# Undergraduate and Applied MSc Projects

Below are the lists of possible undergraduate/Applied MSc Projects, all of which require a combination of designing and/or implementing numerical methods and performing applied analysis.

## Travelling wave solutions for nonlocal bistable reaction-diffusion equation

In this project, travelling waves solutions for the following non-local bistable reaction-diffusion equation $\frac{\partial}{\partial t}u = J\ast u -u +f(u)$ is studied. Here $J$ is a non-negative, continuously differentiable function such that $\int_{\mathbb{R}} J(x)dx=1, \int_{\mathbb{R}} |x|J(x)dx<\infty$ with the typical examples $J(x)=\max(0,1-|x|)$ or $J(x) = \exp(-\pi x^2)$, $J\ast u(x) = \int_{\mathbb{R}} J(x-y)u(y)dy$ is the convolution between $J$ and $u$ and $f(u)=F'(u)$ is the derivative of a double-well potential $F(u)$. Both numberical solutions and qualitative properties of the travelling waves will be investigated.

References:
• X. Chen, "Generation, propagation, and annihilation of metastable patterns", J. Differential Equations 206, 399–437 (2004).
• Bates, Peter W., Paul C. Fife, Xiaofeng Ren, and Xuefeng Wang. "Traveling waves in a convolution model for phase transitions." Archive for Rational Mechanics and Analysis 138, no. 2 (1997): 105-136.

## Applications of scaling symmetry in differential equations

Scale invariance is ubiquitous in mathemtics and physics, as a feature of the underlything qunatity which changes in a simple predictable way when other variables changes by a common factor. In this project, scaling symmetry(or scaling invariance) is explored in several applications related to differential equation: self-similarity and self-similar solutions, conserved quantities and commuting flows of nonlinear equations, exact solutions of equations with scaling symmetries.

References:
• Barenblatt, Grigory Isaakovich. Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics. Vol. 14. Cambridge University Press, 1996.
• Barenblatt, Grigory Isaakovich. Scaling. Vol. 34. Cambridge University Press, 2003.
• Goktas, Unal, and Willy Hereman. "Invariants and symmetries for partial differential equations and lattices." arXiv preprint solv-int/9801024 (1998).
• Dresner, Lawrence. Applications of Lie's theory of ordinary and partial differential equations. CRC Press, 1998.

## Geometric numerical integration of ODEs

In this project, a special class of numerical methods for ordinary differential equations (ODEs) will be studied. These methods focus on preserving the qualitative properties like the conservation of first integrals and large time asymptotic behaviours. The interested student should have taken Numerical Analysis II and be comfortable writing code (MATLAB is preferred).
References:
• Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2002). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag.
• Ben, Leimkuhler; Sebastian, Reich (2005). Simulating Hamiltonian Dynamics. Cambridge University Press
• Iserles, Arieh. A first course in the numerical analysis of differential equations. No. 44. Cambridge university press, 2009.

## Pattern formation of Reaction-diffusion equations with chemotaxis

Many existing equations in pattern formation, like the well-known Gray-Scott system or Gierer-Meinhardt system, usually consist of two component (activator-substrate or activator-inhibitor). In this project, a three-component system proposed in [1] is further studied, using the numerical method proposed in [2], to clarify the following questions:
• The dependence of the pattern on the parameters
• Construction of the spike solutions (see the figures below) using singular perturbation
• Possible reduction of the original equations in [1] to understand the key terms and mechanisms for the the pattern formation

References:
1. R Tyson, SR Lubkin, James D Murray. A minimal mechanism for bacterial pattern formation. Proceedings of the Royal Society of London B: Biological Sciences 266(1416): 299-304. 1999
2. A Chertock, A Kurganov. A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numerische Mathematik 111(2), 169-205

## Random Walks

A random walk is a path consisting of random steps, and is used to model many physical processes like diffusion in natural science. In this project, variants of the standard random walk of fixed step length on a regular lattice are extended: variable step length, general step distributions and long-range correlated walks, etc. The focus is on both analytical properties and numerical experiments. Basic understand of probability and numerical methods are required.
Reference:
• Joseph Klafter, and Igor M. Sokolov. First steps in random walks: from tools to applications. Oxford University Press, 2011.
• Ralf Metzler , and Joseph Klafter. "The random walk's guide to anomalous diffusion: a fractional dynamics approach." Physics reports 339.1 (2000): 1-77.
• Metzler, Ralf, and Joseph Klafter. "The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics." Journal of Physics A: Mathematical and General 37.31 (2004): R161.
• Weiss, G. "Aspects and Applications of the Random Walk (Random Materials & Processes S.)." (2005).

