[Math] The steady state solution of a boundary value problem.

boundary value problempartial differential equations

From a question I posted earlier:

Okay, my question that I have to solve is:

u_tt - ku_xx = x      0<x<1 , t>0

u(0,t) = 1      t>0
u(1,t) = 0

I know that steady state solutions do not depend on time.

I just can't figure out how to start solving this. All of the examples
that we have in our textbook only have u_t and not u_tt. So I don't
really know how I should start this.

I'm not asking for you to solve the problem. I just need help on how
to get it started.

If anyone can help me it would be much apprieciated. Thanks.

Someone's advice was:

A steady state solution would be one for which  ut = 0 . If  ut = 0 , so is  utt . That is what you would use in the partial differential equation, so you now have an ordinary differential equation in  x  to solve. (This is like having a "stationary solution" for an ODE, but here it's a function of  x  , rather than just a number.) –  RecklessReckoner

So using their advice i got:

u"=-x/k and then i integrated it twice to get u = -(1/6k)(x^3)+c but i still need to apply the boundary conditions.

So if i used it from 0 -> 1, then c = 1/6k

but i am like so confused right now. I am clearly doing something wrong here. Because I dont even understand what the answer would be.

Would it be u = -1/6k(x^3-1)???

Please help. My entire assignment is based around this and i cant even get through one.

Best Answer

Hint. If you integrate $u''=-x/k$ once, you obtain $u'=\frac{-x^2}{2k}+C_1$ (with a single integration constant $C_1$). Now integrate this once more. How many integration constants do you have now?

Now apply the boundary conditions...

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