The example above was really not much of a differential equation, because the solution was obtained by straightforward integration. Equations of the form u'(t) = f (t) (B.7)
arise in situations where we can explicitly specify the derivative of the unknown function u. Usually, the derivative is specified in terms of the solution itself. Consider, for instance, population growth under idealized conditions as modeled in Chapter 5.1.4. We introduce the symbol v for the number of individuals at time t (v corresponds to xn in Chapter 5.1.4). The basic model for the evolution of v is (5.9):
As mentioned in Chapter 5.1.4, r depends on the time difference ^r = Tj — Ti_1: the larger ^r is, the larger r is. It is therefore natural to introduce a growth rate a that is independent of ^r: a = r/^r. The number a is then fixed regardless of how long jumps in time we take in the difference equation for vj. In fact, a equals the growth in percent, divided by 100, over a time interval of unit length.
The difference equation now reads
Vj = Vj-1 + a^rVj-1. Rearring this equation we get
Assume now that we shrink the time step ^t to a small value. The left-hand side of (B.9) is then an approximation to the time-derivative of a function V(T) expressing the number of individuals in the population at time t. In the limit ^t ^ 0, the left-hand side becomes the derivative exactly, and the equation reads v' (t) = av(T). (B.10)
As for the underlying difference equation, we need a start value V(0) = v0. We have seen that reducing the time step in a difference equation to zero, we get a differential equation.
Many like to scale an equation like (B.10) such that all variables are without physical dimensions and their maximum absolute value is typically of the order of unity. In the present model, this means that we introduce new dimensionless variables v T
v0 a and derive an equation for u(t). Inserting v = v0u and t = at in (B.10) gives the prototype equation for population growth:
When we have computed the dimensionless u(t, we can find the function v(t) as v(t) = v0u(T/a).
We shall consider practical applications of population growth equations later, but let's start by looking at the idealized case (B.11).
Analytical Solution. Our differential equation can be written in the form du
— = u, dt which can be rewritten as du — = dt, u and then integration on both sides yields ln(u) = t + c, where c is a constant that has to be determined by using the initial condition. Putting t = 0, we have ln(u(0)) = c, c = ln(1) = 0, ln(u) = t, u(t) = et. (B.13)
hence and then so we have the solution
Let us now check that this function really solves (B.7, B.11). Obviously, u(0) = eo = 1, so (B.11) is fine. Furthermore u' (t) = et = u(t), thus (B.7) also holds.
Numerical Solution. We have seen that we can find a formula for the solution of the equation of exponential growth. So the problem is solved, and it is trivial to write a program to graph the solution. We will, however, go one step further and develop a numerical solution strategy for this problem. We don't really need such a method for this problem since the solution is available in terms of a formula, but as mentioned earlier, it is good practice to develop methods for problems where we know the solution; then we are more confident when we are confronted with more challenging problems.
Suppose we want to compute a numerical approximation of the solution of u' (t) = u(t) (B.14)
equipped with the intial condition u(0) = 1. (B.15)
We want to compute approximations from time t = 0 to time t = 1. Let n ^ 1 be a given integer, and define
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