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

CHAPTER 2. SUCCESSIVE APPROXIMATIONS 27
the IVP:
φ(t) =
∞∑
k=0
where the last step follows from Taylor’s theorem.
t k
k! = et (2.12)
Picard iteration is quite easy to implement in Mathematica; here is one possible
implementation that will print out the first n iterations of the algorithm.
Picard[f ,t , t0 , y0 , n ]:=
Module[{i, y=y0}
Print[Subscript["φ", 0], "=", y0];
For[i=0, is}]) ds;
t0
y=ynext;
Print[Subscript["φ", i+1], "=", y];
];
Return[Expand[y]]
]
Function Picard has five arguments (f, t, t0, y0, n) and two local variables ⇐=
(i, y)
Picard[f ,t , t0 , y0 , n ]:=
Module[{i, y=y0},
. . .
]
The local variable y is initialized to the value of the parameter y0 in the list of
variable declarations. This is equivalent to initializing the value of the variable in
the first line of the program. The first line of the program prints the initial iteration
as φ 0 =value of parameter y 0 ,
Print[Subscript["φ", 0], "=", y0];
The output will be displayed on the console in an “output cell.” The next line of
the program is a For loop. A For statement takes on four arguments:
For[initialization,
test,
increment,
statement;
]
.
statement;
The For loop takes the following actions:
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge