Unit information: Applied Partial Differential Equations 2 in 2020/21

Unit name	Applied Partial Differential Equations 2
Unit code	MATH20402
Credit points	20
Level of study	I/5
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Muller
Open unit status	Not open
Pre-requisites	None
Co-requisites	Multivariable Calculus and Complex Functions
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Lecturers: Sebastian Müller and Jens Eggers

Unit Aims

To provide the student with the necessary mathematical tools in order to model a wide variety of different physical problems, ranging from waves on strings, the propagation of signals, the diffusion of heat in solids and chemicals in solution, traffic flow and the vibrations of membranes and surfaces.

Unit Description

Partial differential equations (PDEs) are differential equations involving partial derivatives of functions of several variables. They are essential for understanding many physical processes including the behaviour of ocean waves, the flow of rivers, the diffusion of pollutants, aerodynamics, the operation of musical instruments, atomic physics, and many other branches of science. This unit will give an introduction to simple PDEs and how they arise in physical problems; it will develop techniques for solving them and understanding the behaviour of the solutions.

The unit will develop students' understanding of first year multivariable calculus and linear algebra. It will introduce Fourier series, the Fourier integral, the delta function and other methods for solving linear and nonlinear PDEs, (such as the method of characteristics) and will show how eigenvalues play a central role in applied mathematics. The course emphasises techniques and broad understanding rather than proofs.

Relation to Other Units

This unit is a prerequisite for Mathematical Methods, Fluid Dynamics, Quantum Mechanics and other applied mathematics units. It gives applications of the vector calculus, complex variable methods and other material in Multivariable Calculus and Complex Functions, and includes material (Sturm-Liouville theory) relevant to Ordinary Differential Equations 2, though that course is not a prerequisite.

Intended Learning Outcomes

At the end of the course the student should should be able to:

• Understand the physical models and derive PDE's representing diffusion and wave propagation;
• Identify appropriate boundary conditions for simple linear PDEs;
• Solve linear two-dimensional PDEs on bounded spatial domains by separation of variables and Fourier series;
• Calculate and manipulate Fourier transforms, and use them to solve simple linear PDEs on unbounded spatial domains;
• Use the method of characteristics to solve simple linear and nonlinear first order PDEs;
• Describe some differences between linear and nonlinear PDEs;
• Solve multi-dimensional linear PDE's using separation of variables in a variety of coordinate systems

Transferable Skills:

• Clear thinking; mathematical modelling of physical situations; skill in mathematical manipulation.

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Reading and References

Recommended

• Stanley J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications 1993
• Richard Haberman, Applied Partial Differential Equations, Pearson/Prentice-Hall 2004