivatives req...
05/20/2023

### The BEM and DRBEM schemes for the numerical solution of the two-dimensional time-fractional diffusion-wave equations

In this paper we apply the boundary elements method (BEM) and the dual r...
11/23/2020

### A difference method for solving the nonlinear q-factional differential equations on time scale

The q-fractional differential equation usually describe the physics proc...
06/13/2023

### Investigation of Fractional Compartmental Models with Application to Amiodarone Drug Diffusion in Pharmacokinetics

This paper presents three fractional models formulated from a classical ...
10/06/2020

### Efficient and fast predictor-corrector method for solving nonlinear fractional differential equations with non-singular kernel

Efficient and fast predictor-corrector methods are proposed to deal with...
02/06/2023

### Solving fractional Hantavirus model: A new approach

A three equation differential biodiversity model depicting spread and pr...
08/02/2018

### Machine Learning of Space-Fractional Differential Equations

Data-driven discovery of "hidden physics" -- i.e., machine learning of d...