Steady-State Solution to PDE

Metronome

Junior Member
Joined
Jun 12, 2018
Messages
64
I am trying to solve the following problem...

nAaXVUu.jpg


I know how to use separation of variables to solve the PDE without the forcing function [imath]-g[/imath], so I assume this problem should be solved by adding a particular solution to that homogeneous (no forcing function) solution, but it is not clear that the problem is leading me in that direction. The explanation of steady-state solutions in this chapter is as follows...

ZXfRcOx.jpg

However, this doesn't seem to help with this problem, at least if the boundary conditions referenced in part (a) are the ones later stated in part (b), because these boundary conditions are already homogeneous, so the steady-state solution is [imath]u(x,\ t) = 0[/imath].

I also do not know whether [imath]g[/imath] is a constant (the Newtonian constant of gravitation perhaps?) or something else, so I wouldn't know what to guess in order to use Undetermined Coefficients.

Can I get some clarification on how to start this problem?
 

HallsofIvy

Elite Member
Joined
Jan 27, 2012
Messages
7,685
A "steady state" solution is, as stated here, one that does not change over time. Here the two independent variables are "x", position, and "t", time. The equation is \(\displaystyle u_{tt}= c^2u_{xx}- g\) and, because we want the steady state solution, \(\displaystyle u_{tt}= 0\) so the equation is simply \(\displaystyle c^2u_{xx}- g= 0\) or \(\displaystyle u_{xx}= \frac{g}{c^2}\).
Just integrate, with respect to x, twice.

Yes, of course, c and g are constants.
 

Metronome

Junior Member
Joined
Jun 12, 2018
Messages
64
Okay, so this means the steady state solution is [imath]v = \frac{gx^2 - Lgx}{2c^2}[/imath], so the PDE for the transient part of the solution should be [imath]w_{tt} = c^2 w_{xx} - g, w(0,\ t) = 0, w(L,\ t) = 0, w(x,\ 0) = \frac{Lgx - gx^2}{2c^2}, w_t(x,\ 0) = 0[/imath].

This does not get rid of the forcing function though. Do I need to apply some technique from solving inhomogeneous ODEs (undetermined coefficients, annihilator method, reduction of order, variation of parameters, etc.)?
 
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