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Thread: problems

  1. #1
    Sep 2009


    1) a) find the general solution u(x,t) to

    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
    Mar 2008
    hello mathgeek,

    You can separate this equation using the following assumption:

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

    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:

    $\displaystyle XT'+xX'T=XT$

    After dividing by XT and rewriting this becomes:

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

    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:

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

    The solution is now:

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

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

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