Contact Details
| Room No. | LT1030 (L933) |
|---|---|
| Telephone | +44 (0)141 548 3651 |
| Fax | +44 (0)141 548 3345 |
| m.grinfeld@strath.ac.uk |
Research Interests
Continuum Models of Phase Transitions
I am interested in phase transitions in various types of materials, such as alloys and liquid crystals, and in particular in models with nonstandard diffusion mechanisms and nonlocal terms.
Mathematics in Biology and Medicine
I am trying to work out what mathematics can add to the undersyanding of complex bilogical systems , such as the immune system (in the context of autoimmune disease) and stress-actiovated tissue control systems (in the context of non-targeted irradiation effects).
Mathematical Modelling in Economics
We (R. Cross, H. Lamba and myself) are trying to create a psychologically realistic model of the stock market, whoch takes into account memory effects as well.
Topological Techniques in Dynamical Systems
I have been a long-standing interest in Conley Index Theory and am in a permanet process of writing a book on it. In particular, I am interested in changes in dynamics in gradient systems under a change of metric, and in working out an index theory that applies to integro-differential equations.
Mathematical Modelling in Economics
We (R. Cross, H. Lamba and myself) are trying to create a psychologically realistic model of the stock market, whoch takes into account memory effects as well.
Recent Publications
- Steady State Solutions of a Bistable Quasilinear Equation with Saturating Flux
- Burns M and Grinfeld M
University of Strathclyde Mathematics and Statistics Research Report, 2010, #4, 1-12. - Bifurcation analysis of the twist-Freedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions
- Da Costa FP, Gartland EC, Grinfeld M and Pinto JT
European Journal of Applied Mathematics, 2009, 20, 269-287. - Hysteresis and Economics TAKING THE ECONOMIC PAST INTO ACCOUNT
- Cross R, Grinfeld M and Lamba H
Ieee Control Systems Magazine, 2009, 29, 30-43. - Structural instability in an autophosphorylating kinase switch
- Grinfeld M and Webb SD
Mathematical Biosciences, 2009, 219, 92-96. - Uniqueness in the Freedericksz transition with weak anchoring
- da Costa FP, Grinfeld M, Mottram NJ and Pinto JT
Journal of Differential Equations, 2009, 246, 2590-2600.
Vacancies & Studentships
Coarsening in integro-differential models of materials science
Reference No.: MG1
Supervisor: Dr Michael Grinfeld
Date Advertised: 6th February 2007
Please contact Dr Michael Grinfeld for further information.
Recently, a new class of models to was developed to describe, for example, phase separation in materials such as binary alloys. These take the form of integrodifferential equations.
Coarsening, that is, creation of large scale patterns in such models is poorly understood.
There are partial results that use the maximum principle, while for most interesting problems such a tool is not available.
This will be a mixture of analytic and numerical work and may need tools of functional analysis and semigroup theory.
Modelling in autoimmune diseases
Reference No.: MG2
Supervisor: Dr Michael Grinfeld
Date Advertised: 20th November 2007
Please contact Dr Michael Grinfeld for further information.
In autoimmune diseases, the body reacts to self-antigens.
The object of the project is to understand the dynamics of the species involved (tissue target cells, self-reactive T cells, T-reg cells and APCs) in order to suggest protocols of intervention that would break the vicious positive feedback cycles present in these systems.
Mathematics of gradient systems
Reference No.: MG3
Supervisor: Dr Michael Grinfeld
Date Advertised: 20th November 2007
Please contact Dr Michael Grinfeld for further information.
Many physical systems are assumed to evolve down the gradient of some physically relevant functional, such as free energy.
However, the notion of the gradient is not uniquely defined even in Hilbert spaces.
In that context, using Morse theory, it would be interesting to see how the dynamics depends on the inner product used.
In a Banach space context, all questions to do with dynamics of gradient systems are largely open.
Wave equations with non-sign constant damping
Reference No.: MG4
Supervisor: Dr Michael Grinfeld
Date Advertised: 20th November 2007
Please contact Dr Michael Grinfeld for further information.
Such equations occur, for example, in wind-induced oscillations.
The resulting mathematical object is well-behaved in the sense that solutions remain bounded, but their behaviour is a complete and exciting mystery.
Dynamic behaviour of viscoelastic materials
Reference No.: MG5
Supervisor: Dr Michael Grinfeld
Date Advertised: 10th December 2007
Please contact Dr Michael Grinfeld for further information.
In 1982, Olmstead et al. introduced a caricature model of the motion of a viscoelastic fluid.
If one assumes the Maxwell kernel for viscoelasticity, one ends up with a van der Pol-Duffing equation with diffusion.
This object is tremendously interesting, as though the solutions are bounded, the equation can be shown to undergo an infinite-dimensional Hopf bifurcation.
More precisely, as you increase the Rayleigh number, the homogeneous steady state (no flow) loses stability by an infinite number of pairs of complex eigenvalues crossing the imaginary axis simultaneously.
The question then is to characterise the asymptotioc (in time) behavious of solutions. It is non-trivial even for finite-dimensional truncations of the equations.