Deposition, diffusion and convection: BLUES approximants and some exact results

An analytic procedure for solving nonlinear differential equations, the BLUES function method, is studied. It is first implemented for differential equations that can be reduced to ordinary differential equations (ODEs) with one independent variable. When an inhomogeneous source (or sink) is present in the equation, the BLUES function method provides a natural way to obtain approximate solutions. In this setup, different systems from nonlinear physics and other sciences are investigated, particularly in the context of nonlinear travelling waves within fluid dynamics and biophysics. The BLUES method is applied to a fractional ordinary differential equation (FDE) in the context of heat flow within a semi-infinite rod. An initial comparison with another iterative method is carried out, showing that the BLUES method possesses a larger region of convergence.Next, the method is extended to the realms of partial differential equations (PDEs) and systems of coupled nonlinear differential equations (CDEs). In both of these areas, the method is reformulated slightly to accommodate the particulars of that area, and is consequently studied first by means of simple examples and finally by means of a model for interface growth under the influence of shear flow. Within the field of (coupled) PDEs, the role of the external source is now fulfilled by the initial condition through multiplication with a point source at time t = 0. A comprehensive comparison with preexisting methods is performed and it is shown that in many cases the BLUES function method is an ideal candidate when choosing between iterative methods. When studying systems of coupled ordinary differential equations, the linear operator can often be sensibly chosen in such a way that it includes the fixed points of the nonlinear system, greatly increasing the BLUES function method's region and speed of convergence.Lastly, the hierarchical random deposition model (HRDM) is studied. This deposition process takes place in a viscous medium so that the particles hit the surface in order of size. The size follows a hyperbolic distribution, allowing the larger particles to hit the substrate first. Additionally, square ``holes'' following the same distribution can be excavated. In particular, the connection between the (non)-proliferation of coastal points (level sets) and lateral percolation is investigated by analytical means and numerical simulations. It is found that exactly at the percolation threshold, the number of coastal points (or coastlines in two dimensions) exhibits logarithmic fractal behaviour, increasing linearly with increasing generation. Next, the dynamics of the deposition are adjusted to allow particles to stick sideways to preexisting material, transforming the hierarchical random deposition model into a hierarchical ballistic deposition model (HBDM). The latter is clearly still a random model but the sideways growth introduces lateral correlations between different columns. The surface length increment, the fraction of closed-off voids and the associated porosity are studied by means of numerical simulations and analytical approximations.

Publication year:2021

Accessibility:Open