## ANOMOLOUS DIFFUSION

The diffusion of heat, fluids, particles, and even people, can be modelled by a mathematical equation called the diffusion equation. This equation is also a key building block of more complicated models of, e.g., the weather, and oceans.

Diffusion equations aim to capture the large-scale effect of a large population particles moving randomly. In the standard equation, particles are assumed to undergo Brownian motion, that is, the mean square displacement of a single particle in a given time, t, is linearly proportional to t.

However, more recent empirical studies have shown that many diffusion processes do not satisfy such a scaling law. Instead, the mean-square displacement scales according to tᵖ. These are known as anomalous diffusion processes, that are modelled by fractional diffusion equations.

## FRACTIONAL DERIVATIVES

While derivatives of integer order have standard definitions, there are many different definitions of fractional derivatives; these are not equivalent to each other in general. In this work we largely consider Riemann–Liouville fractional derivatives.

One thing that fractional diffusion operators have in common is that they are non-local, that is, small changes at one point in space affect the solution at points very far away. This makes their solution very challenging.

## SOLVING FRACTIONAL DIFFUSION EQUATIONS

As for standard diffusion equations, most fractional diffusion problems cannot be solved analytically. That is, we cannot somehow manipulate the fractional diffusion equation to extract the solution.

Instead, numerical methods are employed to approximate the solution; these include finite difference, finite volume and finite element schemes. There are many different numerical methods for fractional diffusion equations, but most require the solution of a (large!) set of simultaneous equations. Because fractional diffusion operators are nonlocal, these simultaneous equations are strongly coupled, that is, each equation involves many of the unknowns.

Such systems of simultaneous equations, also known as linear systems, are challenging to solve. This project looks at exploiting certain patterns in the equations in order to solve them efficiently.

## APPLICATIONS

Anomolous diffusion is observed in an increasingly long list of applications. These include: