Virtual element method (VEM) is a newly born baby from Finite element method (FEM) Galerkin approach. In recent years, this method has been extensively studied for a variety of problems such as linear elliptic, parabolic, hyperbolic, stokes problems and so on. We are the first group to work on VEM in India. Our research work mainly focuses on the VEM discretization for the nonlinear equations. The major challenge in devising the computable VEM scheme is due to the fact that the basis functions are not known explicitly.

Phase type (PH) distributions, which are defined on the non-negative real line, became quite popular in the last decades as they have been used in a wide range of applications of stochastic models. It forms a dense subset in the space of all distributions defined on the non-negative real numbers and at the same time it is numerically tractable because of its underlying Markovian structure. We could find a multivariate distribution, defined on any k-orthotope in [0,∞)k , having the analogous properties of the PH class.

Information geometry is a branch of Mathematics that applies the techniques of differential geometry to the field of probability theory. The family of probability distributions which constitutes a statistical model has a rich geometric structure as a manifold with Riemannian metric and dual connections. Using this geometric interpretation one can obtain new insight into the framework of statistical inference and can develop new techniques for inference.

We first consider thin piezoelectric shallow shells with variable thickness and show that the eigenvalues of the three dimensional problem are order square of thickness the corresponding scaled eigensolutions converge to the eigensolutions of a two dimensional model.

Dynamics of micro particles in fluids and the change in rheological parameters due to particle dynamics occur across many industries including many processes relevant to space programs. We developed solutions for some of the problems and the study will be continued. A set of ordinary differential equations governing the migration of an arbitrary forced spheroid in quiescent/ simple-shear/ uniform/ oscillating/ oscillating-shear flows at low Reynolds number is formulated and discussed, assuming a sufficiently diluted suspension to neglect the particle-particle interactions.

To understand the dynamics of WSB trajectories and develop algorithms to obtain WSB trajectories from Earth to Moon and Mars. The role of periodic orbits (PO) and manifolds associated with them in ballistic lunar capture is studied. As a first step towards orbiter missions, planar fly-by trajectories to the Moon were studied in the framework of Restricted three-body problem (RTBP). We attempt the construction of a WSB trajectory toMoon in the restricted four-body problem (RFBP) and its dynamics. An algorithm is developed to obtain WSB trajectories to the moon in a full force model.

Various types of stability analysis of nonlinear dynamical systems in connection with continuous neural network systems are investigated. The LMI tools and the method of Lyapunov functionals are being employed for the analysis.

Controllability is one of the fundamental properties of control systems which guarantees the ability of the system to be steered from any initial state to a desired final state by using suitable control functions. Controllability analysis for stand-alone systems linear systems has been extensively carried out in the literature. However, for nonlinear systems as well as systems with various other effects like delay, impulses, noises, fuzziness etc in state or control are to be explored for investigation.