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Re: Solving 2nd order PDE into Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg115265] Re: Solving 2nd order PDE into Mathematica
*From*: Roland Franzius <roland.franzius at uos.de>
*Date*: Thu, 6 Jan 2011 02:02:13 -0500 (EST)
*References*: <ig1i9d$6ll$1@smc.vnet.net>
Am 05.01.2011 11:48, schrieb tarpanelli at libero.it:
> Hello,
>
> I saw that DSolve can not solve a 2nd order pde like this
>
> pde=-D[f[x,t],x]+D[f[x,t],{x,2}]==D[f[x,t],t]
>
> but I would like to know if someone has already implemented some other
> procedure to solve it
>
The heat/diffusion pde cannot be solved without additonal equations at
boundaries (-oo, oo?) and a starting distribution f[x,0] == g[x].
A formal solution eg
d_t f = 0, f = a + b e^(-x)
is not admissible for physical problems.
The equation is probably the radial diffusion equation in dimension 2
and the general solution is an 2-d-integral over a radial symmetric
function of r=sqrt(x^2+y^2) only, folded with the two dimensional
gaussian kernel
(x,y,t; xi,eta,tau) ->
1/( 2 pi (t-tau ) exp(-((x-xi)^2 +(y-eta)^2))/(2 (t-tau)) (t>tau)
(constants may differ)
--
Roland Franzius
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