### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title LINEAR DIFFERENTIAL OPERATORS IN MATHEMATICAL PHYSICS Code MATH421 Coordinator Dr S Haslinger Mathematical Sciences Stewart.Haslinger@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 7 FHEQ First Semester 15

### Aims

This module provides a comprehensive introduction to the theory of partial differential equations, and it provides illustrative applications and practical examples in the theory of elliptic boundary value problems, wave propagation and diffusion problems.

### Learning Outcomes

(LO1) Apply the basic concepts of mathematical physics, such as generalised functions, fundamental solutions and Green's functions.

(LO2) Apply powerful mathematical methods to problems of electromagnetism, elasticity, heat conduction and wave propagation.

(LO3) Apply mathematical methods to research-centred problems

(S1) Numeracy

(S2) Mathematical software (e.g. Maple, MATLAB)

### Syllabus

• Generalised derivatives. Definition and simple properties of generalised derivatives. Limits and generalised derivatives.

• Laplace's equation and harmonic functions. Dirichlet and Neumann boundary value problems. Elements of potential theory.

• Fundamental solutions of differential equations. Singular solutions of Laplace's equation, the wave equation, the Helmholtz equation and the heat equation.

• Green's functions and Poisson's formulae.

• Spectral analysis for the Dirichlet and Neumann problems for finite domains.

• The heat conduction equation. Maximum principle. Uniqueness theorem.

• The wave equation. Wave propagation and the characteristic cone.

• Cauchy problems for the wave equation and the heat conduction equation.

### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101 Calculus I 2020-21; MATH102 CALCULUS II 2020-21; MATH103 Introduction to Linear Algebra 2020-21

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final exam  120    60
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Homework 1    20
Homework 2    20