About this course
MATH2100 - Applied Mathematical Analysis
Course profile, lecture notes and tutorial sheets.
COURSE CONTENT:
This course is built around four mathematical concepts: Systems of ordinary differential equations (ODEs); Laplace transforms; Fourier series; and Partial Differential Equations (PDEs).
Systems of ODEs generalise the idea of an ODE, as you have seen it covered in MATH1052, for example. Now we have several unknown functions of a single independent variable, say the time t, and we have ODEs linking the unknowns together. We deal mostly with systems of two coupled first-order equations to see the sorts of things that can happen. The notion of the phase-plane is introduced, where the ODEs determine the trajectory of a representative point for the system. Basic notions of stability and instability of equilibrium (critical) points of the system are explored. Illustrative applications are described, such as predator-prey systems, an epidemic model, electrical and mechanical oscillators.
The Laplace Transform is a tool still widely used to deal with linear ODEs and PDEs, especially in engineering and biological applications. We introduce the basic concepts, including applications to simple systems of ODEs.
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 final 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 , 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 differential equations (MATH3403, 4402, 4403, 4406).
The development of fast and reliable methods of solving differential equations approximately on the computer has also become one of the most active and important areas of scientific computation, and if you do MATH2200 with MATH2100, you will be able to go on in this direction too if you wish, through MATH3201.
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 roughly with Chapters 3, 5, 10 and 11 of the book by E. Kreyszig, "Advanced Engineering Mathematics (8th Edition)," which is the set text. Chapters 1, 2, 6, 7, 8 and 9 cover the background knowledge assumed for the course.
WHEN IS IT AVAILABLE:
MATH2100 is offered in second semester every year. The first half, covering systems of ODEs and Laplace Transforms, is also available as MATH2010 in both semesters, and the second half, covering Fourier Series and PDEs, is also available as MATH2011 in second semester.
