Research Projects

Many of the real-time phenomena in various domains, such as physical, chemical or biological, are modeled in terms of advection/convection, diffusion and reaction processes. The mathematical modeling of these types of phenomenon produced parabolic unsteady-state reaction-diffusion-convection (RDC) mathematical models. These models can be categorized as: (i) singularly perturbed linear parabolic (SPLP) models, (ii) nonlinear RDC models such as Chemo-taxis model, Pollutant Transport-Chemistry model, Angiogenesis model, Gierer-Meinhardt model, a weekly stiff RDC system undergoing Turing instability, Navier-Stokes model, Black-Scholes model etc. The main challenges in solving RDC models numerically are as follows: (a) in SPLP models are the impendence of boundary and weak interior layers in their solution, presence of perturbation parameter(s), very diluted zones where the solution, their derivatives or both changes sharply etc. Due to these difficulties, many numerical schemes fail to capture accurate solutions if the singular perturbation parameter is very small. Most of the SPLP models have been studied numerically using FDM and FEM in literature by many groups of researchers. But, the drawback of FDM is that the fully discrete scheme obtained by using the Crank-Nicolson and central finite difference schemes, does not satisfy the discrete maximum principle. Secondly, FEM based schemes for SPLP models are developed with standard Galerkin or streamline upwind/Petrov-Galerkin (SUPG) which has some stability issues. On the other hand, nonlinear RDC models are also challenging to simulate. To overcome the above mentioned challenges of SPLP and nonlinear RDC models, we will develop and analyze robust numerical algorithms based on scale-3 Haar wavelets, radial basis functions (RBFs) and rescaled RBFs. Till date very few works have been done for RDC models by using S3HWs and RBFs. Thus, in current scenario, there is too much scope to develop new robust algorithms based on S3HWs, RBFs and rescaled RBFs to simulate and analyze the unsteady-state RDC models.