## Numerical solution of second order ODEs on the complex plane

In this project, numerical solution of second order ODEs are extended from the real line (treated in standard textbooks) to the complex plane. Special methods based on Padé approximation may be preferred because of the possible poles in the solutions and other singularities like branch points. A family of ODEs called Painlevé transcendents will be studied. Working knowledge on complex variables and numerical methods for ODEs is essential.
References:
• Corliss GF. Integrating ODEs in the complex plane—pole vaulting. Mathematics of Computation. 1980;35(152):1181-9.
• Corliss G, Chang YF. Solving ordinary differential equations using Taylor series. ACM Transactions on Mathematical Software (TOMS). 1982 Jun 1;8(2):114-44.
• Fornberg B, Weideman JA. A numerical methodology for the Painlevé equations. Journal of Computational Physics. 2011 Jul 1;230(15):5957-73.
• Reeger JA, Fornberg B. Painlevé IV: A numerical study of the fundamental domain and beyond. Physica D: Nonlinear Phenomena. 2014 Jul 1;280:1-3.

## Love's integral equation

The Fredholm integral equation o fthe second kind $$f(x) = g(x) + \frac{1}{\pi}\int_{-L}^{L} \frac{f(t)}{(x-t)^2+1}dt$$ arises in the calculation of electrosttic field of two equal circular conducting disks [2]. Existence numerical methods for solving Fredholm integral equations are surveyed, followed by some further studies on properties about the resolvent kernel and eigenvalues/eigenfunctions.

References:
1. L. Fox and E. T. Goodwin. The numerical solution of non-singular linear integral equations. Philos. Trans. Roy. Soc. London. Ser. A. 245, (1953). 501–534.
2. R. R Love. The electrostatic field of two equal circular co-axial conducting disks. The Quarterly Journal of Mechanics and Applied Mathematics 2, no. 4 (1949): 428-451.
3. P. Pastore. The numerical treatment of Love's integral equation having very small parameter. J. Comput. Appl. Math. 236 (2011), no. 6, 1267–1281.
4. K. E. Atkinson, and L. P. Shampine. Algorithm 876: solving Fredholm integral equations of the second kind in Matlab. ACM Trans. Math. Software 34 (2008), no. 4, Art. 21, 20 pp.

## Fractional Differential Equations (Analysis Oriented)

Integrals and derivatives are not restricted to be integer orders, and fractional counterparts like $\frac{d^{1/2}}{dx^{1/2}}$ appear in many applications like groundwater flow or viscoelastic materials. In this project, various notions of fractional integrals and derivatives are reviewed, with an emphasis on the simplest fractional differential equation.

References:
1. Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering V. Academic Press. ISBN 0-12-525550-0.
2. Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN 0-471-58884-9.
3. Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag Telos. ISBN 3-211-82913-X.
4. Podlubny, Igor (1998). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering 198. Academic Press. ISBN 0-12-558840-2.
5. Herrmann, R. (2014). Fractional Calculus - An Introduction for Physicists. Singapore: World Scientific.

## Fractional Differential Equations (Numerical Oriented)

Integrals and derivatives are not restricted to be integer orders, and fractional counterparts like $\frac{d^{1/2}}{dx^{1/2}}$ appear in many applications like groundwater flow or viscoelastic materials. In this project, different finite difference methods for fractional diffusion equations are review.

References:
1. Meerschaert, Mark M., and Charles Tadjeran. "Finite difference approximations for fractional advection–dispersion flow equations." Journal of Computational and Applied Mathematics 172.1 (2004): 65-77.
2. Meerschaert, Mark M., and Charles Tadjeran. "Finite difference approximations for two-sided space-fractional partial differential equations." Applied numerical mathematics 56.1 (2006): 80-90.
3. Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering V. Academic Press. ISBN 0-12-525550-0.
4. Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN 0-471-58884-9.
5. Podlubny, Igor (1998). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering 198. Academic Press. ISBN 0-12-558840-2.

## Past projects

Some past projects are listed here: