(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I need to go into this test with great aplomb.

In each of Problems 1 through 8 find the steady-state solution of the heat conduction equation a^{2}u_{xx}=u_{t}that satisfies the given boundary conditions.

1. u(0,t)=10, u(50,t)=40

....

3. u_{x}(0,t)=0, u(L,t)=0

2. Relevant equations

Will be using separation of variables; so assume u(x,t)=X(x)T(t)

3. The attempt at a solution

After a long time---that is, as t approaches ∞---I anticipate that a steady state temperature distribution v(x) will be reached.

Then u(x,t) will just be a^{2}v''(x)=v'(t). Since v is not a function of t, v'(t) = 0 and v''(x)=0.

v(x) must be a 1^{st}degree polynomial: v(x) = Ax + B.

Notice that the initial conditions to problem one imply that

X(0)T(t)=10, X(50)T(t)=40.

I don't want T(t) to be something trivial, so

X(0)=10, X(50)=40

----> 10= 0 + B ---> B = 10

40=A*50 + 10 ----> A = (40-30) / 50 = 1/5

-----> v(x) = x/5 + 10

That's the steady state heat distribution, which is all the problem asked for.

Now problem 3. Notice the sub x.

Hmmmm..... need to put the pieces of this puzzle together.

u_{x}(0,t)=0 is saying that the rod is insulated at x=0.

So X'(0)=0.

The other initial condition says X(50)=40.

Where do I go from here?

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# Homework Help: A Heat Conduction Problem (Final exam on Monday!)

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