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

CHAPTER 1. CLASSIFYING THE PROBLEM 19
Implementing Numerical Methods with Mathematica
Finally we note that Mathematica contains a full high level programming language
comparable to languages such as C and LISP. We will be using this language to
implement our own versions of various algorithms as they arise.
The following piece of code gives an implementation of Euler’s method using the
traditional C-language like structures in Mathematica. It returns a list of number
pairs {{t 0 , y 0 }, {t 1 , y 1 }, . . . }.
EulersMethod[f , t0 , y0 , tmax , h ] :=
Module[{y, t, solution},
solution = {{t0, y0}};
y = y0;
t = t0;
While[t < tmax,
y = y + h*f[t, y];
t = t + h;
solution = Append[solution, {t, y}];
];
Return[solution];
]
Let us dissect this program. To begin with we have to tell Mathematica that we are
defining a function:
⇐=
EulersMethod[f , t0 , y0 , tmax , h ] :=
Module[{y, t, solution},
· · · stuff · · ·
]
This says that we are defining a new function EulersMethod that takes on 5 arguments
(f, t0, y0, tmax, h) and has 3 local variables (y, t, solution). In the
function declaration the names of the parameters always end with an underscore
character ; this tells Mathematica that these are the names of variables that are
passed to the program. Within the function the underscore is not used. These arguments
and the three local variables are only defined within the scope of the program,
so that any other variable inside EulersMethod can refer to them directly, but no
variable outside of the function definition knows of their existence. So if there is
are global variable f or y , their values are not affected by the function definition.
The next three lines initialize our local variables equal to certain combinations of
argument values.
solution = {{t0, y0}};
y = y0;
t = t0;
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge