Finding the Steady Solution for a PDE Problem

[HalfManHalfAmazing]

I'm wondering if anyone can just run through how this is done. I have the solution so that's now the problem. I just need someone to provide me with the method of finding the steady solution (I can find the transient no problem).

A slender homogeneous conducting bar of uniform cross section lies along the x-axis with ends at x = 0 and x = 1. The lateral surface of the bar radiates heat into the surroundings at temperature zero. The left end is insulated and heat is added through the right end. The initial temperature distributions is u(x,0) = f(x).

Okay so the PDE we have here is:

$$u_t = \alpha^2u_{xx} - u$$
$$u_x(0,t) = 0$$
$$u_x(1,t) = q$$
$$u(x,0) = f(x)$$

And to find the steady solution I do the usual tricks. I know how to get it I just want someone to explain a bit. Thanks.

 

[HalfManHalfAmazing]

Okay well I went ahead and did it myself. I figure I'll explain myself in case anyone wishes:

I start by finding the time indepenent solution - where $$u_t$$ vanishes. I find that I have an DE similar to basic differential equations with solution - $$y = c_1e^{\frac{x}{\alpha}}+c_2e^{-\frac{x}{\alpha}}$$

I want to plug in some boundary conditions and for that I need to differentiate and then plug in.

 

[physics girl phd]

Glad you were able to figure it out!