Computational Methods

This research interest within the Analysis of Differential Equations concentrates on nonlinear problems. The focus is on rigorous analysis of mathematical models motivated from applied mathematics involving nonlinear differential equations, dynamical systems, evolution equations and financial mathematics.

Areas of Interest:

Nonlinear Partial Differential Equations: Nonlinear partial differential equations are one of the key areas of the interaction of mathematics and the sciences. I study qualitative features of equations, such as singularities and oscillations which can manifest themselves as defects, microstructure or blow-up. The opposite phenomena such as regularity, compactness and convergence are of equal interest. In order to extend the mathematical framework to understand these phenomena I pursue a broad spectrum of mathematical ideas. Topics I may consider include:

Blow-up phenomena in parabolic problems

Dynamical systems and evolution equations

Existence, regularity and singularities for nonlinear elliptic problems

Stochastic integral-differential equations

Dynamical Systems: Ordinary differential equations model many important processes in biology, physics, economy and other sciences. This interest studies the qualitative properties and long-time behavior of solutions. I am particularly interested in stability properties of solutions and their basin of attraction. The systems that I consider range from autonomous, non-autonomous, non-smooth and finite-time ordinary differential equations to discrete dynamical systems given by iterations of a map. Research topics include:

Basin of attraction of periodic orbits

Non-autonomous dynamical systems

Attractivity of solutions over a finite time

Non-smooth systems

Numerical methods to determine the basin of attraction

Stochastic Mathematical Finance: Research in the management of financial risk and the pricing and hedging of derivatives, e.g. options, in the use of stochastic methods has become an indispensable tool in recent years. By developing and applying theoretical models of theses and analyze them we gain more insight of the issues in the financial and social world. Research topics in this area include:

Risk assessment computation

Minimization algorithms for finance industries

Option prices and related computations

Interesting Research Topics

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