ABSTRACT
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 14.1 Turning Up the Heat-Semi-Linear Variants of the Heat Equation . . . . . . . . . . 409 14.2 The Classical Wave Equation with Semilinear Forcing . . . . . . . . . . . . . . . . . . . . . . . . 412 14.3 Population Growth-Fisher’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
14.3.1 Two-Dimensional Fisher’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 14.4 Zombie Apocalypse! Epidemiological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 14.5 How Did That Zebra Get Its Stripes? A First Look at Spatial Pattern
Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 14.5.1 Two-Dimensional Gray-Scott Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
14.6 Autocatalysis-Combustion! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
As in the ODE models explored in Part I, external forces are often more complicated than just functions depending on the independent variables. In fact, they often depend on not only time and position, but also on the solution itself and its partial derivatives. More precisely, we shall consider forcing terms of the form f(x, t, z(x, t)). The forcing term can also depend on the partial derivatives of the unknown function (of order equal to at most one less than the order of the PDE). We shall just consider some examples in this chapter, primarily for visualization purposes.
