The Computable Differential Equation Lecture ... - Bruce E. Shapiro

20 CHAPTER 1. CLASSIFYING THE PROBLEM
This sets the local variable solution is going to be a list containing a single point
(which itself is described by a list) {t0, y0} . The local variables y and t are then
set equal to to the parameter values y0 and t0 respectively. Next we follow with a
While loop. The format of a While loop is the following:
While[expression,
statement;
statement;
]
.
statement;
When Mathematica gets to this statement, it looks at the value of expression. If
expression is True then it executes each statement in succession. If expression is
False the rest of the While loop is ignored. After executing the full sequence of
statements, Mathematica returns to the top of the loop and evaluates expression
again. The process is repeated until expression has a value of False. In our case
we have the following loop:
While[t < tmax,
y = y + h*f[t, y];
t = t + h;
solution = Append[solution, {t, y}];
];
Here we repeat the loop as long as the expression t < t max is true. Inside the loop
the numbers y and t are updated according to specific formulas, and then the pair
of numbers {t, y} is added to the end of solution. Since the value of t is updated
each iteration, so long as the initial value of t (which is given by t0) is smaller than
tmax , the loop will eventually terminate. If t0>tmax, then the program will have
an infinite loop. A more robust program would test for this possibility and print an
error message before actually attempting to execute the infinite loop. Finally, when
the loop completes we have one final statement,
Return[solution];
which tells us to return the list of values as the “value” of the function. This will
be a list of numbers such as
{{t 0 , y 0 }, {t 1 , y 1 }, . . . , {t n , y n }}
=⇒
for some integer n = t max /h.
To invoke the function one would first define the function f. For example, to
solve the initial value problem
y ′ = cos t − 2t sin t
y(0) = 1
on the interval [0, 4π] with a step size of 0.5, one would type
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007