Numerical Solutions of Differential Equations

Author

Waleed A A Ali

Introduction

This unit will cover some of the numerical techniques used for solving differential equations and using MATLAB to implement these numerical methods.

Material

All the material will be posted on the Microsoft Teams Page for the unit. Note that is document is regularly being updated so if you find any mistakes or parts missing then do let me know.

Unit Outline

All lectures will be held online on Fridays, 10.00 - 12.00 on Microsoft Teams, to which you should have received a link. The details for the link are as follows:

Meeting ID: 358 213 646 365 0

Passcode: Us2N6hB2

Lecture Date Topic
1 17/10
  • Introduction to NSDE unit for TCC
  • Aims & Objectives of the unit
  • Floating point arithmetic
  • Computational complexity
  • Code timing and profiling
  • Applications for solving linear systems
  • Computational stability
  • Solving linear systems using direct methods
    • Direct substitution
    • Forward/Backward substitution
2 24/10
  • Solving linear systems using direct methods
    • Tridiagonal matrix algorithm
    • Cramer’s Rule
  • Solving linear systems using iterative methods
3 31/10
  • Euler method for IVPs
4 07/11
  • Modified Euler method
  • Runge-Kutta method
  • Backwards Euler method
  • solving stiff IVPs
5 14/11
  • Solving BVPs using the finite difference method
6 21/11
  • Solving MVPs and symmetric BVPs
7 28/11
  • Method of lines
  • Apply MoL for diffusion and/or convection
8 05/12 Stability of the method of lines

Aims & Objectives

The aim for this unit is to be able to understand and derive different numerical techniques for solving differential equations and being able to implement them on MATLAB.

Intended Learning Outcomes:

  • Understand the internal working mechanisms of MATLAB,
  • Solve linear systems using direct and iterative methods,
  • Use different differencing schemes to assess their ability to solve ODEs and PDEs,
  • Assess the stability of different numerical methods.

Questions

For any questions, queries or issues that you see in the material, do not hesitate to contact me on w.a.a.ali@bath.ac.uk.