Research Interests:

  • Surfactant driven thin film flows and biomedical applications.
  • PDE Analysis: thin film equations, spectral stability problems in fluid dynamics.
  • Numerical Analysis: Scientific computations in PDEs
  • Applied Operator Theory: spectral theory of J-selfadjoint operators.
  • Sturm-Liouville problems with spectral parameter dependent boundary conditions.

Undergraduate students research projects

Brief summary of research projects:

2011– ...

  Research project 1:  ”Classes of uniqueness for solutions of the thin film equations. Surfactant driven thin film flows.”

Applications: surfactant replacement therapy and surfactant based drug delivery systems. The surfactant, produced by our alveolar cells, changes the surface tension of the lung, lowering the normal air–water surface tension of approximately 70 mJ/m^2 to values below 5 mJ/m^2. By lowering the alveolar surface tension, the energy required to inflate the lungs during inspiration is minimized, preventing lung collapse during exhaling.

Research in progress: uniqueness for waiting time solutions, local and global in time existence and asymptotic behaviour of nonnegative weak solutions to a coupled system of degenerate parabolic equations, finite speed and waiting time phenomena.
Collaboration: R. Taranets (Institute of Applied Mathematics and Mechanics of NAS of Ukraine and University of Nottingham ), John R. King ( University of Nottingham)

  Research project 2:  ”Finite–time blow up and long–wave unstable thin film equations.”

Applications: such equations arise in the modelling of fluids and materials. For example, the equation with n=m=1 describes a thin jet in a Hele-Shaw cell.
Research in progress: short-time existence, long-time existence, finite speed of propagation, and finite-time blow-up of nonnegative solutions for longwave unstable thin film equations.
Collaboration: R. Taranets (Institute of Applied Mathematics and Mechanics of NAS of Ukraine and University of Nottingham ), M. Pugh (University of Toronto)

2009– 2010

  Research project 3: ”Analytical and Numerical Analysis of Thin Film Equations With Convection.”

Application: Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include coating flows, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy.

Analytical results: existence of nonnegtaive periodic weak solutions was proved for some reasonable restrictions on initial data.
Collaboration: M. Pugh (University of Toronto), R. Taranets (Institute of Applied Mathematics and Mechanics of NAS of Ukraine)

Analytical results: the lower bound on the convergence rate of nonnegative weak solution to the steady state.
Collaboration: A. Burchard (University of Toronto), B. Stephens (University of Washington)

Analytical results: analysis of stability of periodic, solitary, and shock waves in liquid films on vibrating substrates.
Collaboration: E. Benilov (University of Limerick)

2007 – 2008

  Research project 4: ” Nature of Ill-Posedness of Forward-Backward Heat Equation ”

Application: dynamic of a thin film of liquid which is entrained on the inside of a rotating cylinder for example coating of fluorescent light bulbs or distribution of liquid thermosetting plastic in the rotating mould.

Analytical results: completness with the absence of the Riesz basis property was proved for the system of eigenfunctions of the indefinite convection-diffusion operator.

Collaboration: I. Karabash (University of Calgary, Canada), S.G. Pyatkov (Sobolev Institute of Mathematics, Novosibirsk, Russia), V. Strauss (Universidad Simón Bolívar, Venezuela).

  Research project 5: “Eigenvalue Distributions of Quadratic Matrix Pencils with Symmetries”

Application: stability of vortices in multi-dimensional discrete lattices.

Analytical results: upper bound was found for the number of spectrally unstable eigenvalues.

Collaboration: D. Pelinovsky (McMaster University, Canada).

 

2003 – 2007

  Research project 6: “Spectral Stability of Solitary Waves in Dynamical Systems” (Doctoral Thesis, McMaster University)

Application: dynamic of the magneto-acoustic waves in plasma and capillary-gravity water waves, stability of gap solitons in trapped Bose-Einstein condensates, stability of vacuum.

Analytical results: Criteria for stability and instability of solitons were derived in terms of dimensions of sign-definite invariant subspaces using Pontryagin space decomposition method. By one of these criteria it was proved that embedded eigenvalues of negative Krein signature are structurally stable in a linearized fifth – order KdV equation.

Numerical results: the numerical method was constructed to find two-pulse solutions for the fifth-order KdV equation; the MPI (parallel programming library) was used for the implementation of the code

Collaboration: T. Azizov (Voronezh State University, Russia), A. Comech
(Texas A&M University, US), Mason A. Porter (California Institute of Technology)

Advisor: Dr. D. Pelinovsky (McMaster University, Canada).

1994 – 2002

  Research project 7: “Generalized Resolvent Method and Sturm-Liouville Problem with Spectral Parameter Dependent Boundary Conditions”.

Application: transverse vibrations of rotating beam with tip mass and vibrations of string with set of discrete loads.

Analytical results: the asymptotic of eigenvalues was obtained for different type of Sturm-Liouville problems where the familiar Dirichlet-Neumann boundary conditions were replaced by boundary conditions that depend on the spectral parameter; the inverse problem was solved for the Nevanlinna type boundary conditions

Advisor: Dr. A. V. Strauss (Ulyanovsk Teacher's Training State University, Russia).

1990 – 1993

  Research project 8: “Boundary Operators and Entire Extensions of Symmetric Operators”.

Analytical results: entire extension of a second-order differentiation operator was constructed using the boundary operator method.

Advisor: Dr. A. V. Strauss (Ulyanovsk Teacher's Training State University, Russia)

1988 – 1990

  Research project 9: “Automated Construction of Different Sectional Views for a Three-dimensional Model of the Russian Space Station ‘Mir’“. (Master Thesis, Moscow Institute of Physics and Technology)

Numerical results: the numerical algorithm of calculating the area of the shadow of the complicated three-dimensional object was constructed and implemented as a Pascal programming language code; the surfaces of objects (space modules of the station “Mir”) were approximated using triangulation; the dynamical tree – structure was used for a database of the approximated objects.

Advisor: Dr. M. Komarov (M.V.Keldysh Institute of Applied Mathematics, Russia)