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Non-linear differential equations play a fundamental role in the modelling of physical, chemical, and ecological processes and in the understanding of fundamental mathematical problems. The type and form of the differential equations which appear in the modelling of the above mentioned processes are often the same or similar. Modelling, theory and visualisation of such processes are tackled within the group using interdisciplinary techniques. Theoretical investigation of existence, uniqueness and the formation of singularities can give indications of an adequate or inadequate model. On the other hand, numerical simulations of theoretically justified models can give the theorist a graphical representation of the process, and so lead to new insights into the problem and possible directions that may be taken in analysing the model. From a practical point of view, a theoretically sound numerical simulation can often replace expensive experiments.
A Doctor student in the graduate programme should obtain a solid training in theoretical models of physical processes, theoretical and numerical analysis of the model, and visualisation of the results procured through this analysis. The group receives regular visits from experts in their fields, who hold short courses during their stay. Travel expenses are available to members of the group who wish to visit other groups or persons for research purposes.
The interconnecting main themes of the graduate programme are:
Differential equations of continuum and fluid mechanics
(convection flows in semi-conductor smelting,
magnetohydrodynamics und star models,
numerics of the Euler and Navier-Stokes equations,
elasticity theory)
Geometric differential equations, differential geometry
(curvature flow and free boundary problems,
geometric variational problems,
geodesic flow,
differential operators on manifolds)
Modelling and analysis of stochastic processes
(stochastic Hamiltonian systems,
optimal couplings,
mathematics of finance,
statistics of stochastic processes)
Scientific visualisation and interactive analysis (through visualisation
tools) of numerical results and geometrical objects