Author Archives: David Graham

Waves

Wave motion is found everywhere: sound waves, water waves, electromagnetic waves, traffic waves and Mexican waves are just some examples. The mathematical description of wave motion can be fairly straightforward, as with the oscillations of a guitar string for example, but as more physical effects are allowed for the theory becomes more complicated. In this

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Chebychev Polynomials

In this project the aim is to study the mathematical properties of Chebychev polynomials and their role in numerical approximation. The concept of ‘good’ and ‘best’ numerical approximations and the Remes algorithm will be investigated and, if time permits, further applications of Chebychev polynomials could be considered. This project is mainly analytical but would involve

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Numerical methods for PDE’s

Partial differential equations describe how the three/four dimensional world works. Solving such equations is the key to understanding the behaviour of many physical, chemical, biological or financial systems. A large range of numerical methods is available to solve such problems (for example finite difference, finite element, finite volume, boundary element or pseudo-spectral methods to name

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Multifractals in Finance

The standard mathematical model of random asset price variation is Geometric Brownian Motion – where the log of the asset price ‘return’ is normally distributed. However, this under-predicts the frequency of dramatic and large changes in the price. In real data, ‘fat-tails’ are observed, meaning that large deviations from mean behaviour are seen more often

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Mathematical Finance

Many topics in mathematical finance can make good project topics. The key to understanding lies in the modelling of the random behaviour seen in prices of shares, oil, foreign exchange rates and the like. Projects in this area usually require solving either stochastic differential equations (SDE’s) or partial differential equations (PDE’s) analytically and/or numerically. A

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