MAS222 Differential Equations

Both semesters, 2017/18 20 Credits
Lecturer: Prof Elizabeth Winstanley uses MOLE Timetable Reading List
Aims Outcomes Teaching Methods Assessment Full Syllabus

The module aims at developing a core set of advanced mathematical techniques essential to the study of applied mathematics. Topics include the qualitative analysis of ordinary differential equations, solutions of second order linear ordinary differential equations with variable coefficients, first order and second order partial differential equations, the method of characteristics and the method of separation of variables.

Prerequisites: MAS110 (Mathematics Core I); MAS111 (Mathematics Core II)
Corequisites: MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:
MAS212Scientific Computing and Simulation
MAS280Mechanics and Fluids
MAS316Mathematical modelling of natural systems
MAS320Fluid Mechanics I
MAS377Mathematical Biology
MAS414Mathematical Modelling of Natural Systems
MAS422Magnetohydrodynamics


Outline syllabus

  • First order ordinary differential equations.
  • Planar first order autonomous systems, linearisation of nonlinear planar systems.
  • Stability of equilibrium points.
  • Second order linear ordinary differential equations, power series solutions, ordinary and singular points.
  • Sturm-Liouville problems.
  • Second order partial differential equations: wave equation, heat equation, Laplace's equation, separation of variables.
  • First order partial differential equations, method of characteristics.
  • Method of characteristics for second order hyperbolic partial differential equations.

Office hours

In semester 2 office hours will operate on an appointment basis, with appointments booked on google calendar. The times will vary from week to week. Students will be notified by e-mail of the times of office hours and a link for booking appointments.



Aims

  • To learn the qualitative analysis of ordinary differential equations
  • To learn how to solve second order oridnary differential equations with variable coefficients
  • To learn how to solve first order partial differential equations using the method of characteristics
  • To learn how to solve second order partial differential equations using the method of separation of variables
  • To learn the properties of the solutions of the classical partial differential equations

Learning outcomes

  • To be able to characterise systems of ordinary differential equation qualitatively.
  • To be able to solve second order ordinary differential equations with variable coefficients using various techniques
  • To be able to solve first order partial differential equations using the method of characteristics
  • To be able to solve second order partial differential equations using the method of separation of variables and the method of characterisics

Teaching methods

Lectures, tutorials, problem solving


40 lectures, 10 tutorials

Assessment

One 1 hour exam on computer at the end of Semester 1 (10%)
One 2.5 hour written examination at the end of Semester 2 (90%)

Full syllabus

Semester 1

  • Revision of ordinary differential equations (ODEs)
  • Qualitative analysis of first order ODEs: direction fields, autonomous equations, equilibrium points, phase lines.
  • Planar first order autonomous systems: equilibrium points, trajectories, nullclines, classification of equilibrium points for linear systems, phase portraits in the neighborhood the equilibrium points
  • Linearisation of nonlinear planar system
  • Stability of equlibrium points: linear systems, effects of nonlinear terms
  • Phase portraits of planar first order autonomous systems
  • Second order linear ordinary differential equations with variable coefficients: boundary value problems, normal form, reduction of order
  • Power series solution and Frobenius series solution: ordinary points and singular points, Hermites' equation, Airy equation, Bessel's equation, Legendre equation, and Laguerre's equation (as examples or tutorial questions)
Semester 2
  • Introduction and basic definitions
  • Separation of variables for homogeneous problems: the heat equation, wave equation, Laplace's equation
  • Separation of variables for inhomogeneous problems: the heat equation, wave equation Laplace's equation
  • Method of characteristics for first order PDEs
  • Method of characteristics for second order hyperbolic PDEs
  • D'Alembert's solution of the one-dimensional wave equation.

Reading list

Type Author(s) Title Library Blackwells Amazon
B Boyce and Diprima Elementary Differential Equations and Boundary Value Problems Blackwells Amazon
B D. W. Trim Applied Partial Differential Equations Blackwells Amazon
B King, Billngham, and Otto Differential Equations: Linear, Nonlinear, Ordinary, Partial Blackwells Amazon
B Simmons Differential Equations with Applications and Historical Notes Blackwells Amazon
C A. Jeffrey Applied Partial Differential Equations: An Introduction Blackwells Amazon
C Logan Applied Partial Differential Equations Blackwells Amazon
C Steven Stogatz Nonlinear dynamics and chaos Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

Timetable (semester 2)

Tue 16:00 - 16:50 lecture   Hicks Lecture Theatre 1
Thu 13:00 - 13:50 lecture   Hicks Lecture Theatre 1
Thu 15:00 - 15:50 tutorial (group 3b) (even weeks) Hicks Lecture Theatre 4
Thu 16:00 - 16:50 tutorial (group 1b) (even weeks) Hicks Lecture Theatre 4
Thu 16:00 - 16:50 tutorial (group 4b) (even weeks) Hicks Lecture Theatre 10
Fri 09:00 - 09:50 tutorial (group 2b) (even weeks) Hicks Seminar Room F20
Fri 09:00 - 09:50 tutorial (group 5b) (even weeks) Hicks Lecture Theatre 9