From: jahnke@crl.crl.com (Frank Jahnke)
Newsgroups: sci.math,sci.physics,sci.engr.chem,sci.engr.mech.fluids
Subject: Re: Help:solution for pde.
Date: 6 Apr 1998 15:28:07 -0700

Check the "usual sources," namely, Crank, "The mathematics of diffusion," 
or Craslaw and Jaeger, "conduction of heat in solids."

If you want to solve it yourself, however, give Laplace transforms a 
shot.  Inversion formulae are in the CRC math handbook, or erdelyi (sp?).

Frank Jahnke, Ph.D.
==============================================================================

From: "W. R. Smith" <no_spam_please@aol.com>
Newsgroups: sci.math,sci.physics,sci.engr.chem,sci.engr.mech.fluids
Subject: Re: Help:solution for pde.
Date: Tue, 07 Apr 1998 07:26:44 -0400

Good advice.  I'd also add Jost's classic book on diffusion (the title of
which I forget at the moment) .  The situation described (with a finite space
domain) is a little more involved than other commonly considered cases.

Frank Jahnke wrote:

> Check the "usual sources," namely, Crank, "The mathematics of diffusion,"
> or Craslaw and Jaeger, "conduction of heat in solids."
>
> If you want to solve it yourself, however, give Laplace transforms a
> shot.  Inversion formulae are in the CRC math handbook, or erdelyi (sp?).

-- W. R. Smith, PhD, P. Eng., Senior Scientist, Mathtrek Systems --
EMail(replace "_at_" by "@", "_dot_" by "."): support_at_mathtrek_dot_com
--------------------- http://www.mathtrek.com ---------------------
-Mathtrek Systems - Home of EQS4WIN Chemical Equilibrium Software -
==============================================================================

From: shaw4@popcorn.llnl.gov (Henry Shaw)
Newsgroups: sci.math,sci.physics,sci.engr.chem,sci.engr.mech.fluids
Subject: Re: Help:solution for pde.
Date: Mon, 06 Apr 1998 23:20:17 GMT

venkatra@engr.sc.edu wrote:

>I need to solve
>dC/dt=d^2C/dx^2
>(Diffusion equation)
>IC t=0,C = 0
>and BCs,
>x = 0, dc/dx = 0
>x = 1, dc/dx = 1
>
>I have difficulties in solving this by separation of variables. Can I find
>solution of this anywhere?

What happened to the diffusion coefficient?

Your best bet is to use a Laplace transform to solve this equation.

Or....

Carslaw and Jaeger, 1959, Conduction of Heat in Solids, Oxford Univ.
Press, gives the solution in section 3.8.i,
 and 
Crank, 1979, The Mathematics of Diffusion, Oxford Univ. Press, gives
the solution as equation 4.55 in section 4.3.7 (you have to recognize
that your  boundary conditions are equivalent to setting dC/dx=1 at
both x=1 and x=-1

If the prescribed flux at x=L is F (in your case L = 1 and F = 1), the
initial concentration is C0 (in your case C0=0), and the diffusion
coefficient is D, the solution is:

C-C0 = (F*L/D) * [D*t/L^2 + (3x^2-L^2)/6L^2 - 2/pi^2*SUM(n=1 to
infinity) of {(-1)^n/n^2 * exp(-D*n^2*pi^2*t/L^2)*cos(n*pi*x/L)}]

Carslaw and Jaeger also give a series solution in terms of integrals
of the error-function complement.



C(x,t) =