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Math Help - the heat equation

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    41

    Question the heat equation

    A cold steel bar of length
    L = 1 is held in mid-air by two posts which can be considered as a heat leak to the lab with a reference temperature Tref = 6. Initially (time t = 0), the temperature distribution across the steel bar is given by
    T(t = 0; x) = -3x + 17x^2- 10x^3 0<= x <= 1

    The question which i am struggling on is:
    Q) Show that
    T(x; t) = Tref is a solution of the heat equation including the boundary conditions. This is the state the steel bar will approach for large times. How much deviates the average temperature from Tref at t = 0:5?

    Any help will be great.
    Thanks


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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    This looks ill-posed to me: the initial condition and boundary conditions contradict each other. And therefore, while the solution u(x,t)=6 is a solution to the DE and boundary conditions, it is not a C^2 solution which satisfies the given IVP. Also, looks better if you write it as:

    u_t=k u_{xx},\quad 0\leq x\leq 1,\quad t>0

    u(0,0)=6,\quad u(1,0)=6

    u(x,0)=-3x+17x^2-10x^3
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