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Programme & Unit Catalogues

## MA50181: Mathematical methods 1

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 1
Assessment Summary: CW 25%, EX 75%
Assessment Detail:
• Coursework (CW 25%)
• Examination (EX 75%)
Supplementary Assessment: Mandatory extra work (where allowed by programme regulations)
Requisites:
Description: Aims:
To furnish the student with a range of methods for the solution of linear systems, ODEs and PDEs. Students should be able to obtain the solution of certain ODEs and PDEs. They should also be aware of certain analytic properties associated with the solution e.g. uniqueness and by considering a variety of examples, to appreciate why these properties are important.

Learning Outcomes:
Students should learn a set of mathematical techniques in a variety of areas and be able to apply them to either solve a problem or to construct an accurate approximation to the solution. They should demonstrate an understanding of both the theory and the range of applications (including the limitations) of all the techniques studied.

Skills:
Problem solving methods and their analysis, for models involving Differential Equations arising in applications (T, A); approximation techniques (T, A).

Content:
Sturm-Liouville theory: Reality of eigenvalues. Orthogonality of eigenfunctions. Expansion in eigenfunctions. Approximation in mean square. Statement of completeness. Fourier Transform: As a limit of Fourier series. Properties and applications to solution of linear systems defined by differential equations. Frequency response of linear systems. Characteristic functions. Quasi-linear first-order PDEs in two independent variables: Characteristics. Integral surfaces. Uniqueness, envelopes and domains of definition. Linear and quasi-linear second-order PDEs in two independent variables: Cauchy-Kovalevskaya theorem (without proof). Classification as elliptic, parabolic, and hyperbolic. Different standard forms. Constant and nonconstant coefficients. One-dimensional wave equation: d'Alembert's solution. Uniqueness theorem for corresponding Cauchy problem (with data on a space-like curve).
Asymptotic analysis, scaling arguments (via the Newton polygon), self-similarity. Applications to algebraic equations and two-point boundary value problems.
Programme availability:

#### MA50181 is Compulsory on the following programmes:

Department of Mathematical Sciences

#### MA50181 is Optional on the following programmes:

Department of Mathematical Sciences

Notes:
* This unit catalogue is applicable for the 2015/16 academic year only. Students continuing their studies into 2016/17 and beyond should not assume that this unit will be available in future years in the format displayed here for 2015/16.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.