## Partial Differential Equations and Vector Calculus A

 Unit code: MATH20401 Credit Rating: 20 Unit level: Level 2 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

#### Additional Requirements

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.

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

#### Learning outcomes

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

• convert Cartesian coordinates into cylindrical and spherical polar coordinates and solve problems expressed in these coordinate systems;
• understand functions of several variables, their partial differentiation, integration, and their geometrical interpretation;
• understand the basic concept of orthogonal functions;
• compute Fourier series, Fourier sine series and Fourier cosine series of piecewise continuous functions;
• solve first order parial differential equations using the method of characteristics;
• recognise classical PDEs describing physical processes such as diffusion, wave propagation and electrostatics;
• choose appropriate boundary and initial conditions for PDEs;
• classify second-order PDEs as elliptic, hyperbolic or parabolic;
• solve analytically, using 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 convection-diffusion equations numerically using upwind finite difference schemes;
• compute elements of surface and volume in different coordinate systems;
• evaluate line, surface and volume integrals over domains using various coordinate systems;
• use grad, div and curl operator notation and relate key identities to properties of vector fields;
• understand, and use the Divergence, Green's and Stokes' theorems.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Continuous assessment worth 20% consisting of 6 electronic coursework throughout the semester
• 3 hours end of semester examination on ALL material; worth 80%

#### 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. [6 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, extension to n variables, 1st order system of PDEs. Characteristics. Cauchy problem, well-posedness, choice of boundary and initial conditions [3 lectures]
• Section 4: First order PDEs. Scalar first order pdes in two variables. Linear constant coefficient, Dâ€'Alembert. Method of characteristics for semi-linear and quasi-linear equations. [5 lectures]
• Section 5: Fourier series. Motivation via œtrial 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. Special functions: circular functions, Bessel functions, Legendre polynomials, Chebyshev polynomials, Frobenius' method. [12 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. [12 lectures]

#### Recommended reading

• Vector analysis. Schaumâ€'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

#### Feedback methods

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.

#### Study hours

• Lectures - 44 hours
• Tutorials - 22 hours
• Independent study hours - 134 hours

#### Teaching staff

Simon Cotter - Unit coordinator

Raphael Assier - Unit coordinator