Lecture 40

# Lecture 40 - Example RK4 Method dy = t y y(0 = 1 0 t 1 dt...

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1 0 0 0 4 2 3 3 0 1 0 0 0 0 1 2 ( , ) 0 1.0 0 ( 0.5 , 0.5 ) (0.25, 1.0) (0.25 1) ( 0.5 , 0.5 ) (0. 0.25769 0.2 25, 1.0625) 0.25 1.0625 ( , ) (0.5, 1.128847) 0.5 1.1 0.53123 2884 5 7 6 4 k f t y f t h y k h f f t h y h f f k t h y h f y y k k k k = = = = + + = = = = + + = = = = + + = = = = [ ] 4 0 1 t 2 3 1 1 ( 2 2 ) 1 0 2( ) 2( ) ( 0.5312 0.5) 6 6 1.128885 0.2 ( 0.0018% 57694 3 ) 6 0.25 k h k k k ε + + + + = + + + + = = , (0) 1, 0 1 dy t y y t dt = = ≤ ≤ Example: RK4 Method h = 0.5 f ( t , y )= t * y ^0.5 First step, t 1 = 0.5 t i t i + h/2 t i + h k 4 2 1 3 1 ( 2 2 ) 6 k k k k ϕ = + + + φ k 2 k 3 k 1

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1 1 1 2 1 1 1 3 1 1 2 4 1 1 3 ( , ) 0.5 1.128885 0.531236 ( 0.5 , 0.5 ) (0.75, 1.261697) 0.75 1.261697 ( 0.5 , 0.5 ) (0.75, 1.339495) 0.75 1.339495 ( , 0 ) (1.0, 1.5628 0.842439 .8680 6 9 242 k f t y k f t h y k h f k f t h y k h f k f t h y k h f = = = = + + = = = = + + = = = = + + = [ ] 3 1 1 t 2 0 4 7) 1.0 1.562897 1 ( 2 2 ) 6 1 1.128885 0.531 0.8680 236 2( ) 2( ) (0.5)
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Lecture 40 - Example RK4 Method dy = t y y(0 = 1 0 t 1 dt...

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