Overview

Overview#

Unit Title:

Computational Methods in Ordinary Differential Equations

  • Runge-Kutta Methods and Linear Algebra – Jon Shiach

  • Linear Multistep Methods and Sparse Matrices – Zhihua Ma

What will we learn?#

Our main focus is to learn and apply a range of numerical methods to solve a single or a system of First-Order Ordinary Differential Equation(s) given as

  • A single equation: \(\displaystyle \diff{y}{t}=f(t, y)\)

  • A system of equations: \(\displaystyle \diff{\mathbf{Y}}{t}=\mathbf{F}(t, \mathbf{Y})\)

We will learn to derive linear multistep methods and analyse their properties including:

  • accuracy,

  • consistency,

  • stability,

  • convergency.

We will also touch on graph theory and sparse matrix computation to:

  • Form adjacency matrix for a graph,

  • Reorder a graph to reduce the bandwidth and fill-ins of its adjacency matrix.

Why are we learning about Ordinary Differential Equations?#

Ordinary differential equations have wide applications in science and engineering:

  • Physics

    • Mechanics

    • Electrical Engineering

  • Biology

    • Population Dynamics (Level 5: Numerical Methods and Modelling)

    • Disease Dynamics

  • Chemical Kinetics

  • Economics

  • Finance

  • Life Insurance

Cultivating skills in dealing with Ordinary Differential Equations opens a door for mathematicians to solve real problems in lots of areas.

Why are we learning Computational Methods?#

In general there are two ways to solve ordinary differential equations:

  • analytical methods

  • computational methods

Analytical methods only work for a very limited number of simple cases, for most of the realistic complex problems we need to solve them numerically.

Learning Objectives#

  • Apply a range of practical computational methods in ODEs and determine their limitations and advantages through stability properties of the methods.

  • Apply a range of matrix computational methods and perform re-ordering of matrices for the treatment of large sparse matrices.

  • Use a range of mathematical software and numerical algorithms.

  • Solve IVPs, BVPs and linear systems of equations, plot results, perform comparison analysis and determine the most suitable methods for the solution of a given problem. Prepare an academic report.

  • Model and solve real-world systems of linear and ordinary differential equations.

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