# NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

MAT4PDE

2018

Credit points: 15

## Subject outline

Many quantities of practical interest in continuum mechanics, environmental mechanics and population ecology, are commonly represented as smooth functions of space and time that are governed by nonlinear partial differential equations. Such equations arise also in differential geometry of smooth manifolds. Some understanding of nonlinear partial differential equations is considered to be essential for any further development of applicable analysis or continuum geometry, with impacts on many other branches of mathematics, physics, chemistry, finance and biology.

Topics covered include (i) local formulation of conservation laws, (ii) classical and weak solutions of hyperbolic, parabolic and elliptic equations, (iii) method of characteristics (iv) viscosity solutions of Burgers' equation, (v) reversible and irreversible processes, (vi) integrable nonlinear diffusion equations, (vii) curvature-driven flows, (viii) reaction-diffusion equations and (ix) solitons.

SchoolSchool Engineering&Mathematical Sciences

Credit points15

Subject year levelYear Level 4 - UG/Hons/1st Yr PG

Exchange StudentsYes

## Subject particulars

### Subject rules

Prerequisites MAT2VCA and 30 credit points of 3rd year Mathematics subjects (MAT3 or STM3)

Co-requisitesN/A

Incompatible subjectsN/A

Equivalent subjectsN/A

Special conditionsN/A

## Graduate capabilities & intended learning outcomes

01. Demonstrate method of characteristics for solving first-order and second-order nonlinear wave equations.

Activities:
Interactive lectures and text-book research guided by assignments.
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Discipline -Specific Knowledge and Skills (Discipline-Specific Knowledge and Skills)

02. Solve Burgers' equation with standard boundary conditions, relate these to the travelling wave solution and the viscosity solution of inviscid Burgers' equation

Activities:
Interactive lectures and text-book research guided by assignments.
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Discipline -Specific Knowledge and Skills (Discipline-Specific Knowledge and Skills)

03. Construct entropy measures for nonlinear diffusion equations.

Activities:
Interactive lectures and text-book research guided by assignments.
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Discipline -Specific Knowledge and Skills (Discipline-Specific Knowledge and Skills)

04. Solve nonlinear boundary value problems for diffusion equations.

Activities:
Interactive lectures and text-book research guided by assignments.
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Discipline -Specific Knowledge and Skills (Discipline-Specific Knowledge and Skills)

05. Distinguish nonlinear dynamical behaviour from linear dynamical behaviour.

Activities:
Interactive lectures and text-book research guided by assignments. Student seminars.
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)
Discipline -Specific Knowledge and Skills (Discipline-Specific Knowledge and Skills)

06. Recognize the role of women and various cultures in the science of nonlinear partial differential equations.

Activities:
Biographical notes in lectures Student seminars.
Literacies and Communication Skills (Writing,Speaking,Quantitative Literacy)
Inquiry and Analytical Skills (Critical Thinking,Creative Problem-solving,Inquiry/Research)

## Subject options

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Start date between: and    Key dates

## Melbourne, 2018, Semester 1, Day

### Overview

Online enrolmentNo

Maximum enrolment sizeN/A

Enrolment information