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# BSc Mathematics with Financial Mathematics / Course details

Year of entry: 2021

## Course unit details:Partial Differential Equations and Vector Calculus A

Unit code MATH20401 20 Level 2 Semester 1 Department of Mathematics No

### Overview

The first half of this course equips students with the fundamental tools required in order to solve simple partial differential equations (PDEs). This includes important aspects of vector calculus (curvilinear coordinates and integral theorems) and an understanding of how to classify PDEs and what this classification means physically. The method of characteristics is then introduced in order to solve First order semi and quasi-linear PDEs.

The second half of the course focuses on solving second order PDEs (mainly Laplace's equation, the heat equation and the wave equation), first analytically by employing separation of variables and Fourier series and then numerically by introducing the topic of finite difference methods.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Calculus and Vectors A MATH10121 Pre-Requisite Compulsory
Calculus and Vectors B MATH10131 Pre-Requisite Compulsory
Level 2 students only

### Aims

On completion of the course students should

• understand more about applied mathematics and see the importance of mathematical modelling,
• see that PDEs describe a number of important phenomena,
• understand that vector calculus has a fundamental importance in applied mathematics,
• understand that one can use analytical and/or numerical methods (usually a combination of both) in order to solve PDEs that arise in real world applications,
• be well prepared for future courses in applied mathematics and numerical analysis.

### Learning outcomes

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

• 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,
• use the formulae for the grad, div, curl and Laplacian to express these operators in a given (potentially unseen previously) set of orthogonal curvilinear coordinate system and prove this formula for grad,
• 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,
• classify PDEs of any order and determine the type of second order semi-linear PDE,
• recognise classical PDEs describing physical processes such as diffusion (heat equation), wave propagation (wave equation) and electrostatics (Laplace's equation) and choose appropriate boundary and initial conditions for these PDEs,
• solve second order semi-linear PDEs using the four-step method of transformation to reduce them to their canonical forms,
• solve first order semi-linear PDEs using the 3-step method of transformation AND using the method of characteristics,
• define orthogonality of functions on an interval, and use orthogonality principles to determine series coefficients in integral form,
• define and sketch the periodic extensions for full-range, sine and cosine Fourier series, and compute these series for piecewise continuous functions,
• 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 rectangular or circular domains,
• 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,
• 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,
• 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.

### Syllabus

Section 1: Introduction and motivation. What are PDEs? Why study them? Some examples and applications. [2 lectures]

Section 2: Vector calculus in curvilinear coordinates. Introduction to general formalism of switching from Cartesian to curvilinear coordinate systems. Basis vectors, line, surface and volume elements. Grad, div, curl and transforming to curvilinear coordinates. Surface and volume integrals in three dimensions. Gauss (divergence) and Stokes' theorems in three dimensions. [8 lectures]

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. Method of Characteristics. Cauchy problem, choice of boundary and initial conditions [4 lectures]

Section 4: First order PDEs. Scalar first order pdes in two variables. Method of characteristics for semi-linear and quasi-linear equations. [4 lectures]

Section 5: Fourier series. Motivation via “trial separation of variables solution for homogeneous heat equation in 1D with homogeneous boundary conditions, general initial profile. General concepts of eigenvalues/eigenfunctions, orthogonality. Fourier series, sine and cosine series and associated Fourier (Dirichlet) theorem regarding piecewise-smooth functions, orthogonality. Differentiating and integrating. [4 lectures]

Section 6: Separation of variables for second order PDEs. Separation of variables for homogeneous heat and wave equation in curvilinear coordinates 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. [6 lectures]

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, upward differencing. [10 lectures]

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Continuous assessment worth 20% consisting of 6 electronic coursework throughout the semester
• End of semester examination on ALL material; worth 80%

### Feedback methods

Feedback will be given in four different ways. Weekly examples classes will allow the students to discuss their work and ask questions to the lecturers and/or Teaching Assistants. The electronic coursework automatically provides immediate feedback on the student’s answers. Finally feedback on the marks received at the coursework will be given in lectures. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

- 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

### Study hours

Scheduled activity hours
Lectures 44
Tutorials 10
Independent study hours
Independent study 146

### Teaching staff

Staff member Role
Alice Thompson Unit coordinator
Raphael Assier Unit coordinator