BSc Computer Science and Mathematics

The main topics to be explored are: Fourier series, partial differential equations, analytical and numerical methods for solving classical PDEs (Laplace's equation and the heat and wave equations) and several topics in vector calculus, including surface and volume integrals. The course covers similar material to MATH20401 but contains a reduced range of topics and with fewer details, where appropriate. The methods employed in the course will prove essential for all of the applied mathematics and numerical analysis options in the remaining semesters of the Joint Honours BSc and MMath degree programmes.

This course introduces students to analytical and numerical methods for solving partial differential equations (PDEs) and builds on the first year core applied mathematics courses to develop more advanced ideas in differential and integral calculus.

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

• convert Cartesian coordinates into cylindrical and spherical coordinates and sketch surfaces expressed in all of these coordinate systems;
• perform partial differentiation of functions of more than one variable and interpret these derivatives physically;
• recognise classical PDEs describing physical processes such as heat diffusion;
• interpret boundary and initial conditions physically;
• define orthogonal functions and list and apply their properties;
• compute Fourier series, Fourier sine series and Fourier cosine series of some piecewise continuous functions;
• classify second-order PDEs as being either elliptic, hyperbolic or parabolic;
• solve, analytically, via the method of separation of variables, the heat and wave equations (in one space variable) and Laplace's equation (in two space variables), on rectangular and circular domains;
• solve, numerically, via finite difference schemes, the heat equation in one space variable;
• solve, numerically, convection-diffusion equations using upwind finite differencing;
• compute elements of surface and volume in different coordinate systems;
• evaluate line, surface and volume integrals over a range of domains, using transformations to other coordinate systems where appropriate;
• define and apply grad, div and curl operators and relate key identities to physical properties of vector fields;
• recall, apply, and interpret physically, the classical Divergence, Green's and Stokes' theorems.

• Introductory material. [3 lectures]

Cartesian, cylindrical and spherical coordinates. Functions of several variables, surfaces. Partial derivatives, chain rule. Partial differential equations, boundary and initial conditions. Integrals of functions of several variables.

• Fourier series. [4 lectures]

Orthogonality. Fourier series and Fourier coefficients. Periodic, even and odd functions. Fourier's theorem. Fourier sine and cosine series.

• Partial Differential Equations. [2 lectures]

Linearity, homogeneity and order of PDEs. Classification of second-order equations. Introduction to the classical equations: Laplace's, heat and wave equations.

• Analytical Solution of PDEs. [4 lectures]

The method of separation of variables. Solving, exactly, initial-value problems for the heat and wave equations.  Solving Laplace's equation in both Cartesian and plane polar co-ordinates.

• Numerical Solution of PDEs. [4 lectures]

Solving, approximately, the reaction-diffusion and convection-diffusion equations (ODEs) via finite difference methods.

Solving, approximately, the heat equation in one space variable (PDE). Explicit and implicit numerical schemes.

• Vector Calculus. [5 lectures]

Surfaces, unit vectors, elements of surface/volume. Line, surface and volume integrals. Scalar and vector fields: differential and integral calculus. Grad, div and curl operators and related identities. Classical theorems: Divergence,  Stokes' theorems.

• Coursework: an in-class test, weighting within unit 20%
• End of semester examination: weighting within unit 80%

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.

• Morton, K.W., Mayers, D.F, Numerical solution of partial differential equations, Cambridge University Press, 2005.
• James Stewart, Calculus, Early Transcendentals, Thomson, fifth edition (international student edition), 2003.
• R Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, (Third edition) Prentice-Hall, 1998.
• Schey, H. M. Div, Grad, Curl, and all that : an Informal Text on Vector Calculus, New York : W. W. Norton, various editions.

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.