MATH424 Techniques in Applied Mathematics I

Overview

Techniques for solving problems in applied mathematics, including exact, approximate, and iterative methods for solving linear and nonlinear ODEs, linear PDEs, boundary value problems, and integral equations.

Techniques for solving problems in applied mathematics, including exact, approximate, and iterative methods for solving linear and nonlinear ODEs, linear PDEs, boundary value problems, and integral equations.

About this paper

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 problems in applied mathematics. The topics will include techniques for solving ordinary differential equations, partial differential equations, difference equations, and integral equations. We will also describe how mathematical models for real-world phenomena are derived.

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

1. Modelling using differential equations; discrete versus continuum domains; ordinary versus partial differential equations; how to construct a mathematical model; solution of differential equations arising from various applications.
2. Exact solution methods for ordinary differential equations; linear and nonlinear separable equations; first order linear equations, second order constant coefficient equations; initial value problems.
3. Second order variable coefficient ordinary differential equations; Cauchy-Euler equations; fundamental solutions and linear independence; the Wronskian; inhomogeneous second order equations; reduction of order; variation of parameters.
4. Approximate and iterative solution methods for ordinary differential equations; Taylor series solutions; Taylor polynomial approximations; methods for solving discrete / difference equations; small parameter perturbation theory.
5. Boundary value problems in one variable; Sturm-Liouville theory; eigenvalues and eigenfunctions; construction of an orthonormal basis; singular Sturm-Liouville problems.
6. Partial differential equations on unbounded domains; method of characteristics, d'Alembert's formula for the wave equation on the real line; the Fourier transform; heat kernels and the diffusion equation on the real line; hyperbolic versus parabolic versus elliptic PDEs.
7. Boundary value problems in multiple variables; general separation of space and time variables on bounded space domains; spectral methods and their convergence, Laplace's equation and Poisson's equation on bounded space domains; the diffusion equation on bounded space domains; the wave equation on bounded space domains; the Schrödinger equation on bounded space domains.
8. Diffusion processes on semi-infinite domains; self-similar solutions on the half-line; self-similar solutions for piecewise initial conditions on the real line.
9. Integral equations; Fredholm integral equations; Volterra integral equations; Neumann series; iterated kernels and the resolvent method.

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, Research, Information literacy, Communication.
View more information about Otago's graduate attributes.

Learning Outcomes

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

1. Solve linear and nonlinear ordinary differential equations of relevance in applied mathematics.
2. Solve linear partial differential equations and boundary value problems of relevance in applied mathematics.
3. Solve integral equations of relevance in applied mathematics.
4. Combine multiple aspects of an applied mathematics problem to formulate a correct solution.
5. Ability to properly write-up solutions to mathematical problems and then interpret what such a solution means.

Timetable