ABSTRACT
This chapter contains nonlinear models of epidemics, image restoration and value of option contracts. All three applications have diffusion-like terms, and so mathematically they are similar to the models in the previous chapters. In the epidemic model the unknown concentrations of the infected populations will depend on both time and space. A good reference is the second edition of A. Okubo and S. A. Levin [19]. Image restoration has applications to satellite imaging, signal processing and fish finders. The models are based on minimization of suitable real valued functions whose gradients are similar to the quasi-linear heat diffusion models. An excellent text with a number of MATLAB codes has been written by C. R. Vogel [26]. The third application is to the value of option contracts, which are agreements to sell or to buy an item at a future date and given price. The option contract can itself be sold and purchased, and the value of the option contract can be modeled by a partial differential equation that is similar to the heat equation. The text by P. Wilmott, S. Howison and J. Dewynne [28] presents a complete derivation of this model as well as a self-contained discussion of the relevant mathematics, numerical algorithms and related financial models.
