inverse laplace transform

[nduman]

please someone should help me out of this.

solve the heat equation using inverse laplace transform.

dw/dt=c^2d^2w/dx^2.

subject to the d following boudary conditions W(x,0)=To

W(0,t)=0

W(x,t) goes to 0 as X tends to infinity.

[Bignose]

nduman,

 

we tend not to just solve or do work for other people here. Please post what you have done so far, and where you have gotten stuck and we'll try to guide you from there.

 

you may want to look into using this forum's LaTeX capabilities, too, to make your post much easier to read. http://www.scienceforums.net/topic/3751-quick-latex-tutorial/

Edited October 15, 2012 by Bignose

[mathteacher025]

Hello!

I would try to help you.

So what I got is

1)d = dx*sqrt(dw/(dt*w))/c 2)second solution d = -dx*sqrt(dw/(dt*w))/c

You can check the answers right here:

http://www.numberemp...&var=d&answers=

Good Luck!

Edited November 13, 2012 by mathteacher025

[Bignose]

Hello!

I would try to help you.

So what I got is

1)d = dx*sqrt(dw/(dt*w))/c 2)second solution d = -dx*sqrt(dw/(dt*w))/c

You can check the answers right here:

http://www.numberemp...&var=d&answers=

Good Luck!

 

Wow, couldn't be wronger. This is not an algebraic problem, but a differential equation.

[Enthalpy]

This is the diffusion equation, it applies to heat and more situations.

 

Its solution is like erfc[x/(2*sqrt(t)]. You have to arrange it a bit (replace w by 1-w after solving) because the usual condition is exactly the opposite, with matter having temperature w=0 before t=0 where a constant temperature begins at the surface.

 

Yes, you can solve it by a Laplace transform. It is done, but is uncommon among engineers, because it involves half-integer powers of s (written p in some countries), the formal Laplace variable. Sacadura does it, probably in French only (Initiation aux transferts thermiques).

 

The old Fourier did it with his transform for a finite bar of material, so a Fourier series was enough. A semi-infinite bar would need a Fourier transform instead - which is nearly identical with a Laplace transform for real W.

 

A different way is Green's method, more expressive to engineers, which convolves

- a Gaussian function thats starts perfectly sharp (Dirac) and spreads out in the material over time (Gaussian functions are such favourable solutions of the diffusion equation, they extend and decrease as sqrt(t))

- with the proper, user-found function of the position that lets the convolution fit the boundary conditions.

The nice part is that at t=0 the convolution of many Dirac functions is trivial... If the initial condition is a known temperature distribution, Green's method boils down to a sum of Gaussian functions.

Sometimes you have to distinguish x=0- from x=0+ and t=0- from t=0+.

 

A lot there:

http://en.wikipedia.org/wiki/Heat_equation