MATH426 Techniques in Applied Mathematics II

Overview

Techniques for solving problems in applied mathematics, including dynamical systems theory, perturbation theory, multiple scales methods, boundary layer theory, solitons and nonlinear waves, and methods for solving stochastic differential equations.

Techniques for solving problems in applied mathematics, including dynamical systems theory, perturbation theory, multiple scales methods, boundary layer theory, solitons and nonlinear waves, and methods for solving stochastic differential equations.

About this paper

Corequisite

MATH 424

Restriction

MATH 401-412

Limited to

BA(Hons), BSc(Hons), PGDipArts, PGDipSci, MA (Thesis), MSc, MAppSc, PGDipAppSc, PGCertAppSc

Contact

Mathematics 400-level programme coordinator: Dr Fabien Montiel fabien.montiel@otago.ac.nz

Teaching staff

Dr Robert Van Gorder

Paper Structure

This paper is an introduction to techniques for solving more involved problems in applied mathematics. One focus will be understanding the behaviour of systems of differential equations. In many realistic applications, there are more than one dependent variable, and in this paper, we will focus on techniques for studying systems of ordinary or partial differential equations.

For systems of ordinary differential equations this will involve a study of dynamical systems and nonlinear dynamics. A key feature will be the qualitative analysis of differential equations, where we make qualitative predictions about solutions of nonlinear differential equations without actually solving the equations.

We develop perturbation theory as a tool for solving nonlinear differential equations, discussing more advanced methods for the asymptotic approximation of solutions to certain nonlinear differential equations such as multiple scales approaches and boundary layer theory.

We consider the topic of stochastic differential calculus, highlighting the Itô integral and solving certain stochastic differential equations.

We then extend our knowledge of linear PDEs from the first paper to include certain exactly solvable nonlinear PDEs, developing nonlinear wave theory and the notion of a "soliton."

There will be three lectures per week for six weeks. The lectures will cover the following topics:

1. Systems of ordinary differential equations; steady states and linear stability analysis; dynamical systems arising in real-world applications; transcritical bifurcation; saddle-node bifurcation; pitchfork bifurcation.
2. Planar dynamics; phase plane analysis; periodic orbits; Hopf bifurcation; limit cycles.
3. Regular and singular perturbation theory; Poincaré-Lindstedt method; perturbation theory for eigenvalue problems; method of multiple scales.
4. Boundary layers; method of matched asymptotic expansions; interior layers.
5. Stochastic calculus; the Itô stochastic integral; stochastic differential equations (SDEs); exact solutions of linear SDEs; Itô's Lemma for nonlinear SDEs.
6. Exactly solvable nonlinear PDEs; nonlinear waves, traveling waves, and solitons

Teaching Arrangements

18 lectures (50 minutes each).

Weekly assessments will comprise problem sets assessing the material of that week (100% of final mark).

Graduate Attributes Emphasised

Critical Thinking, Lifelong Learning, Scholarship, Self-motivation, Information Literacy, Research.
View more information about Otago's graduate attributes.

Learning Outcomes

On completion of the study of this paper, students are expected to:

1. Solve systems of nonlinear ordinary differential equations of relevance in applied mathematics.
2. Solve nonlinear ordinary and partial differential equations using perturbation methods.
3. Solve stochastic differential equations of relevance in applied mathematics.
4. Solve special nonlinear partial differential equations admitting wave solutions.
5. Combine multiple aspects of an applied mathematics problem to formulate a correct solution.
6. Properly write-up solutions to mathematical problems and then interpret what such a solution means.

Timetable