General informationThis course is taught by Dr Stefan Güttel. My office hour is Monday 15–16 in ATB 2.114. The teaching times are as follows:
- Monday 12–13, lecture in Simon Bld, Theatre D
- Tuesday 17–18, lecture in Simon Bld, Theatre D
- Monday 13–14, tutorial A in ATB G.209
- Tuesday 13–14, tutorial B in the ATB G.205
Lecture notesThe lecture notes should be used to supplement your own notes taken in lectures. (It is highly recommended that you take your own notes.)
- Section 1: Introductory Material (week 1)
- Section 2: Fourier Series (week 2–3)
- Section 3: Introduction to PDEs (week 3-4)
- Section 4: Separation of Variables (week 4–6)
- Section 5: Finite Difference Methods (week 7–8)
- Section 6: Vector Calculus (week 9–11)
Example sheetsThe example sheets below contain exercises for you to attempt before the tutorials. As a guideline, there is one example sheet to complete for each week. The tutorials are intended to answer your questions on these exercises. The solutions will be posted only when all exercises have been attempted.
- Sheet 1: Introduction (Solutions)
- Sheet 2: Orthogonality (Solutions)
- Sheet 3: Fourier Series (Solutions)
- Sheet 4: Partial Differential Equations (Solutions)
- Sheet 5: Separation of Variables A (Solutions)
- Sheet 6: Separation of Variables B (Solutions)
- Sheet 7: Finite Difference Methods A (Solutions)
- Sheet 8: Finite Difference Methods B (Solutions)
- Sheet 9: Vector Calculus A (Solutions)
- Sheet 10: Vector Calculus B (Solutions)
Handouts and MATLAB codesThe handouts below contain additional material to be read between lectures. They are meant to be used as a preparation for the lectures and should be read before the week indicated in each bullet point. The MATLAB codes will further help you to experiment with PDEs.
- Week 0: Classical PDEs [heateq_demo.m]
- Week 1: Orthogonal Vectors
- Week 2: Fourier Series [FourierN_demo.m]
- Week 3: Separation of Variables
- Week 4: Bessel functions
- Week 5: Finite Differences for Reaction-Diffusion [trisolve.m, reac_diff_1D.m]
- Week 6: Finite Differences for Convection-Diffusion Equation [conv_diff_1D.m]
- Week 7: Explicit Finite Differences for 1D Heat Equation [heat_eq_explicit_fd.m]
- Week 8: Implicit Finite Differences for 1D Heat Equation [heat_eq_implicit_fd.m]
- Week 9: Vectors and Div, Grad, Curl
- Week 10: Line integrals
Material from the webThis section collects some optional material, ordered by topic, which I thought you might find interesting and/or fun.
- Coordinate systems: lecture on conversion between Cartesian (there called
rectangular), cylindrical, and spherical coordinates
- Fourier series: videos on a visual introduction to Fourier series and the harmonic analyzer, a mechanical machine to compute Fourier coefficients!
- Separation of variables: We have used this method to solve the wave equation on a 1D string, and also on a circular 2D drum. But what about other 2D shapes? Can one infer the drum's shape from the Fourier expansion? Here are some papers on that question: Kac 1966, Protter 1987, Gordon, Webb, Wolpert 1992.
- Vector calculus: videos on the visualisation of divergence and curl and Stokes' theorem
TextbooksThe above course materials are self-contained and you do not need to buy any textbooks. We will study several basic topics in calculus and applied mathematics, which are covered in hundreds of available texts in the library. The following books all contain some material you will meet in the course.
- J Stewart: Calculus, Early Transcendentals, Thomson, fifth edition (international student edition), 2003. (Useful for the first part of the course and vector calculus.)
- R Haberman: Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, (Third edition) Prentice-Hall, 1998. (Useful for the section on Fourier Series and introduction to PDEs.)
- KW Morton and DF Mayers: Numerical solution of partial differential equations, Cambridge University Press, 2005. (Useful for the sections of finite difference methods and numerical analysis.)
- HM Schey: Div, Grad, Curl, and all that: an Informal Text on Vector Calculus, New York: W. W. Norton, various editions. (Useful for the final few weeks of the course when we tackle vector calculus.)