Applied Nonlinear Partial Differential Equations

Lecturers

Professor Philip Broadbridge, La Trobe University
Dr Dimetre Triadis, La Trobe University

Synopsis

Partial differential equations (PDEs) model an enormous variety of continuum dynamic processes, including fluid dynamics, elasticity, solute and heat diffusion, subterranean hydrology, population dynamics, electromagnetic fields and gravity. Nonlinearity (or dependence of PDE coeffcients on dependent variables) is essential to explain some familiar phenomena such as thermal ignition and wave breaking. Some helpful techniques will be applied to develop a conceptual understanding of nonlinear waves and nonlinear diffusive processes, with minimal theory of function spaces. These techniques will include the method of characteristics, asymptotic approximations, symmetry reduction and integrable models.

Course Overview

Part A (nonlinear waves; 11 lectures and 2 problem-solving sessions)

  • The theory of characteristics for first-order PDEs.
  • Weak solutions for first-order PDEs in conservation form.
  • Hyperbolic, parabolic and elliptic second-order PDEs.

Part B (nonlinear diffusion; 11 lectures and 2 problem-solving sessions)

  • Role of Burgers’ equation in gas dynamics and soil-water flow.
  • Initial-boundary value problems with Burgers’ equation.
  • Nonlinear heat conductivity and nonlinear diffusivity.
  • Initial-boundary problems with integrable nonlinear diffusion models.
  • Reaction-diffusion equations with blow-up. Similarity reductions.
  • Reaction-diffusion equations with logistic source. Stable travelling waves.
  • Theory of Lie symmetry reductions.

Prerequisites

  • Any one-semester subject on vector calculus.
  • Any one-semester subject that introduces linear partial differential equations.
  • Qualification for enrolment in honours applied mathematics or honours mathematics or honours theoretical physics.

Assessment

  • Best 3 of 4 assignments 15% each.
  • Examination 55%.

Resources/pre-reading (if available)

  • Course notes will be provided.
  • Reference works that may be useful:
    • Paul Garabedian, Partial Di fferential Equations, 1964
    • Fritz John, Partial Di fferential Equations, Springer, circa 1978.
    • J. Kevorkian, Partial di fferential equations: analytical solution techniques, 2nd. ed., Springer, 2000.
    • G. R. Fulford and P. Broadbridge, Industrial Mathematics, Cambridge Univ.Press, 2001.

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Professor Philip Broadbridge,
La Trobe University

Phil Broadbridge was educated at University of Adelaide and University of Tasmania. He has previously worked as a high school teacher and as a CSIRO research scientist. He served as a professor for 28 years at Wollongong, Delaware, AMSI/Melbourne and La Trobe. His research has mostly been on various applications of partial differential equations in physics, hydrology, heat conduction, materials science and biology. He now has honorary professor positions at Wollongong, Kyushu and La Trobe.

Dr Dimetre Triadis,
La Trobe University

Dimetre is a research fellow employed jointly at the Department of Mathematics and Statistics, La Trobe University, and the Institute of Mathematics for Industry, Kyushu University, Japan. His research is focused on exact solutions for nonlinear partial differential equations occurring in elastic contact problems, soil water infiltration, nonlinear heat conduction, and other applications.