MATU9LB - Modelling with Differential Equations

SCQF Level: 10
Course Prerequisite: MATU9M2
Credit Value: 20 (1 module)

Aims

The primary objective of the module is to explore both the qualitative and quantitative aspects of linear and nonlinear ordinary and partial differential equations. Model-building skills will be emphasised with the use of appropriate software packages.

Learning Outcomes

At the end of this module the student should be able to:

• solve first order ordinary differential equations and apply the techniques to analyse mathematical models, solve linear differential equations with sinusoidal forcing and to find resonant frequencies and amplitudes;
• find and classify equilibria of nonlinear differential equations, draw local phase portraits, and extend these to global phase portraits in elementary cases;
• analyse the bifurcations of families of ordinary differential equations;
• obain solutions to the diffusion, heat and wave equations in Cartesian coordinates in rectangular domains;
• obtain Fourier series and apply them to boundary value problems.

Content

A1: Linear differential equations, forced damped oscillator, resonance.
A2: Linear systems, classification of critical point, global behaviour.
A3: Nonlinear systems, linearization, predator-prey and competition models.
B1: Partial differential equations, classification of second order equations.
B2: The three classical partial differential equations: diffusion, Laplace and wave equations. Solution through separation of variables in Cartesian coordinates.
B3: Fourier series and solution of boundary value and initial value problems for partial differential equations.

Transferable Skills

Formulation of practical problems, investigative skills, experience with various software packages.

Bibliography

G. ZILL and M. R. CULLEN "Differential equations with boundary-value problems", 4e, ITP 1997, ISBN 0-534-95580-0.
D.K. ARROWSMITH, C.M. PLACE "Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour", Chapman & Hall, 1991, ISBN 0412 090809

Teaching Format

3 one-hour lectures and one 1.5 hour tutorial per week.

Assessment

1/3 coursework (2 class tests) and 2/3 examination.

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