The project proposes a complex study of certain processes encountered in environment and life sciences which are governed by diffusion type equations. The mathematical models studied here will be established by starting to examine attentively the diffusion properties of the medium, because these properties induce the prevailing features in the diffusion equations. Thus, linear and nonlinear, continuous and discrete, either slow or fast diffusive type behaviours will be selected. The project will focus particularly on nonlinear diffusion models involving nonlinearities both in equation coefficients and in boundary conditions. Besides diffusion, the mathematical models considered in our study can also describe other significant physical phenomena accompanying diffusion, like transport processes and/or production of mass. This study will focus especially on certain models arising in:
a)Environmental diffusion: water and solute diffusion in soils
b)Biological and ecological diffusive systems.
The phenomena in these domains allow similar mathematical modelling and the investigation of these models can lead to relevant extrapolations to other fields. The present project will focus on qualitative study of each type of model (existence, unicity, properties of the solutions) numerical methods in solving the equations and validation of the results. Moreover, a comparative analysis will be carried on between continuous and discrete aspects of certain diffusion phenomena.
The principal objectives and the specific objectives of our project can be written as follows:
  1. Modelling and qualitative study of diffusive models.
    1. Classification of diffusion types and physical models
    2. Mathematical modelling and qualitative study of the envisaged diffusive models.
  2. Numerical techniques, algorithms, numerical simulation and validation.
    1. Discrete approximation of nonlinear diffusion equation.
    2. Numerical methods to solve algebraic equations.
    3. Theoretical and numerical validation.
  3. Comparative analysis of the discrete and continuous diffusion processes.
    1. Kinetic models.
    2. Spatially discrete form of continuous diffusion equation.