ABSTRACT
Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 12.4 Autocatalysis Combustion! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 12.5 Money, Money, Money-A Simple Financial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
12.5.1 Black and Scholes Equation and the Heat Equation . . . . . . . . . . . . . . . . . . 394
As seen in Chapters 1 and 3, the rate at which certain quantities change is dependent on the amount present at a given instant in time. This concept led to the model{
du(t) dt = au(t)
u(0) = u0 (12.1)
where a is a non-zero real number and u0 is a positive real number. The solution to (12.1) is an exponential function which was useful in some situations, but not realistic in others. For example, if we wanted to model the interaction of two chemical or biological species then an unbounded solution, like the exponential, is not meaningful. We should expect some type of limited growth or decline to account for resources, such as space and nutrients.
