About this course
MATH2011 - Analysis of Partial Differential Equations
COURSE CONTENT:
This course is built around two mathematical concepts: Fourier series; and Partial Differential Equations (PDEs).
Fourier series were introduced as a tool for solving linear PDEs, but are important in their own right, and contain the germ of the ideas underlying the most advanced forms of linear analysis. The essence of the idea is to expand an arbitrary periodic signal in terms of harmonics. Again, we introduce the basic ideas with illustrative examples rather than detailed theory.
The second topic in the course is an introduction to PDEs. Here we deal with functions (fields) depending on several variables such as x, y, z and t. Many important applications are described by PDEs, and we look at some of these to introduce the three main types of linear PDEs in two independent variables: heat conduction and molecular diffusion, waves on a stretched string, and steady temperature distributions in 2-dimensions. In particular, we will introduce Fourier's method of separation of variables and superposition as a key solution method in each case.
WHERE IS IT USED:
An understanding of these concepts is important for anyone who wants to apply mathematics to the real world. DEs are still the most widely used tools in mathematical modelling. They are used extensively in theoretical physics, mathematical biology and ecology, human movement studies, and all branches of engineering. In particular, they inevitably arise when we try to model anything that moves continuously in time or varies continuously in space. In this course and MATH2010, we lay the groundwork for more advanced modelling courses (MATH3101, 3102, 3104, 4102, 4104).
Because of their fundamental importance, the theory of differential equations is still one of the most active areas of pure mathematics research. After completing this course, you will be well-placed to go on if you wish to more advanced courses in the theory of partial differential equations (MATH3403 & 4403).
WHO IS INTERESTED:
Students with interests in the applications of mathematics to the real world, or with the development of the tools needed for such applications, will need a good understanding of the basic ideas introduced here. This includes those interested in research careers in theoretical physics, engineering, applied mathematics, pure mathematics (analysis), and numerical analysis.
WHAT DO I NEED:
You will need to know basic ideas about ODEs from MATH1052 and MATH2000, as well as some linear algebra and vector analysis from MATH2000. The course deals with Chapters 10 and 11 (approximately) of the book by E. Kreyszig, "Advanced Engineering Mathematics (8th Edition)," which is the set text. Chapters 1, 2, 8 and 9 cover the background knowledge assumed for the course.
WHEN IS IT AVAILABLE:
MATH2011 is offered in second semester every year.
