# Research lines

1. Evolution problems with memory terms

Many problems of interest can be modelled by evolution equations with memory terms (J. Prüss, J., Evolutionary Integral Equations and Applications. Basel, Birkhäuser Verlag, 1993). Some of the researchers of the group have experience in the numerical treatment of Volterra type equations, in particular in the construction of efficient time integrators. The linear Volterra equation is the natural framework of the models and includes fractional time derivative problems. In analogy with the abstract diferential equations, the simplest approach for it is the starting point for the study of more general situations involving new memory terms (with lack of temporal homogeneity, semi-linear terms, nonlinear operators, etc).

2. Systems of nonlinear nonlocal dispersive wave equations for internal waves

The aim of this part of the proposed research is the computational study of some systems that appear in the mathematical modelling for the propagation of internal waves. The extensive bibliography (see K.R. Helfrich and W.K. Melville, Long nonlinear internal waves, Annu. Rev. Fluid Mech., 38 (2006), 395–425 and references therein) on observations of internal waves in oceans offers a challenge from the point of view modelling, as well as of mathematical and numerical analysis. The diversity, observed experimentally, of modes of propagation for these internal waves leads to the formulation of different approaches for the corresponding models, which must take into account the parameters of the physical regime associated with the observed data and entail the formation of internal solitary waves. One of the simplest representations assumes that the waves propagate along the interface between two layers of fluids of different density. From this idealized system, different physical hypotheses generate corresponding models, in the form of systems of partial differential equations for the deviation of the interface and the velocity, along with initial and boundary conditions.

3. Nonlinear complex diffusion problems in image processing

One of the mathematical tools to model image filtering processes consists of the restoration of an initial, noisy image given by a function u on some bidimensional domain as the consequence of an evolution from it which is governed by a differential, parabolic type problem along with Neumann or Dirichlet type boundary conditions. The choice of the type of equation depends on the goals of the model (mainly noise cleaning and contour detection) and should attend to three main points: a) Well-posedness of the problem and its numerical treatment (with the aim of controlling the stability of the filtering process); b) the satisfaction of some scale-space properties; c) finally, the choice of right hand side F of the equation should allow to adjust the diffusion according to regularity regions of the image and to detect their contours in an efficient way.

One of the approaches is based on taking this as

F=div(D∇u)

where D is the diffusion tensor. From the linear, isotropic case, corresponding to D=σI, the nonlinear alternatives are focused on the idea reducing the influence of D in regions with large ||∇u|| and with inclusion of anisotropy. Typically rhe numerical simulation here makes use finte elements and volumes along with suitable time intregrators.

Another point of view was introduced by Gilboa and collaborators, (G. Gilboa, N. Sochen and Y. Y. Zeevi, Image enhancement and denoising by complex diffusion processes, IEEE Trans. Pattern. Anal. Mach. Intell., 26(2004), 1020--1036), where a complex diffusion model is proposed with F=e^iθ div(D∇u), for a small phase parameter θ, in such a way that the image u is now represented by a complex function. For small θ, |Im(u)| is large on the contours, justifying the dependence D=D(|Im(u)|), with a positive, decreasing, Perona-Malik type function D. The resulting model is able to detect contours without imposing a dependence of the diffusion tensor on the size of ∇u.