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Find the lowest order solution for a boundary value problem

  1. Mar 6, 2013 #1
    Hello,
    I need help in solving the problem:

    " find the lowest order uniform approximation to the boundary value problem εy''+y'sinx+ysin(2x)=0. y(0)=(pi), y(pi)=0. "

    what I did:

    y(out)=Ʃ(ε^n)y(n)
    εy''(out)+y'(out)*sinx+y(out)*sin(2x)=0
    for order 0: y'(out)*sinx+y(out)*sin(2x)=0
    with the B.C y(pi)=0---> y(out)=0.

    for the inner region:
    x=εX
    y(x)=Y(in)(X)
    Y(in)=Ʃ(ε^n)Y(n)
    Y''(in)+Y'(in)*sin(εX)+Y(in)*sin(2εX)=0
    sin(εX)≈εX+ O(ε^3)
    for order 0: Y''(in)=0
    Y(in)=AX+B
    with the B.C Y(0)=pi---> Y(out)=AX+(pi).
    the matching:
    x=δz, X=x/ε
    ε-->0
    Y(in)=A*δz/ε+(pi) -->∞
    y(out)=0.
    can't do the matching and find B!

    Where is my mistake???

    Thank you very much!!
     
  2. jcsd
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