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Math Help - Advanced Ordinary Differential Equation

  1. #1
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    Advanced Ordinary Differential Equation

    Consider the boundary value problem
    Ey"+y'=2t, y(0) =y(1)=1

    For the function y=y(t), where E is a small parameter, 0<E<<1. You are given that the function y has a boundary layer close to the point t=0.

    Q, Find the lowest order term, yo(t), in the outer expansion:

    y(t) = yo(t) = Ey1(t) + ... (*)

    A, Ok here's my workings...

    Subbing (*) into boundary value problem i get:

    E(yo" + Ey1" + ...) + (yo'+Ey1' + ...) = 2t

    y(0) = yo(0) + Ey1(0) + ... = 1
    y(1) = yo(1) + Ey1(1) + ... = 1

    Therefore equating coefficients:

    E^0: yo' = 2t, yo(0) = yo(1) = 1

    Integrating yo' = 2t gives me: yo = t^2 + C

    So then to work out the constant I then apply the initial conditions either, either y(0) = 1 or y(1)=1?, but obviously this would give me two different values for c? In my solutions they use y(1) = 1 which makes C = 0, so that yo = t^2. Could anyone shed any light upon which this is though, would be much appreciated.

    Thanks.
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  2. #2
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    Quote Originally Posted by bower87 View Post
    Consider the boundary value problem
    Ey"+y'=2t, y(0) =y(1)=1

    For the function y=y(t), where E is a small parameter, 0<E<<1. You are given that the function y has a boundary layer close to the point t=0.

    Q, Find the lowest order term, yo(t), in the outer expansion:

    y(t) = yo(t) = Ey1(t) + ... (*)

    A, Ok here's my workings...

    Subbing (*) into boundary value problem i get:

    E(yo" + Ey1" + ...) + (yo'+Ey1' + ...) = 2t

    y(0) = yo(0) + Ey1(0) + ... = 1
    y(1) = yo(1) + Ey1(1) + ... = 1

    Therefore equating coefficients:

    E^0: yo' = 2t, yo(0) = yo(1) = 1

    Integrating yo' = 2t gives me: yo = t^2 + C

    So then to work out the constant I then apply the initial conditions either, either y(0) = 1 or y(1)=1?, but obviously this would give me two different values for c? In my solutions they use y(1) = 1 which makes C = 0, so that yo = t^2. Could anyone shed any light upon which this is though, would be much appreciated.

    Thanks.
    I found this for you. It has an example close to the one you're working (page 4).

    http://www.maths.ox.ac.uk/filemanager/active?fid=9030
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