BSc Actuarial Science and Mathematics

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Syllabus:
Section 1: Introduction and motivation. What are PDEs? Why study them? Some examples and applications.

Section 2: Vector calculus in orthogonal curvilinear coordinates. Introduction to general formalism of switching from Cartesian to orthogonal curvilinear coordinate systems. Basis vectors, line, surface and volume elements. Grad, div, curl and transforming to orthogonal curvilinear coordinates. Surface and volume integrals in three dimensions. Gauss (divergence) and Stokes' theorems in three dimensions. 

Section 3: Classification of PDEs. Classification as order, scalar/vector, homogeneous/inhomogeneous, linear/semi-linear/quasi-linear/nonlinear. PDE type: 2nd order in two independent variables (elliptic, hyperbolic, parabolic, mixed), canonical forms.  Choice of boundary and initial conditions.

Section 4: Fourier series. Motivation via trial separation of variables solution for homogeneous heat equation in 1D. General concepts of orthogonality. Fourier series, sine and cosine series and associated Fourier (Dirichlet) theorem regarding piecewise-smooth functions.

Section 5: First order PDEs. Scalar first order PDEs in two variables. Method of transformations; Cauchy problem.  Method of characteristics for semi-linear and quasi-linear equations.

Section 6: Separation of variables for second order PDEs. General concepts of eigenvalues/eigenfunctions, Separation of variables for homogeneous heat and wave equation with homogeneous BCs and inhomogeneous initial conditions. Separation of variables for Laplace's equation with inhomogeneous BCs. Link with Sturm Liouville eigenvalue problems. Some ideas of special functions.

Section 7: Numerical solution of PDEs. Finite difference methods. Link with solutions obtained in Section 4, 6 and 7. Explicit and implicit schemes and the theta method, truncation error, stability and convergence, Crank Nicholson, convection-diffusion problems, upwind differencing.

The unit aims to: 
further develop the methods underlying much of modern applied mathematics and provide students with a solid foundation for the applied units in Years 3 and 4.
 

• ILO 1    compute the scale factors, the unit basis vectors and the surface and volume elements for a given set of (potentially unseen previously) orthogonal curvilinear coordinates and check that the resulting basis is right-handed and orthogonal
• ILO 2    state precisely and use the Divergence theorem and Stokes' theorem in three dimensions and compute the (curved) surface integrals (using either parametrisation or projection) as well as the volume and line integrals involved
• ILO 3    classify PDEs of any order and determine the type of second order semi-linear PDE
• ILO 4    solve second order semi-linear PDEs using the method of transformation to reduce them to their canonical forms
• ILO 5    solve first order semi-linear PDEs using the 3-step method of transformation AND using the method of characteristics
• ILO 6    define orthogonality of functions on an interval, and use orthogonality principles to determine series coefficients in integral form
• ILO 7    define and sketch the periodic extensions for full-range, sine and cosine Fourier series, and compute these series for piecewise continuous functions
• ILO 8    use the method of separation of variables to solve the heat and wave equation (in one spatial and one time variable) and Laplace's equation (in two spatial dimensions) on simple domains
• ILO 9     construct finite-difference schemes of a given order and template to approximate derivatives of a function of a single variable, and analyse their order of consistency
• ILO 10    use finite difference methods to solve initial and boundary value problems in a single spatial or time variable, and analyse the stability of the numerical scheme.
• ILO 11    use the finite difference schemes based on the method of lines to solve the heat equation in one spatial variable, and use upwind finite difference schemes to solve advection-diffusion equations.
   

End of year examination

- 3 hours

General feedback is available after the exam is marked

80%

6 electronic courseworks throughout the year

Individual feedback available online after the coursework closes    

20%

- Vector analysis. Schaum's outlines. Editors: M.R. Spiegel, S. Lipschutz and D. Spellman. 2nd edition. 2009

- Div, Grad, Curl and all that: an informal text on vector calculus. H.M.Schey. W.W. Norton and Co. 4th Edition. 2005.

- An introduction to partial differential equations. Y. Pinchover and J. Rubinstein. Cambridge University Press. 2005

- Applied Partial Differential Equations with Fourier Series and Boundary Value Problems. R. Haberman, Pearson, 5th edition, 2012

- Numerical Solution of Partial Differential Equations. K.W. Morton and D.F. Mayers, Cambridge University Press. 2nd Edition. 2005

- Essential Partial Differential Equations. D.F. Griffiths, J.W. Dold, D.J. Silvester, Springer, 2015