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Math Help - Working with initial and boundary conditions in PDEs

  1. #1
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    Working with initial and boundary conditions in PDEs

    Hello,

    I have this problem:

    A homogeneous body occupying the solid region D is completely insulated. Its initial temperature is f(x). Find the steady-state temperature that it reaches after a long time. (Hint: No heat is gained or lost)



    From what I understand the initial condition is written as u(x, t_o) = f(x)
    and since the region is insulated the neumann condition is interpreted to be \partial{u}/\partial{n} = 0

    At this point, how would i a derive a means to define a steady-state temperature?
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  2. #2
    Junior Member
    Joined
    Mar 2008
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    Quote Originally Posted by dlbsd View Post
    Hello,

    I have this problem:

    A homogeneous body occupying the solid region D is completely insulated. Its initial temperature is f(x). Find the steady-state temperature that it reaches after a long time. (Hint: No heat is gained or lost)



    From what I understand the initial condition is written as u(x, t_o) = f(x)
    and since the region is insulated the neumann condition is interpreted to be \partial{u}/\partial{n} = 0

    At this point, how would i a derive a means to define a steady-state temperature?
    Start from the basic conductivity equation (in 1D). This is:
    \frac{\partial u}{\partial t}=\frac{\lambda}{\rho c_p}\left(\frac{\partial^2 u}{\partial x^2} \right)
    Steady state means:
    \frac{\partial u}{\partial t}=0
    giving now:
    \frac{\partial^2 u}{\partial x^2}
    Because this is only depending on x, we can write ordinary derivatives instead of partials and the solution is:
    u=A\cdot x+B

    Applying now the boundary condition, you will have a simple result. Can you do this and interpret physically what it is? After that how would you calculate this value if f(x) was given? Expanding it to more dimensions will be fairly straightforward once you solve this 1D case.
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