Long insulated copper rod with 2 temperatures find T(x,t)

[samee]

Homework Statement

A long copper rod with insulated lateral surface has its left end maintained at a temperature of 0C and its right end, at x=2m , maintained at 100C . Determine the temperature T as a function of time and coordinate if the initial condition is given by

T(x,0)={ 100x, 0<x<1
________100, 1<x<2

Homework Equations

So I'm assuming this question is also actually just a diffusion equation or a wave equation, since that's what the rest of our homework was on. Alpha2uxx=ut
and
u(x,t)=X(x)T(t)=(C₁coskx+C₂sinkx)e^-K2alpha2t+C₃+C₄x

The Attempt at a Solution

I asked another question like this one... So here I have a rod with a temperature difference. These differences are being maintained, so I guess what I'm doing i assuming that it's in equilibrium and the temperature isn't shifting at all. I'm asked for the temperature as a function of time and coordinate, but that doesn't make sense if time isn't a factor. In the initial condition it certainly is not a factor. So I guess I need to set boundary conditions as given?

I know u(x,0)={X(x)100x 0<x<1
_____________X(x)100 1<x<2

and 0<x<2 for the whole problem.

Now, I really need to set my boundary conditions, right? I guess I can say that since the rod is finite, and 0<x<2, anything less than zero or greater than 2 is zero. Therefore;

u(0,t)=0 and u(2,t)=0.

If u(0,t)=(C₁)e^-K2alpha2t+C₃=0
This would mean that C₁=0 and C₃=0, so

u(x,t)=(C₂sinkx)e^-K2alpha2t+C₄x=X(x)T(t)If u(2,t)=(C₂sink2)e^-K2alpha2t+C₄2=0
This would mean that C₄=0 as well. Also,
(C₂sink2)e^-K2alpha2t=0
If C₂ IS NOT 0, then sin2k=0, and 2k=npi. Therefore k=npi/2 and...

u(x,t)=C₂sin(xnpi/2)e^{-(npialpha/2)2t}

SOOOO! IF I did my boundary conditions right, I can then plug it back into the original equation and get;

u(x,t)={ X(x)=(1/100x)C₂sin(xnpi/2)e^{-(npialpha/2)2t} for 0<x<1
________X(x)=(1/100)C₂sin(xnpi/2)e^{-(npialpha/2)2t} for 1<x<2NOW what do I do next?

 

[rude man]

?

Your rod is not in thermal equilibrium unti t = ∞. It starts out per your initial condition
T(x,0) and obviously ends up with T varying linearly with x, assuming temperature-independent conductivity, density and specific heat.

I haven't done the problem but I would assume separation-of-variables solution of the heat equation as you state. And hope like hell!

 

[LCKurtz]

You likely need more help than I can give you here and I'm not sure I should bother with this since you abandoned the last thread I tried to help you. But I will see how it goes...

Homework Statement

A long copper rod with insulated lateral surface has its left end maintained at a temperature of 0C and its right end, at x=2m , maintained at 100C . Determine the temperature T as a function of time and coordinate if the initial condition is given by

T(x,0)={ 100x, 0<x<1
________100, 1<x<2

Homework Equations

So I'm assuming this question is also actually just a diffusion equation or a wave equation,

So you are going to just guess which it is?

since that's what the rest of our homework was on. Alpha2uxx=ut
and
u(x,t)=X(x)T(t)=(C₁coskx+C₂sinkx)e^-K2alpha2t+C₃+C₄x

Where did that "solution" come from? It doesn't look correct. Do you know what is missing? Please show your steps to get that or, if you copied it from somewhere, fix it.

The Attempt at a Solution

I asked another question like this one... So here I have a rod with a temperature difference. These differences are being maintained, so I guess what I'm doing i assuming that it's in equilibrium and the temperature isn't shifting at all. I'm asked for the temperature as a function of time and coordinate, but that doesn't make sense if time isn't a factor.

But it isn't in equilibrium. The problem clearly states to determine u(x,t) as a function of time $t$ and position $x$.

In the initial condition it certainly is not a factor. So I guess I need to set boundary conditions as given?

I know u(x,0)={[STRIKE]X(x)[/STRIKE]100x 0<x<1
_____________[STRIKE]X(x)[/STRIKE]100 1<x<2

and 0<x<2 for the whole problem.

That is the initial condition. I have crossed out what shouldn't be there.

Now, I really need to set my boundary conditions, right? I guess I can say that since the rod is finite, and 0<x<2, anything less than zero or greater than 2 is zero.

Any value of $x$ outside of [0,2] is irrelevant to this problem. There is no rod there.

Therefore;

u(0,t)=0 and u(2,t)=0.

If u(0,t)=(C₁)e^-K2alpha2t+C₃=0
This would mean that C₁=0 and C₃=0, so

Why? When I put $t=0\$in there I get $C_1+C_3 = 0$ which just means $C_1=-C_3$.

u(x,t)=(C₂sinkx)e^-K2alpha2t+C₄x=X(x)T(t)


If u(2,t)=(C₂sink2)e^-K2alpha2t+C₄2=0
This would mean that C₄=0 as well. Also,
(C₂sink2)e^-K2alpha2t=0
If C₂ IS NOT 0, then sin2k=0, and 2k=npi. Therefore k=npi/2 and...

u(x,t)=C₂sin(xnpi/2)e^{-(npialpha/2)2t}

SOOOO! IF I did my boundary conditions right, I can then plug it back into the original equation and get;

u(x,t)={ X(x)=(1/100x)C₂sin(xnpi/2)e^{-(npialpha/2)2t} for 0<x<1
________X(x)=(1/100)C₂sin(xnpi/2)e^{-(npialpha/2)2t} for 1<x<2


NOW what do I do next?

Have you studied Fourier Series? Do you know what is missing if you are going to use Fourier Series? There may be other issues but that's enough for now. Please respond to each comment if you want to continue.