Lecture Notes in Differential Equations - Bruce E. Shapiro

366 LESSON 33. NUMERICAL METHODS
Figure 33.4: Result of the forward Euler method to solve y ′ = −100(y −
sin t), y(0) = 1 with h = 0.001 (top), h = 0.019 (middle), and h = 0.02
(third). The bottom figure shows the same equation solved with the backward
Euler method for step sizes of h = 0.001, 0.02, 0.1, 0.3, left to right
curves, respectively
.
1
0.8
0.6
0.4
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1
0 0.5 1 1.5 2 2.5 3
0.5
0
0.5
1
2
1.5
1
0.5
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0.5
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1
0.8
0.6
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0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Improving Euler’s Method
All numerical methods for initial value problems of the form
variations of the form
y ′ (t) = f(t, y), y(t 0 ) = y 0 (33.40)
y n+1 = y n + φ(t n , y n , . . . ) (33.41)
for some function φ. In Euler’s method, φ = hf(t n , y n ); in the Backward’s
Euler method, φ = hf(t n+1 , y n+1 ). In general we can get a more accurate