Modelling with ODEs

Ordinary Differential Equations (ODEs) are a powerful tool for modelling a wide range of physical systems. Unlike purely data-driven models, ODEs are based on the underlying physics, biology, or chemistry of the system being modelled. This makes them particularly useful for predicting the behaviour of a system under conditions that have not been observed. In this section, we will introduce the basics of ODE modelling, and illustrate their use with a series of examples written using the DiffSol crate.

The topics covered in this section are:

• First Order ODEs: First order ODEs are the simplest type of ODE. Any ODE system can be written as a set of first order ODEs, so libraries like DiffSol are designed such that the user provides their equations in this form.
  • Example: Population Dynamics: A simple example of a first order ODE system, modelling the interaction of predator and prey populations.
• Higher Order ODEs: Higher order ODEs are equations that involve derivatives of order greater than one. These can be converted to a system of first order ODEs, which is the form that DiffSol expects.
  • Example: Spring-mass systems: A simple example of a higher order ODE system, modelling the motion of a damped spring-mass system.
• Discrete Events: Discrete events are events that occur at specific times or when the system is in a particular state, rather than continuously. These can be modelled by treating the events as changes in the ODE system's state. DiffSol provides an API to detect and handle these events.
  • Example: Compartmental models of Drug Delivery: Pharmacokinetic models describe how a drug is absorbed, distributed, metabolised, and excreted by the body. They are a common example of systems with discrete events, as the drug is often administered at discrete times.
  • Example: Bouncing Ball: A simple example of a system where the discrete event occurs when the ball hits the ground, instead of at a specific time.
• DAEs via the Mass Matrix: Differential Algebraic Equations (DAEs) are a generalisation of ODEs that include algebraic equations as well as differential equations. DiffSol can solve DAEs by treating them as ODEs with a mass matrix. This section explains how to use the mass matrix to solve DAEs.
  • Example: Electrical Circuits: Electrical circuits are a common example of DAEs, here we will model a simple low-pass LRC filter circuit.
• PDEs: Partial Differential Equations (PDEs) are a generalisation of ODEs that involve derivatives with respect to more than one variable (e.g. a spatial variable). DiffSol can be used to solver PDEs using the method of lines, where the spatial derivatives are discretised to form a system of ODEs.
  • Example: Heat Equation: The heat equation describes how heat diffuses in a domain over time. We will solve the heat equation in a 1D domain with Dirichlet boundary conditions.
  • Example: Physics-based Battery Simulation: A more complex example of a PDE system, modelling the charge and discharge of a lithium-ion battery. For this example we will use the PyBaMM library to form the ODE system, and DiffSol to solve it.