The mission of Analysis, and Simulation of Some Evolutionary Phenomena (MASE) Research Group is mathematical models of such evolutionary phenomena linear and nonlinear cases Modeling, Analysis, and Simulation of Some Evolutionary Phenomena focuses on the analytical theory of such equations (existence, uniqueness, qualitative behavior) and on the development and implementation of some new algorithms and their numerical analysis.
In chemical interactions we have to model the chemical phenomena into graphical structure. The main idea to compute some topological indices for these graphical structures which are being used by researchers in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies to predict the physico-chemical properties of molecules.
1. Mathematical modeling of patterns in Islamic design
Arts is essential for humanity in all cultures, both as spiritual and meditative forms. Islamic arts use complex geometric patterns and shapes. In Wood for ornamental patterns, in bricks of buildings, in brass for decorations, paper, tiling, plaster, glass, etc. In this project we study a new model for simulating an interesting class of Islamic designs. Based on periodic sequences in one-dimensional manifolds "Circle", and from emerging numbers, we construct closed graphs with edges on the unit circle. Coloring the subareas of these graphs leads to the construction of very nice class of geometric patterns of so-called Islamic design.
Moreover, we mathematically characterize and analyze some convergence properties of the used up-down sequences.
2. Long-time behavior of partially damped systems modeling degenerate plates with piers
The complex composition of a suspension bridge yields two major difficulties: firstly it appears aerodynamically quite vulnerable, secondly it appears very hard to describe its behavior through simple and reliable mathematical models. We consider a partially damped nonlinear beam-wave system of evolution PDE's modeling the dynamics of a degenerate plate. The system has two degrees of freedom and, consequently, the solution has two components. We show that the component from the damped beam equation always vanishes asymptotically while the component from the (undamped) wave equation does not. In case of small energies we show that the first component vanishes at exponential rate.
3. On the stability of some wave and viscoelastic systems with non-standard nonlinearities
With the advancement of sciences and technology, many physical and engineering models require more sophisticated mathematical functional spaces to be well understood. For example, in fluid dynamics, the electrorheological fluids (smart fluids) have the property that the viscosity changes (often dramatically) when exposed to an electrical field. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems as well as other models like the image processing. In this project we consider several wave and viscoelastic "weakly" damped problems with variable-exponent nonlinearities having time-dependent coefficients and discuss the stability in the presence and absence of forcing terms. Our results, if established, will extend some known stability results in the constant-variable case to the variable-exponent case. We also intend to obtain explicit decay rates and give numerical applications to illustrate our theoretical results.
4. Mathematical analysis of Maxwell-Navier-Stokes system
The main idea of this project is to study a coupled system of equations consisting of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. We are interested in the global existence of time of the solution under the minimal assumptions on the initial data at the level that are required for the uniqueness of the solutions. Moreover, build some numerical simulation to show the behavior of plasma flow speed in a magnetic field.
5. Molecular Descriptors for Special Nanostructures
Molecules and molecular compounds are often modeled by molecular graphs. A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds.
- The focus of this project is to characterize the structure or chemical properties of a molecule in terms of the numerical value (which called topological index).
- The exact formulae for certain edge version of vertex-degree-based topological indices of certain nanostructures will be determined.