Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform help ;)

[chrissimpson]

Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform... help! ;)

Hi

I'm trying to model the temperature profile of an inertia friction welding during and after welding. I have the welding outputs and have come up with a net heat flow wrt time during the process.

I now want to put this information into the heat equation and have become fairly stuck and was hoping for a little help

u(x,t) is temperature as a function of position and time
a is thermal diffusivity

PDE: ut=auxx

BC (Neumann?):
ux(L,t) = Q/kA - Q is heat input in watts, k is thermal conductivity, A is area
ux(0,t)=0

IC:
u(x,0) = 0


My plan was to form an auxiliary problem with constant boundary conditions:

PDE: wt=awxx

BC (Neumann?):
wx(L,t) = 1 - Q is heat input in watts, k is thermal conductivity, A is area
wx(0,t)=0

IC:
u(x,0) = 0

The next step would be to produce the Laplace transform of this, solve the PDE, invert the transform and to then sub the 'simple' w(x,t) into the 'Duhamel's principle' equation, thus giving me my u(x,t).

----------

I'm basing this methodology on a similar problem to mine from the PDE book by Farlow. In this problem the boundary conditions are in the dirichlet form, and the auxiliary problems constant boundary conditions are of the from:

w(0,t)=0
w(L,t)=1

(where L is 1 and a was 1)

as opposed to wx(0,t) and w(L,t).

Nevertheless the general form of the problem is the same and the laplace transform that is produced - W(x,s) - is similar to mine:

1s. W(x,s)=(1/s)(sinh(x*(s)^0.5)/(sinh(s^0.5)) - The Laplace transform for 'Farlow's problem'

There is then a jump to:

1t. w(x,t)= x + (2/pi)*Ʃ ((-1)^n)/n) * e^(-t*(n*pi)^2) * sin(n*pi*x)


So, I was hoping to do the same with my problem, which is given below (and I believe is correct):

2. W(x,s)=(1/s)*(1/(s*a)^0.5)*(cosh(x*(s*a)^0.5))/(sinh(L*(s*a)^0.5)) - The Laplace transform for 'my problem'


The problem is that I have no idea how to go about solving either my own or Farlow's invert transform and wondered if there was something obvious that I was doing wrong.

I used the Taylor series of sinh(x) to attempt the problem but ended up with a silly answer.

In another example I dug up there was some mention of asymptotic series and term by term inversion but I really don't know where to go with that.


So, in conclusion, does anyone have a good idea of how I can invert my Laplace transform (or Farlow's one) and/or whether I am going about this problem in the correct way.

Thank you for taking the time to read this! I am kinda going out of my mind with this one and any help would be going a long way towards preserving my sanity ;).

Cheers

Chris

 

[chrissimpson]

So, to solve that inverse laplace transform I should use the Bromwich integral. This all seems a bit more difficult (and therefore time consuming) than I was hoping.

I can go down this path but would appreciate any advice on whether there is a better methodology to go about solving this differential.

Chris

 

[hunt_mat]

For your problem I would take the Laplace transform in t, to obtain a second order ODE and then you can apply your boundary conditions easily. All this you have done but you have a horrid function to invert and if you were going to invert it you would need to include branch cuts with your Bromwhich contour.

You can do a term by term inversion of your transform and you can decide which terms are small and which are important.

 

[hunt_mat]

There is an example of this in Priestly, An introduction to complex analysis.

 

[chrissimpson]

Thanks for that - that's brilliant! I'll get my hands on a copy of the book tomorrow.

I can also get a free copy of Maple - could this churn out an answer to the transform? I've tried with MATLAB but it didn't want to play ball!

 

[hunt_mat]

Matlab and MAPLE share the same symbolic package, you could try mathematica but a doubt that would work, I found examples by googling. Dean Duffy has a whole book on Laplace transforms and inverting them.

 

[chrissimpson]

Thanks for your help! I'll have a play around with this over the next couple of days. It might take me a little while but when I get to the final answer I'll post it up here.

 

[hunt_mat]

I would be interested, see if you can get a few plots, I would be interested to see them.