Deterministic solutions of fractal growth

We study the problem of pattern formation on a lattice. A maximum-likelihood growth algorithm is formulated to supplement the integral solution of Laplace's equation. This allows us to study diffusion-limited aggregation and viscous flow in porous media. Our deterministic solution for the aggregation process displays the same fractal dimension (D\ensuremath{\approxeq}1.7) as Monte Carlo diffusion-limited aggregation. We also compare our solution to Laplace's equation with the experimentally observed pattern in one limit of a Hele-Shaw cell experiment. Lastly, we present a second deterministic algorithm which results in a pattern with tips which appear parabolic.