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 Calculus`DSolveIntegrals` to find special solutions." This way <<Calculus`DSolveIntegrals` 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 |