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
- Applications of scaling symmetry in differential equations
- Geometric numerical integration of ODEs
- Pattern formation of Reaction-diffusion equations with chemotaxis
- Pattern formation of Reaction-diffusion equations with sea shells or sand dune
- Random Walks
- Numerical solution of second order ODEs on the complex plane
- Love's integral equation
- Fractional Differential Equations (Analysis Oriented)
- Fractional Differential Equations (Numerical Oriented)
- Past projects

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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.

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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.

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.

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- 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:

- 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 - A Chertock, A Kurganov.
A second-order positivity preserving central-upwind scheme for
chemotaxis and haptotaxis models.
*Numerische Mathematik*111(2), 169-205

- examinine the underlying modelling assumptions
- analyse the bifurcation of steady patterns
- investigate the selection of length scales

References:

- Meinhardt, Hans. The algorithmic beauty of sea shells. Springer Science & Business Media, 2009.
- Hoyle, Rebecca B. Pattern formation: an introduction to methods. Cambridge University Press, 2006.

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).

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.

- 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. - 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. - P. Pastore.
*The numerical treatment of Love's integral equation having very small parameter*. J. Comput. Appl. Math. 236 (2011), no. 6, 1267–1281. - 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.

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- 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.
- 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.
- Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag Telos. ISBN 3-211-82913-X.
- 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.
- Herrmann, R. (2014). Fractional Calculus - An Introduction for Physicists. Singapore: World Scientific.

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- 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.
- 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.
- 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.
- 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.
- 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.

- MSc Project, Travelling Wave Solutions , by Rami Achouri (Manchester 2016)
- MSc project, Stationary densities of stable Levy flights in external potentials, by M. Leval (Imperial College 2015)
- Second year group project, Gangs and Graffiti: A Statistical Mechanical Approach, by S. Dutt, B. Kinger, A. Leidiger and L. Mineh (Imperial College 2015)
- Third year joint Math and Computer Science project, Finite Volume Schemes for Non- Linear Diffusion Equations, by H. Ostwal (Imperial College 2015)
- Second year group project, Ising Model for Gang and Graffiti, by O. Ayouche, E. Lau, W. Liu and A. Magli (Imperial College 2014)