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Math Help - problems

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    2

    problems

    1) a) find the general solution u(x,t) to
    ut+xux=u

    b) what is the soln that satisfies the auxilliary conidtion u(x,0)=x^2?

    2) suppose fx is a conitinuously differentiable vector field in R3 that satisfies the bound

    |fx|<= 1/1+|x|^3

    show that the triple integral (radiant).(Fdx)=0.

    3) a solid, homogeneous object occupies the region D is contained R^3, and is completely insulated. its initial temperature in given by u(x,0)=f(x), for some function f. so u satisfies the heat equation ut=K().(delta u) with boundary condition partial of u/partial n=0 on partial of D. after a long time, the object reaches a steady, uniform temperature. in terms of f, what is this temperature?
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  2. #2
    Junior Member
    Joined
    Mar 2008
    Posts
    61
    hello mathgeek,

    You can separate this equation using the following assumption:

    u(x,y)=X(x)\cdot T(t)

    Meaning that we assume that the solution can be written as the product of two functions X and T, each respectively depending on only x and t. After substituting this in the equation you get:

    XT'+xX'T=XT

    After dividing by XT and rewriting this becomes:

    x\frac{X'}{X}=1-\frac{T'}{T}

    Because the left and right hand side are only depending on x and t respectively, these must be equal to a constant, say [imath]\alpha[/imath] This gives rise to two ordinary differential equations which can be solved fairly easy. We have for these:

    X(x)=K_2x^{\alpha}
    T(t)=K_1e^{(1-\alpha)\cdot t}

    The solution is now:

    u(x,t)=K\cdot x^{\alpha}\cdot e^{(1-\alpha)\cdot t}

    Can you take it from here to solve the next part of the question?

    coomast
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