My research area is in mathematical physics and mathematical aspects of materials science. In addition to the theoretical aspects, I am very keen on finding real world applications. I am interested in a variety of problems that range from kinetic theory of matter to macroscopic evolution of phase boundaries to phase transitions and critical phenomena.
CURRENT RESEARCH: Improved Solving Methods for Coupled-Chemistry Climate Model Computations
An integral and expensive part of coupled climate model simulations is the gas phase chemistry in the atmosphere which involve dozens of species and hundreds of reactions which give rise to hundreds of coupled differential equations (DEs). While a standard way to handle these large systems is by using Newton-Raphson type algorithms the amount of computational costs is rather high due to the iterative structure of the method. Alternatively, we propose an alternative approach for such DE solvers which improves their convergence and robustness. We achieve this by combining the Newton-Raphson methods with Quasi-Newton methods. We test our approach with numerical experiments and show that our modification improves the robustness of the solver and speeds up the overall run-time with negligible loss in the accuracy.
RECENT PAST RESEARCH: Droplet Evaporation and the Coffee Stain Effect
One side of my recent research activity involved modelling of evaporation phenomena in droplets. In particular, I studied the motion of colloid particles under the influence of the fluid flow in the bulk and interfacial currents on the surface of the droplet. The flows are greatly affected by external sources (such as heating) and the intrinsic properties of the system (such as non-zero size of particles). I believe, the ideas that I develop here can be helpful in understanding the colloid transfer phenomena and can find applications in 3D printing.
PAST RESEARCH: Collisionless Kinetic Equations in Plasma Physics
It is known for the Vlasov-Poisson system that the solutions may not be as nice as in the case of whole space. I am working on the stationary solutions of the Vlasov-Poisson system with boundary conditions under a variety of settings. I am also investigating the stability of the statioanary solutions.
Another area that I am working on is the applications of renormalization group and scaling methods in physics and mathematics. Such techniques can be used beyond critical phenomena for example in obtaining the time dependency of physically relevant quantities (such as characteristic length) of self similarly evolving systems or in calculating blow up and extinction exponents of certain non-linear parabolic equations.
DOCTORAL RESEARCH: Evolution of Phase Boundaries and Effect of Microscopic Non-local Interactions on Macroscopic Properties
My doctoral research was on phase transitions and free boundary problems from the diffuse interface point of view (phase field) which is also related to density functional theories for interfaces. The main goal was to understand development of spatial macroscopic patterns in anisotropic systems. This involved developing new non-local diffuse interface models that capture anisotropic interaction effects and using asymptotic analysis to get the macroscopic behavior.