## Essential Partial Differential Equations

 Unit code: MATH36041 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

MATH36041 pre-requisites

Students must have taken MATH20401 OR MATH20411

#### Aims

This course builds on MATH20401 (PDEs and vector calculus) to further develop the rigorous study of PDEs using tools from analysis and numerical analysis.

#### Overview

We study the well posedness of the classical PDEs by semigroup and weak approximation and rigorously develop numerical approximation by the Galerkin and finite difference method. The module is theoretical and has the flavour of a pure module; proofs are given.

#### Learning outcomes

On completion of this unit successful students will be able to:

• Derive the weak form of an elliptic partial differential equation,
• Prove existence and uniqueness of solution for specified  boundary value problems,
• Derive a Galerkin approximation,
• Formulate and solve a finite element approximation for an elliptic boundary value problem,
• Formulate and solve finite difference approximations for time domain problems.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework worth 20%
• Two hour end of semester examination; Weighting within unit 80%

#### Syllabus

• Introduction. Review of PDEs (elliptic, hyperbolic, parabolic). Finite difference method and convergence (by maximum principle) [4 lecture]
• Elliptic PDEs and weak solutions. Hilbert spaces, inner product, Cauchy-Schwarz, and L2. Definition of weak derivative and weak solution. Examples. Riesz representation theorem and Lax Milgram Lemma. Proof of existence and uniqueness for model diffusion problem. More general models. [6 lectures]
• Galerkin method. Best approximation in the energy norm. Finite element and spectral Galerkin. Rates of convergence. Iteration methods. Comparison with finite difference method. [6 lectures]
• Semigroups of operators. Examples (linear test equation, heat equation on bounded domain, wave equation). Definition by Fourier analysis. Mild solutions. Semilinear equations, existence and uniqueness by contraction mapping. Reaction diffusion equation. Method of lines. Proof of convergence. [6 lectures]

• Endre Suli and David Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003.
• Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005.
• J. Robinson, Infinite Dimensional Dynamical Systems, Cambridge University Press, 2001.
• K. Morton and Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 2005.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

William Lionheart - Unit coordinator