y′(t)=f(t,y), a<=t<=b, y(a)=alpha

Brandon Monroe 2022-08-07 Answered
y ( t ) = f ( t , y ) , a t b , y ( a ) = α
apply for the extrapolation technique
we let h 0 = h 2 and use Euler method with w 0 = α
w 1 = w 0 + h 0 f ( a , w 0 )
and then midpoint method
t i 1 = a , t i = a + h 0 = a + h 2 to produce
approximation to y ( a + 2 h 0 )
w 2 = w 0 + 2 h 0 f ( a + h 0 , w 1 )
y 1 , 1 = 1 2 [ w 2 + w 1 + h 0 f ( a + 2 h 0 , w 2 ) ]
then why this form results O ( h 0 2 ) approximation to y ( t 1 )
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Answers (2)

Michelle Chavez
Answered 2022-08-08 Author has 13 answers
This looks more like an order-blackuced RK3 method. Using the standard Runge-Kutta notation, you get
k 1 = h f ( a , w 0 ) , w 1 = w 0 + 1 2 k 1 k 2 = h f ( a + 1 2 h , w 0 + 1 2 k 1 ) , w 2 = w 0 + k 2 k 3 = h f ( a + h , w 0 + k 2 ) y 1 , 1 = 1 2 ( w 1 + w 2 + 1 2 k 3 ) = w 0 + 1 4 k 1 + 1 2 k 2 + 1 4 k 3
This then gives the Butcher tableau
c 1 c 2 a 21 c 3 a 31 a 32 b 1 b 2 b 3 = 0 1 2 1 2 1 0 1 1 4 1 2 1 4
where the first and second order conditions
b 1 + b 2 + b 3 = 1 c 1 b 1 + c 2 b 2 + c 3 b 3 = 1 2
are both satisfied.
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Trevor Rush
Answered 2022-08-09 Author has 6 answers
The solution to the initial value problem has the series
y ( a + h ) = α + f ( a , α ) h + f t ( a , α ) + f ( a , α ) f y ( a , α ) 2 h 2 + O ( h 3 )
where f t = f / t and f y = f / y. On the other hand, compute the series for y 11 and you should also find
y 1 , 1 = α + f ( a , α ) h + f t ( a , α ) + f ( a , α ) f y ( a , α ) 2 h 2 + O ( h 3 )
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