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

44 CHAPTER 3. APPROXIMATE SOLUTIONS
Since y ′ (t) = f(t, y), we can approximate the left hand side of (3.7) by
y ′ n(t n ) ≈ f(t n , y n ) (3.8)
and hence
y n+1 = y n + h n f(t n , y n ) (3.9)
which is the iteration formula for the Forward Euler Method.
It is often the case that we use a fixed step size h = t j+1 − t j , in which case we
have
t j = t 0 + jh (3.10)
In this case the Forward Euler’s method becomes
y n+1 = y n + hf(t n , y n ) (3.11)
The Forward Euler’s method is sometimes just called Euler’s Method. The application
of Euler’s method is summarized in Algorithm 4.2.
An alternate derivation of equation (3.9) is to expand the solution y(t) in a
Taylor Series about the point t = t n :
y(t n+1 ) = y(t n + h n ) = y(t n ) + h n y ′ (t n ) + h2 n
2 y′′ (t n ) + · · · (3.12)
= y(t n ) + h n f(t n , y( n )) + · · · (3.13)
We then observe that since y n ≈ y(t n ) and y n+1 ≈ y(t n+1 ), then (3.9) follows immediately
from (3.13).
If the scalar initial value problem of equation (3.1) is replaced by a systems of
equations
y ′ = f(t, y), y(t 0 ) = y 0 (3.14)
then the Forward Euler’s Method has the obvious generalization
y n+1 = yn + hf(t n , y n ) (3.15)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007