Parabolic equation in DSolve

• To: mathgroup at smc.vnet.net
• Subject: [mg13660] Parabolic equation in DSolve
• From: Vitaliy Ababiy <vinni at chair12.phtd.tpu.edu.ru>
• Date: Mon, 10 Aug 1998 12:34:03 -0700 (GMT+7)
• Sender: owner-wri-mathgroup at wolfram.com

Widely know that the solution
u[x,t]=1/(2K Sqrt[\pi t])Exp[-x^2/(4K^2 t)] is analitical result for
parabolic equation D[u[x,t],t]==K D[u[x,t],{x,2}]
u[x,0]== DiracDelta[x]

or in general formula
u[x,t]=1/(2K Sqrt[\pi t])Integrate[f[xx]Exp[-(x-xx)^2/(4K^2 t)],
{xx,-Inf,+Inf}]

but there is no way to find it in analytical.

I try
DSolve[D[f[x, t], t] ==K D[f[x, t], {x,2}], f[x,t], {x,t}]
and etc. with any initial and boundary conditions
{f[x,t]==DiracDelta[x], f[-Inf,t]==0,[+Inf,t]==0} don't give result or
give
"Partial differential equation may not have a general solution. Try
loading
CalculusDSolveIntegrals to find special solutions."

This way
<<CalculusDSolveIntegrals
CompleteIntegral[
Derivative[0,1][f][x,t] == K Derivative[2,0][f][x,t],f[x,t], {x,t}] do
not give result too.

There is some result in numeriacal analys (from man) K=1;
solution=NDSolve[{D[f[x, t], t] ==K D[f[x, t], {x,2}],
f[x, 0] == Exp[-x^2], f[-5, t] == f[5, t]},f, {x, -5, 5}, {t, 0,
5}]; Plot3D[Evaluate[f[x, t] /. First[solution]],{x,-5,5},{t,0,5},
PlotRange->All,PlotPoints->30];
but is any way fo find it result in analytic?

Thanks.

|       Vitali Ababi            |
| Physical Technical Department |
| Tomsk Polytechnic University  |
|     634004 Tomsk, Russia      |



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