Monash University Handbook

The mathematical modelling of physical systems is based upon differential equations and linear algebra. This unit will introduce fundamental techniques for studying linear systems and differential equations, focusing on applications to physical systems. The topics in linear algebra to be considered include: eigenvalues and eigenvectors, diagonalisation of square matrices, matrix functions, LU-decomposition, applications. The topics in optimisation include: Lagrange multipliers, the method of least-squares, linear programming, applications. Finally, the topics in differential equations include: matrix solutions of constant coefficient systems of ordinary differential equations, conservative systems, hase-planes of simple non-linear ODEs, solution of first order partial differential equations by the Method of Characteristicds applications. Students will be introduced to the Mathematica computer package, and learn how to use it for analytic and numerical calculations and graphics. It will be integrated into most activities.

Outcomes

On completion of this unit students will be able to:

Apply differential equations and linear algebra to the modelling of real-world systems;
Solve linear systems and calculate the eigenvalues and eigenvectors of square matrices;
Calculate the solution of difference and differential equations using matrix functions;
Apply optimisation techniques to the solution of real-world problems;
Solve constant coefficient ordinary differential equations;
Apply the Method of Characteristics to the solution to first order PDEs;
Understand the phase-planes for second-order differential equations describing oscillating systems and interacting populations.
Use the Mathematica software package for the solution and presentation of mathematical problems.
Present clear mathematical arguments in both written and oral forms.