Engineering and the Environment, University of Southampton

Knowledge and understanding
Having successfully completed the module, you will be able to:

• Describe the use of PDEs to model physical processes.
• Explain the difference between solutions of the wave equation (hyperbolic), diffusion equation (parabolic) and Laplace equation (elliptic).
• Solve second-order linear PDEs using the method of separation of variables.
• Explain what is a complex function?
• Determine whether a complex function is analytic in a prescribed region of the complex plane.
• Evaluate complex integrals.
• Utilize methods of complex analysis to solve some types of real integrals.

Cognitive (thinking) skills
Having successfully completed the module, you will be able to:

• Recognize different types of PDEs.
• Understand one type of method which may be applied to solve PDEs.
• Recognize that problems only involving real variables may be solved by utilizing methods of complex analysis.
• Improved ability to read and interpret scientific textbooks.

Practical, subject specific skills

Having successfully completed the module, you will be able to:

• Recognize and characterize different types of PDEs.
  
• Recognize and characterize different types of boundary and initial conditions.
  
• Solve linear PDEs (using the method of separation of variables).
  
• Solve complex integrals using residue calculus.
  
• Link problems involving real variables to problems involving complex variables.

Key transferable skills

• The assignments include report writing.
  
• Synthesis of theory from different mathematical fields (e.g. Fourier series, ODEs, PDEs, vector calculus).
  
• Improved ability to read and understand scientific textbooks.
  
• General mathematical skills.

The aim of this module is to introduce some of the fundamental theory of Partial Differential Equations and Complex Analysis.

Revision

• To review: Fourier series; Methods to solve ODEs; Integration by parts.

Partial Differential Equations

• To introduce the three fundamental types of second-order linear PDE (wave equation, diffusion equation, Laplace equation).
• To explain how the type of solution is different for each PDE.
• To outline why boundary and initial conditions are crucial to physical processes.
• To demonstrate how second-order linear PDEs may be solved by using the method of separation of variables (homogeneous problems).
• To show how inhomogeneous problems also may be solved using separation of variables.

• To introduce the concept of a complex function.
• To define analytic functions, and complex differentiation.
• To introduce the concept of singularities, poles and residues.
• To derive Cauchy's theorem and the residue theorem.
• To show how Cauchy's theorem and the residue theorem may be utilized to solve complex integrals.
• To demonstrate how some types of real integral may be solved by transforming the integral to a complex (contour) integral.

Part 0: Revision.

• Fourier series.
  
• Ordinary Differential Equations.
  
• Integration by Parts.
  
• Trigonometric functions.
  
• Summary of key results.

• Introduction.
  
• Second-order linear Partial Differential Equations.
  
• Classification of Partial Differential Equations.
• Boundary and Initial Conditions.
• Solution of Partial Differential Equations: Separation of Variables.

• Introduction.
  
• Complex Differentiation.
  
• Complex power series, singularities, poles and residues.
  
• Complex Integration.

Three lectures per week. About half the lecture time will be used to present the theory and worked examples. The remaining time will be used for problem classes.

Lecture notes are provided which, in general, cover the theory listed in the module syllabus. In addition to explaining the theory outlined in the notes, a set of worked examples are presented during the lectures. Students should make their own notes for these worked examples.

Each week at least one lecture will be a problem class, when students will attempt examples provided, following the lecture notes and worked examples already discussed in previous lectures.

Problem sheets and specimen answers are provided, and solutions are discussed (generally near the end of the course).

Two brief assignments are set. These assignments are marked by the lecturer providing the students with both formative and summative feedback during the course.

At the end of the course there are revision lectures, which include a review of past exam questions.

Students are encouraged to attempt the problem sheets and past exam papers.

Private study: students are expected to consult relevant textbooks, in order to further read about the theory presented during lectures.

Core text

Advanced Modern Engineeing Mathematics, G. James, Pearson Education Ltd.

Secondary text

Advanced Engineering Mathematics, E. Kreyszig, John Wiley * Sons, Inc.