MATH34032 - 2009/2010
Specification
Aims
To introduce students to Green's functions, linear integral equations and the calculus of variations and to develop associated mathematical methods used in applied mathematics.
Brief Course Description
This course unit consists of methods of solving various mathematical problems which arise in science. The method of Green's functions is a powerful tool in solving linear ordinary and partial differential equations, and the course starts with an introduction to this topic. There are situations where physical laws are better expressed as integral equations and the course continues starts with Fredholm and Volterra integral equations and their methods of solution. The final section of the course considers some simple problems of the calculus of variations - e.g. showing the shortest path between two points on a plane is a straight line.
Learning Outcomes
On successful completion of this course students will:
- Be familiar with the concepts of integral operator and functional.
- Have acquired sound knowledge of Green's functions, Fredholm and Volterra integral equations and of the calculus of variations.
- Have solved representative problems in applied mathematics using the above, eg solving ordinary and partial differential equations using Green's functions, obtaining and solving linear integral equations and solving typical problems in the calculus of variations, e.g. the isoperimetric problem.
Syllabus
- Green's functions: Definition and basic properties. [3 lectures]
- Application to the solution of ordinary differential equations. [2]
- Application to the solution of partial differential equations. [2]
- Fredholm and Volterra equations of the 1st and 2nd kinds. [2]
- Fredholm equations with separable kernels. [3]
- Neumann series. [3]
- Euler-Lagrange equation(s). [3]
- 1st and 2nd functional derivatives. [2]
- Lagrange multipliers. [2]
Textbooks
- Clive R Chester, Techniques in Partial Differential Equations, McGraw-Hill
- R Courant and D Hilbert, Methods of Mathematical Physics, Vols. I and II, Interscience.
Teaching and learning methods
Two lectures per week plus one weekly examples class. In addition students should expect to do at least four hours private study each week for this course unit.
Assessment
- Coursework: Test in week 8 or 9 worth 20%
- End of semester 2 hour Examination 80%