School of Mathematics

Jitse Niesen — Undergraduate projects

I supervise the following topics for MATH3422/3423 (Project in Applied Mathematics, 15 credits) and MATH5003/5004 (Assignment in Mathematics, 30/40 credits):

• Numerical solution of ordinary differential equations.
• Periodic solutions of the N-body problem.
• The Magnus expansion.
• The Evans function (only for 4th-year students).

You can find more details on these projects below. All projects (except possibly the third one) require you to do some computations, so they provide good opportunities to enhance your computer skills. You are free to choose the environment you are most comfortable with; the computations project. Computations can be done in a computer algebra system like Maple, a numerical environment like Matlab, or a programming language like C/C++, Python, or R; in some cases Excel is also a possibility.

Please contact me if you're interested or you want more information. You can find my contact details on my homepage.

Numerical solution of ordinary differential equations

As you have (hopefully) learnt from the courses you have already taken, many phenomena can be described by differential equations. Most differential equations in the exams can be solved exactly. However, this is no longer the case when you leave the realm of exam questions, and then you need methods for computing approximate solutions. This project is about numerical methods for the solution of ordinary differential equations.

An important concept is the error: the difference between the computed solution and the real solution. Starting from there, the project can develop in various directions. Error estimates are often used to automatically adapt the method; the idea is that when the method commits a large error it better move slowly and carefully. Alternatively, you may want to look how to actually compute the error, so that you get an idea of how accurate the solution is, or you may study more quantitative aspects, making a connection to the theory of dynamical systems.

Other possible topics that students have tackled in the past are: Pricing options using stochastic differential equations, and the finite element method.

No knowledge of differential equations beyond first-year analysis is required. It would be useful if you took MATH2600 (Numerical Analysis). There is also an overlap with the later numerical courses (especially MATH3475).

Further reading:

• Lecture notes from MATH2600 (Numerical Analysis).
• Numerical ordinary differential equations on Wikipedia.
• Arieh Iserles, A first course in the numerical analysis of differential equations, Cambridge University Press.

Periodic solutions of the N-body problem

The (gravitational) N-body problem studies a number of masses moving under their mutual gravitational forces. When there are only two masses, this problem can be solved exactly (think of the planets moving in an ellipse around the Sun). The problem is much more complicated when there are three or more masses and only a few solutions are known. One, due to Euler, has one mass stationary and the other two masses moving in a circle around the stationary mass. Another solution was found a few years ago in which the three masses chase each other on a figure-8 curve (see this movie).

The project is to study this solution. You should probably start by constructing the figure-8 solutions. There are several possibilities from here, depending on your interests. For a theoretical project, you could start by studying tbe existence proof and the connection with geometry. For a more experimental project, you could try to find similar solutions (a search on the internet should give you some inspiration about the many possibilities). The stability of this and other solutions is very interesting from an applied point of view; again, this can be studied both theoretically and numerically.

Different areas of mathematics are used depending on the direction the project takes. The experimental parts use techniques from MATH2600 (Numerical Analysis) and possibly MATH2431 (Fourier series). Stability is introduced in MATH2391 (Nonlinear Differential Equations) and further in MATH3396 (Dynamical Systems) and MATH3397 (Nonlinear Dynamics). The theoretical parts uses techniques from MATH2650 (Calculus of Variations) and there are possibilities to tie in with MATH2051 (Geometry of Curves and Surfaces) and MATH3071 (Groups and Symmetry).

Further reading:

• Lecture notes from MATH2600 (Numerical Analysis).
• N-body choreographies on Scholarpedia.
• Richard Montgomery. A new solution to the three-body problem. Notices Amer. Math. Soc. 48 (2001), no. 5, 471-481.

The Magnus expansion

Autonomous linear ODEs are equations of the form y' = Ay where A is a matrix. These can be solved using the matrix exponential. The Magnus expansion is the analogous – but more complicated – solution in the case that the matrix A is time-dependent and the equation is thus not -autonomous.

Your task is firstly to understand the Magnus expansion and how it is derived. Then you'll have to apply it in some examples. For a more theoretical project, you may want to study the algebraic theory in the background which will lead you to Lie algeras.

Perhaps surprisingly, the solution of autonomous linear differential equations is treated in MATH2391 (Nonlinear Differential Equations). The numerical solution of all kinds of differential equations is discussed in MATH2600 (Numerical Analysis). The module MATH3385 (Quantum Mechanics) is useful background to understand why the Magnus expansion is useful. If you want to take the project in a theoretical direction, you need some abstract algebra as discussed in MATH2033 (Rings, Polynomials and Fields) and MATH3193 (Algebras and Representations).

Further reading:

• Magnus expansion on Wikipedia.
• Arieh Iserles, Expansions that grow on trees, Notices Amer. Math. Soc. 49 (2002), no. 4, 430–440.
• S. Blanes, F. Casas, J.A. Oteo and J. Ros, The Magnus and expansion and some of its applications, Physics Reports 470 (2009), 151–238.

The Evans function

Note: This topic is only available for MATH5003/5004 (fourth-year assignments) due to the amount of theory you need to master.

Partial differential equations may have many interesting solutions like pulses and fronts which may be stationary or moving. Given such a solution, we are interested in whether it is stable: will any perturbation die out or will it grow and annihilate the pulse or front? The Evans function is a technique to assess the stability of pulses and fronts.

This project requires you to first understand what the Evans function is and how it is used. First, we will try to tackle a fairly simple example, which can be approached both analytically and numerically. After that, you may try some other examples in which analytic techniques cannot be used, or you may delve in more theoretical aspects of the Evans function construction.

The concept of stability is introduced in MATH2391 (Nonlinear Differential Equations) and further in MATH3396 (Dynamical Systems) and MATH3397 (Nonlinear Dynamics). The computational part uses techniques from MATH2600 (Numerical Analysis). Useful theoretical background is MATH3024 (Homotopy and Surfaces) and MATH3215 (Hilbert Spaces and Fourier Analysis).

Further reading:

• Todd Kapitula, The Evans function: A Primer, talk given at a workshop, 2006.
• Björn Sandstede, Stability of travelling waves, review article, 2002.