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

CHAPTER 4. IMPROVING ON EULER’S METHOD 85
delta = yvaln-yvalp;
If[Abs[delta] < tol, Break[]];
];
yval = yvaln;
AppendTo[r, {time + h, yval}];
];
Return[r];
]
✬
✩
Algorithm 4.2. Backward Euler Method Using Newton’s
Method To solve the initial value problem
y ′ = f(t, y), y(t 0 ) = y 0
on an interval [t 0 , t max ] with a fixed step size h.
1. input: f(t, y), t 0 , y 0 , h, t max , tol
2. output: (t 0 , y 0 )
3. let t = t 0 , y = y 0
4. while t < t max
(a) let i = 0, y (0)
n
(b) Repeat:
until |y (i+1)
n
y (i+1)
n
(c) let y = y (i+1)
n
(d) let t = t + h
= y
i = i + 1
(e) let t n = t, y n = y
= y (i)
n
− y (i)
n | < tol
− y(i) n
(f) output: (t 0 , y 0 ), . . . , (t n , y n )
− y n−1 − hf(t, y (i)
n )
1 − hf y (t, y (i)
n )
✫
✪
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge