(adsbygoogle = window.adsbygoogle || []).push({}); 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 kindagoing out of my mindwith this one and any help would be going a long way towards preserving my sanity ;).

Cheers

Chris

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

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