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Math Help - I need help with 2-step Adams-Moulton method please!?

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    I need help with 2-step Adams-Moulton method please!?

    Hey Everyone I could really use some help answering this question as I have tried to answer it several times and I can't seem to understand exactly what to do. so if someone could please show me a step by step calculation

    The initial value problem y'= f(x,y), y(X0)=Y0 is to be solved numerically using the 2-step Adams-Moulton method

    Yi+1 - Yi = h/12 [ (5 Fi+1) + (8 Fi - Fi -1)

    where Fi= f(x, y) . Find the local truncation error of this method and hence determine its order.

    Thanks for your help everyone, much appreciated :-)
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    Quote Originally Posted by hazeleyes View Post
    Hey Everyone I could really use some help answering this question as I have tried to answer it several times and I can't seem to understand exactly what to do. so if someone could please show me a step by step calculation

    The initial value problem y'= f(x,y), y(X0)=Y0 is to be solved numerically using the 2-step Adams-Moulton method

    Yi+1 - Yi = h/12 [ (5 Fi+1) + (8 Fi - Fi -1)

    where Fi= f(x, y) . Find the local truncation error of this method and hence determine its order.

    Thanks for your help everyone, much appreciated :-)
    You need to expand all of these in a taylor series.

    If h is your stepsize then t_{i+1}=t_i+h

    so

    y(t_{i+1})=y(t_i+h)=y(t_i)+hy'(t_i)+\frac{h^2}{2!}  y''(t_i)+\frac{h^3}{3!}y'''(t_i)+\mathcal{O}(h^4)

    Now use the definition of the ODE to get

    5f_{i+1}=5y'(t_i+h)=5\left(y'(t_i)+hy''(t_i)+\frac  {h^2}{2!}y'''(t_i)+\frac{h^3}{3!}y^{(4)}(t_i)+\mat  hcal{O}(h^5) \right)


    -f_{i-1}=-y'(t_i-h)=-\left(y'(t_i)-hy''(t_i)+\frac{h^2}{2!}y'''(t_i)-\frac{h^3}{3!}y^{(4)}(t_i)+\mathcal{O}(h^5) \right)

    8f_{i}=8y'(t_i)

    Now plug all of this in and see how well it matches

    On the RHS I get

    hy'(t_i)+\frac{h^2}{2!}y''(t_i)+\frac{7h^3}{24}y''  '(t_i)+\mathcal{O}(h^4)

    so they only match up to the 2nd orderterms
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