Is any direct Adams Moulton formula or how to related those two formula which i am mention. Are

Ali Marshall

Ali Marshall

Answered question

2022-04-21

Is any direct Adams Moulton formula or how to related those two formula which i am mention.
Are those formulas the same? When I am programming for Adams Moulton fourth order methods I use the 2nd formula which I mentioned in the 2nd image. Is it correct?
If correct, please explain me.
Yn+1=Yn+h24[9f(xn+1,yn+1)+19f(xn,yn)5f(xn11yn1)+f(xn2,yn2)

Answer & Explanation

zavesiljid

zavesiljid

Beginner2022-04-22Added 21 answers

STEP 1
The 5th order Adams-Moulton formula is
yn+1=yn+h720[ 251·f(xn+1,yn+1)+646·f(xn,yn) 264·f(xn1,yn1)+106·f(xn2,yn2) 19·f(xn3,yn3)]
This is an implicit equation for yn+1. One could iterate it directly as
yn+1[k+1]=yn+h720[251·f(xn+1,yn+1[k])+Rn],   Rn= 646·f(xn,yn)264·f(xn1,yn1) +106·f(xn2,yn2)19·f(xn3,yn3)
or try to accelerate this by capturing the majority of the linear behavior of f close to (xn,yn) in some Jacobian Jn, to get the linear system
STEP 2
(Ih·251720·Jn)yn+1[k+1]=yn+h720[251·[f(xn+1,yn+1[k])Jn·yn+1[k]]+Rn]
For some special case of f this might result in your second formula.

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