At heart I am a parabolic PDE person. Currently, my favourite equations and problems all take this form.

My research program has three main components:

Applied Math Modelling
My work in this area focusses on partial differential equation models for physical systems arising in continuum mechanics. I am particularly interested in asymptotics and simulation for higher-order PDEs and especially in finite time blow-up phenomena. Lately, I have also begun working on problems from image processing.

Relevant papers:

• Volumetric image registration methods based on solving the Monge-Ampere equation. Joint with M. Sulman, R.D. Russell & M.F. Beg. To appear in the Canadian Applied Mathematics Quarterly special edition to celebrate the 30th anniversary of the Canadian Applied and Industrial Mathematics Society.
• A perturbation solution for bacterial growth and bioremediation in a porous medium with bio-clogging. Joint with M. Chapwanya and S. O'Brien. Journal of Computational and Applied Mathematics 234 (2010) 2709–2723.
• On the phase diagram for microphase seperation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional. Joint with R. Choksi and M.A. Peletier. SIAM J. Appl. Math. 69-6 (2009) pp. 1712-1738.
• Canonical bifurcations in the Swift-Hohenberg equation. Joint with L.A. Peletier. SIAM J. Appl. Dyn. Syst. 6 (2007), no. 1, 208--235.
• Multibump blow-up solutions in CGL. With C.J. Budd and V. Rottschaffer. SIAM J. Appl. Dyn. Syst. 4 (2005), no. 3, 649--678.
• Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity. JD Evans, VA Galaktionov and JF Williams. SIAM J. Math. Anal. 38 (2006), no. 1, 64--102.
• On very singular similarity solutions of a higher-order semilinear parabolic equation. V.A. Galaktionov and JF Williams. Nonlinearity vol. 17. (2004), pp 1075-1099.
• Self-similar blow-up in higher-order semilinear parabolic equations. With C.J. Budd and V.A. Galaktionov. SIAM J. Appl. Math. (2004), Vol. 64, No. 5, pp 1775-1809
• Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory. V.A. Galaktionov and JF Williams. EJAM vol. 14 (2003), pp 745-764.
  

Numerical Analysis
The PDEs I typically study require adaptive numerical methods in time and space. This has lead to work on Parabolic Monge-Ampere equations, r-adaptivity and scaling invariance.

Relevant papers:

• Optimal Mass Transport for Higher Dimensional Adaptive Grid Generation. Joint work with M. Sulman and R.D. Russell. To appear in Journal of Computational Physics (accepted 2011).
• On the Convergence of the Parabolic Monge-Ampere equation for grid generation. Joint work with M. Sulman and R.D. Russell. To appear in Applied Numerical Mathematics (accepted 2010).
• How to adaptively resolve evolutionary singularities in differential equations with symmetry. Joint with C.J. Budd. J. Engineering Mathematics, 66 (2010), no 3, 217--236.
• Convergence of de Boor’ for generation equidistributing meshes. Joint with X. Xu, W. Huang and R.D. Russell. To appear in IMA J. Num. Anal. (2010) (accepted Nov. 2009).
• M. Sulman, JF Williams, and R. D. Russell, Monge Kantorovich Approach for Grid Generation, Proceedings of International Conference of Numerical Analysis and Applied Mathematics, American Institute of Physics Conf. Proc., 1168 25-28 (2009).
• Moving mesh generation using the Parabolic Monge-Ampere equation. Joint with C.J. Budd. SIAM J. Sci. Comput. 31 (2009), 3438--3465.
• A comparison of direct discretization of 4th order problems versus system reduction. Joint with R.D. Russell & X. Xu. Some topics in industrial and applied mathematics, 195--205, Ser. Contemp. Appl. Math. CAM, 8, Higher Ed. Press, Beijing, 2007.
• MovCol4: a high resolution moving collocation method for evolutionary PDEs. Joint with R.D. Russell and X. Xu. SIAM J. Sci. Comput. 29 (2007), no. 1, 197--220.
• Parabolic Monge-Ampere methods for blow-up problems in several space dimensions.. Joint with C.J. Budd. Phys. A 39 (2006), no. 19, 5425--5444.
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Preprints

The following papers are in revision and are available on request.

• "2D Phase Diagram for Minimizers of a Cahn-Hilliard Functional with Long-range Interactions. Joint work with M. Maras and R. Choksi. In revision with SIAM Applied Dynamical Systems.

The following papers have been recently submitted:

• (In-)Stability of Singular Equivariant Solutions to the Landau-Lifshitz-Gilbert Equation. Joint work with Jan Bouwe van den Berg. Submitted to Communications in Pure and Applied Mathematics (2010).

Industrial Mathematics
I both run and participate in a variety of academic/industrial workshops aimed at bringing high level research tools to bear on industrial problems.
Study groups provide a platform for mathematicians and people working in industry to come together and work on problems in a focussed environment. Students and academics alike get to learn and apply new skills directly to problems of immediate interest and also to develop new connections and possible future industrial collaboration. And, of course, a good time is generally had by all. For a history of the study group concept, go to the source.

Industrial Reports:
(Note: Industrial reports have multiple (possibly 10!) authors.  For brevity these have been omitted here.)

1. Sodium Flux during Haemodialysis. Proceedings of the OCCAM–Fields–MITACS Biomedical Problem Solving Workshop, 2009.
2. Proceedings of the 2008 MITACS Industrial Mathematics Summer School. (Editor.)
3. Proceedings of the 2007 MITACS Industrial Mathematics Summer School. (Editor.)
4. Proceedings of the 10th PIMS Industrial Problem Solving Workshop. (Editor.)
5. Cavity formation in loosely packed sandstone. Internal report of the third Schlumberger Mathematical Study Group. (2006)
6. Modelling the spread of SARS over intercontinental travel routes. Proceedings of the First Fields-MITACS Industrial Math Workshop. (2006)
7. Mathematical techniques for neuromuscular analysis. Proceedings of the Fifty-Second European Study Group with Industry, 109–127, CWI Syllabi, 55. CWI, Amsterdam, 2006.
8. Proceedings of the Fifty-Second European Study Group with Industry. Edited by Jan Bouwe van den Berg, Sandjai Bhulai, Joost Hulshof, Ger Koole, Corrie Quant and JF Williams. CWI Syllabi, 55. Centrum voor Wiskunde en Informatica, Amsterdam, 2006. 129 pp. (Editor.)
9. Droplet breakup in polymeric liquids. Proceedings of the 45th European Study Group with Industry. 2003.
10. Spider silk spinning. Proceedings of the 46th European Study Group with Industry. 2003.
11. The Artis Problem. Proceedings of the 42nd European Study Group with Industry. 2002
12. Magma Design Automation: Component placement on chips; the “holey cheese” problem. Proceedings of the 42nd European Study Group with Industry. 2002.
13. Modelling crystal growth. Proceedings of the fifth PIMS Industrial Problem Solving Workshop. 2001
14. Temperatures in cold rooms. Proceedings of the 40th European Study Group with Industry. 2001
15. Induction heating in oil flow problems. Proceedings of the fourth PIMS Industrial Problem Solving Workshop. (2000)
16. Thermal modelling in polymer extrusion. Proceedings of the 39th European Study Group with Industry. (2000)
17. Contaminant transport in a municipal water system. Proceedings of the third PIMS Industrial Problem Solving Workshop. (1999)