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Math Help - Transformation/change of variables in differential equation

  1. #1
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    Transformation/change of variables in differential equation

    Maybe my background is just weak...I was thinking about this for almost 1.5 hours already, but I still end up totally confused. Perhaps this is because I was never able to understand the ideas of a function and change of variables completely...perhaps I have a serious conceptual flaw.
    ========================================

    Consider the following partial differential equation with initial values(IV) and boundary conditions(BC):
    u_t - k u_xx = x + 2t, 1<x<7, t>0
    BC: u(1,t) = u(7,t) = 0
    IV: u(x,0) = x+5
    Our goal is to transform the above to the interval [0,6].
    Let w = x-1.
    Transform the whole problem to the interval w E [0,6]. (write in terms of w)
    ================================================

    u_x = u_w dw/dx = (u_w) (1) = u_w
    u_xx = ...(apply chain rule again) = u_ww
    [On the left side, think of u as u(x,t). On the right side, think of u as u(w,t)]

    BC:
    x=1 <=> w=0
    x=7 <=> w=6
    So the boundary conditions get transformed to u(0,t)=u(6,t)=0 [here think of u as u(w,t)]

    IV:
    We know u(x,0) = x+5
    => u(w,0) = w+5
    [I believe the logic in this step cannot be wrong, consider e.g. f(4z)=cos(4z), now how do we find f(4z-y)? Of course, f(4z-y)=cos(4z-y). How do we find f(z)? Surely, f(z)=cos(z). Right??]

    So my final answer is: [here think of u as u(w,t)]
    u_t - k u_ww = w+1+2t, 0<w<6, t>0
    BC: u(0,t) = u(6,t) = 0
    IV: u(w,0) = w+5

    However, I really have some bad feeling that the result u(w,0) = w+5 is wrong, but I don't know where the mistake is.
    I tried to calculate it in a different way and the answer is the same.
    u(w,0)
    = u(x-1,0) = (x-1)+5
    => u(w,0) = w+5

    Can someone please kindly explain why and where my mistake is? What is the correct answer?

    Any help is greatly appreciated!

    [also under discussion in sos math cyberboard]
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  2. #2
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    Quote Originally Posted by kingwinner View Post
    Maybe my background is just weak...I was thinking about this for almost 1.5 hours already, but I still end up totally confused. Perhaps this is because I was never able to understand the ideas of a function and change of variables completely...perhaps I have a serious conceptual flaw.
    ========================================

    Consider the following partial differential equation with initial values(IV) and boundary conditions(BC):
    u_t - k u_xx = x + 2t, 1<x<7, t>0
    BC: u(1,t) = u(7,t) = 0
    IV: u(x,0) = x+5
    Our goal is to transform the above to the interval [0,6].
    Let w = x-1.
    Transform the whole problem to the interval w E [0,6]. (write in terms of w)
    ================================================

    u_x = u_w dw/dx = (u_w) (1) = u_w
    u_xx = ...(apply chain rule again) = u_ww
    [On the left side, think of u as u(x,t). On the right side, think of u as u(w,t)]

    BC:
    x=1 <=> w=0
    x=7 <=> w=6
    So the boundary conditions get transformed to u(0,t)=u(6,t)=0 [here think of u as u(w,t)]

    IV:
    We know u(x,0) = x+5
    => u(w,0) = w+5
    [I believe the logic in this step cannot be wrong, consider e.g. f(4z)=cos(4z), now how do we find f(4z-y)? Of course, f(4z-y)=cos(4z-y). How do we find f(z)? Surely, f(z)=cos(z). Right??]

    So my final answer is: [here think of u as u(w,t)]
    u_t - k u_ww = w+1+2t, 0<w<6, t>0
    BC: u(0,t) = u(6,t) = 0
    IV: u(w,0) = w+5

    However, I really have some bad feeling that the result u(w,0) = w+5 is wrong, but I don't know where the mistake is.
    I tried to calculate it in a different way and the answer is the same.
    u(w,0)
    = u(x-1,0) = (x-1)+5
    => u(w,0) = w+5

    Can someone please kindly explain why and where my mistake is? What is the correct answer?

    Any help is greatly appreciated!

    [also under discussion in sos math cyberboard]
    Wouldn't it be

    u(w,0) = w+6?

    Check the values at the boundaries ( x = 1, w = 0 and x = 7, w = 6)
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  3. #3
    Senior Member
    Joined
    Jan 2009
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    Maybe it's u(w,0) = x+6. But can you please tell me where my mistake is? Also, HOW did you get u(w,0) = x+6?

    We know u(x,0) = x+5
    => u(w,0) = w+5 ?
    I believe the logic in this step cannot be wrong, consider e.g. f(4z)=cos(4z), now how do we find f(4z-y)? Of course, f(4z-y)=cos(4z-y). How do we find f(z)? Surely, f(z)=cos(z). Right??


    u(w,0)
    = u(x-1,0) = (x-1)+5
    => u(w,0) = w+5 ?
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