## MATH2375 Linear Differential Equations and Transforms

### 15 creditsClass Size: 360

Module manager: Professor Alexander Mikhailov; Dr P.P.Dechant
Email: A.V.Mikhailov@leeds.ac.uk;P.P.Dechant@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2024/25

### Pre-requisite qualifications

(MATH1005 and MATH1025 and MATH1026) or (MATH1050 and MATH1055 and MATH1400 and MATH1331) or (MATH1050 and MATH1055 and MATH1400 and MATH1060), or equivalent.

This module is not approved as a discovery module

### Module summary

This module introduces a variety of techniques for the solution of initial and boundary value problems for linear ordinary differential equations and basic partial differential equations of mathematical physics, which describe such ubiquitous phenomena as waves and diffusion. It also provides an introduction to contour integration in the complex plane with applications to the Fourier transform.

### Objectives

On completion of this module, students should be able to:

a) obtain power series solutions of 2nd order linear ordinary differential equations;
b) put a 2nd order linear differential operator into the Sturm-Liouville form and discuss orthogonality of eigenfunctions;
c) solve the standard partial differential equations of mathematical physics in Cartesian coordinates subject to given boundary conditions by the method of separation of variables;
d) use Fourier series and Fourier transform techniques including solution of linear partial differential equations;
e) use contour integration in the complex plane and apply Cauchy’s residue theorem in simple cases.

### Syllabus

a) Power series solution of ordinary differential equations.
b) Inner products, Sturm-Liouville operators and orthogonality of eigenfunctions.
c) Bessel and Legendre functions, their basic properties and application.
d) Fourier series and transforms, separation of variables, with applications to initial boundary value problems for the standard partial differential equations of mathematical physics.
e) Introduction to methods for functions of complex variables, including contour integration and applications of Cauchy’s residue theorem.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 33 1.00 33.00 Private study hours 107.00 Total Contact hours 43.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular examples sheets.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated