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MA50181: Mathematical methods 1

[Page last updated: 03 August 2022]

Academic Year: 2022/23
Owning Department/School: Department of Mathematical Sciences
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Masters UG & PG (FHEQ level 7)
Semester 1
Assessment Summary: CW 25%, EX 75%
Assessment Detail:
  • Coursework (CW 25%)
  • Examination (EX 75%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
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.

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.

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 Optional on the following programmes:

Department of Mathematical Sciences


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