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

Chapter 2
Successive Approximations
2.1 Picard Iteration
The Method of Successive Approximations or Picard Iteration
y ′ = f(t, y) (2.1)
y(t 0 ) = y0 (2.2)
through a sequence of recursive iterations. Any function y = φ(t) that satisfies 2.1
must also solve the integral equation
The method is summarized below in Algorithm 2.1.
✬
∫ t
φ(t) = y 0 + f(s, φ(s))ds
t 0
(2.3)
Algorithm 2.1. Picard Iteration To solve the initial value problem
y ′ = f(t, y), y(t 0 ) = y 0
for the function y(t)
1. input: f(t, y), t 0 , y 0 , n max
2. let φ 0 = y 0
3. For i = 1, 2, . . . , n max
✩
∫ t
let φ i+1 (t) = y 0 + f(s, φ i (s))ds
t 0
4. output: φ i (t)
✫
✪
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