Dr Simon J.A. Malham

• Room No. : CM T.21
• Address :    Mathematics Department   
                      Heriot-Watt University 
                      Edinburgh EH14 4AS 
                      Scotland
• E-Mail Address : S.J.A.Malham [at] hw.ac.uk

Research

• The Navier-Stokes equations. Do solutions of the Navier-Stokes equations develop a singularity in finite time? This is one of the Clay Millenium Prize problems worth a million dollars.
• Reaction-diffusion phenomena: Travelling waves; Autocatalysis; Combustion.
• Computational spectral theory: Evans function; Geometric methods; Grassmannian--Riccati methods; Multi-dimensional stability.
• Stochastic differential equations: Lie group integration; Efficient simulation; Shuffle and quasi-shuffle Hopf algebras; Heston model.
• Nonlinear partial differential equations: Direct linearisation, Fredholm Grassmannian flows and Poppe algebras.
• Coagulation phenomena: Smoluchowksi equation; Non-associative algebras; Binary trees; Simulation.

If you would like to know more or are interested in doing a PhD, email me (see address above).

Publications

1. Bartuccelli, M.V., Gibbon, J.D. and Malham, S.J.A. 1993, Lattice methods and the pressure field for solutions of the Navier-Stokes equations (pdf), Nonlinearity 6(1), pp. 79-92.
2. Bartuccelli, M.V., Doering, C.R., Gibbon, J.D. and Malham, S.J.A. 1993, Length scales in solutions of the Navier-Stokes equations (pdf), Nonlinearity 6(4), pp. 549-568.
3. Bartuccelli, M.V., Doering, C.R., Gibbon, J.D. and Malham, S.J.A. 1993, Length Scales and the Navier-Stokes Equations, Proc.III Potsdam–V Kiev Int. Workshop on Nonlinear Processes in Physics, Aug. 1991., Ed. by D. Kaup, pp. 172–175.
4. Bartuccelli, M.V., Doering, C.R., Gibbon, J.D. and Malham, S.J.A. 1993, Length scales in solutions of dissipative PDEs, NATO Adv. Sci. Inst. Ser. B Phys. 312, pp. 97–100.
5. Malham, S. and Xin, J. 1998, Global solutions to a two-dimensional Boussinesq combustion system with front data on finite and infinite strips, Proceedings of the Mittag-Leffler Institute 1995, pp. 1–17.
6. Malham, S. and Xin, J. 1998, Global solutions to a reactive Boussinesq system with front data on an infinite domain (pdf), Comm. Math. Phys. 193(2), pp. 287-316.
7. Balmforth, N.J., Craster, R.V. and Malham, S.J.A. 1999, Unsteady fronts in an autocatalytic system (pdf), Proc. R. Soc. Lond. A 455, pp. 1401-1433.
8. Malham, S.J.A. and Oliver, M. 2000, Accelerating fronts in autocatalysis (pdf) , Proc. R. Soc. Lond. A 456, pp. 1609-1624.
9. Ford, R., Malham, S.J.A. and Oliver, M. 2002, A new model for shallow water in the low Rossby-number limit (pdf) , JFM 450, pp. 287-296.
10. Aparicio, N.D., Malham, S.J.A. and Oliver, M. 2005, Numerical evaluation of the Evans function by Magnus integration (pdf), BIT 45, pp. 219-258.
11. Lord, G., Malham, S.J.A. and Wiese, A., Efficient strong integrators for linear stochastic systems (arxiv link), SIAM J. on Numerical Analysis 46(6) (2008), pp. 2892-2919.
12. Malham, S. and Niesen, J., Evaluating the Evans function: order reduction in numerical methods (arxiv link), Math. Comp. 77 (2008), pp. 159-179.
13. Malham, S.J.A. and Wiese, A., Stochastic Lie group integrators (arxiv link), SIAM J. Sci. Comput. 30 (2) (2008), pp. 597-617.
14. Ledoux, V., Malham, S.J.A. and Thummler, V., Grassmannian spectral shooting (arxiv link), Math. Comp. 79 (2010), pp. 1585-1619.
15. Malham, S.J.A. and Wiese, A., Chi-square simulation of the CIR process and the Heston model (arxiv link), International Journal of Theoretical and Applied Finance 16(3) (2013).
16. Ledoux, V., Malham, S.J.A., Niesen, J. and Thummler, V., Computing stability of multi-dimensional travelling waves (arxiv link), SIAM J. on Applied Dynamical Systems 8(1) (2009), pp. 480-507.
17. Malham, S.J.A. and Wiese, A., Stochastic expansions and Hopf algebras (arxiv link), Proc. R. Soc. A 465 (2009), pp. 3729-3749. doi:10.1098/rspa.2009.0203
18. Malham, S.J.A. and Wiese, A., Efficient almost-exact Levy area sampling (arxiv link), Statistics and Probability Letters 88 (2014), pp. 50-55.
19. Ebrahimi-Fard, K., Lundervold, A., Malham, S.J.A., Munthe-Kaas, H. and Wiese, A., Algebraic structure of stochastic expansions and efficient simulation (arxiv link), Proc. R. Soc. A 468 (2012), pp. 2361-2382.
20. Curry, C., Ebrahimi-Fard, K., Malham, S.J.A. and Wiese, A., Levy processes and quasi-shuffle algebras (arxiv link), Stochastics 86(4) (2014) pp. 632-642.
21. Beck, M. and Malham, S.J.A. 2014, Computing the Maslov index for large systems (arxiv link), Proc. Am. Math. Soc. (doi.10.1090/S0002-9939-2014-12575-5).
22. Karambal, I. and Malham, S.J.A., Evans function and Fredholm determinants, (arxiv link), Proc. R. Soc. A:2015471 20140597;DOI: 10.1098/rspa.2014.0597. Published 14 January 2015.
23. Ebrahimi-Fard, K., Malham, S.J.A., Patras, F. and Wiese, A., Flows and stochastic Taylor series in Ito calculus, (arxiv link), 2015 J. Phys. A: Math. Theor. 48: 495202 doi:10.1088/1751-8113/48/49/495202
24. Ebrahimi-Fard, K., Malham, S.J.A., Patras, F. and Wiese, A., The exponential Lie series for continuous semimartingales, (arxiv link), Proc. R. Soc. A :2015 471 20150429; DOI: 10.1098/rspa.2015.0429.
25. Curry, C., Ebrahimi-Fard, K., Malham, S.J.A. and Wiese, A., Algebraic structures and stochastic differential equations driven by Levy processes, (arxiv link), Proceedings of the Royal Society A 2018.
26. Beck, M., Doikou, A., Malham, S.J.A., and Stylianidis, I., 2019, Grassmannian flows and applications to nonlinear partial differential equations, (arxiv link), Proceedings of the Abel Symposium 2016.
27. Beck, M., Doikou, A., Malham, S.J.A., and Stylianidis, I., 2018, Partial differential systems with nonlocal nonlinearities: Generation and solutions (arxiv link), Phil. Trans. R. Soc. A 376: 20170195.
28. Doikou, A., Malham, S.J.A., and Wiese, A., 2019, Stochastic analysis and discrete quantum systems, (arxiv link), Nuclear Physics B 945: 114658.
29. Doikou, A., Malham, S.J.A., and Wiese, A., 2020, Time evolution in quantum systems and stochastics, (arxiv link), Centre de Recherches Mathematiques Series.
30. Malham, S.J.A., Shen, J. and Wiese A., 2021, Series expansions and direct inversion for the Heston model, (arxiv link), SIAM J. Financial Mathematics 12(1).
31. Doikou, A., Malham, S.J.A. and Stylianidis, I. 2021, Grassmannian flows and applications to non-commutative non-local and local integrable systems, (arxiv link), Physica D: Nonlinear Phenomena 415: 132744.
32. Malham, S.J.A. 2022, Integrability of local and nonlocal non-commutative fourth order quintic NLS equations, (arxiv link), IMA J. Appl. Math..
33. Malham, S.J.A. 2022, The non-commutative Korteweg-de Vries hierarchy and combinatorial Poppe algebra, (journal link), Physica D 434.
34. Doikou, A., Malham, S.J.A., Stylianidis, I. and Wiese A., 2023, Poppe triple systems and integrable equations, (journal link), PDEs in Applied Mathematics 8, 100565.
35. Doikou, A., Malham, S.J.A., Stylianidis, I. and Wiese A., 2023, Applications of Grassmannian flows to coagulation equations, (journal link), Physica D451, 133771.
36. Blower, G., Malham, S.J.A. 2023, The algebraic structure of the non-commutative nonlinear Schrodinger and modified Korteweg-de Vries hierarchy, (journal link), Physica D456, 133913.
37. Malham, S.J.A. 2023, Coagulation, non-associative algebras and binary trees, (arxiv link), submitted.

Book

• Berkshire, F.H., Malham, S.J.A, and Stuart, J.T., Introductory incompressible fluid mechanics, CUP (2022).

Unpublished manuscripts/reviews/notes/thesis

1. Malham, S.J.A., Regularity assumptions and length scales for the Navier-Stokes equations, PhD Thesis (July 1993), 129 pages.
2. Malham, S.J.A., An introduction to asymptotic analysis, Lecture notes (Nov 2005), 56 pages.
3. Malham, S.J.A., Differential equations and linear algebra, Lecture notes (March 2008), 95 pages.
4. Ledoux, V. and Malham, S.J.A., Spectral shooting is Schubert calculus, Unpublished working manuscript (Nov 2009), 14 pages.
5. Ledoux, V. and Malham, S.J.A., Introductory Schubert calculus, Review notes (v1: Jan 2010; v2: Sept 2010), 18 pages.
6. Malham, S.J.A. and Wiese, A., An introduction to SDE simulation, Lecture notes (Feb 2010), 27 pages.
7. Malham, S.J.A., An introduction to Lagrangian and Hamiltonian mechanics, Lecture notes (Aug 2016), 87 pages. (Complete detailed solutions available on request.)
8. Doikou, A., Malham, S.J.A., Stylianidis, I. and Wiese A., Applications of Grassmannian flows to integrable systems (2022), (arxiv link), detailed preprint to PDEAM article above.

Summer School

• LMS-EPSRC Short Course: Theoretical Fluid Dynamics (2011).

Miscellaneous

• Surfing.