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 Post subject: Partial Differential EquationsPosted: Fri, 28 Oct 2011 07:24:29 UTC
 S.O.S. Oldtimer

Joined: Mon, 28 Dec 2009 00:16:28 UTC
Posts: 228
Can anyone help with these problems? I have no idea where to start. What is the general approach?

Determine the solution of ∂ρ/∂t = (sin x)ρ which satisfies ρ(x,0) = cos x.
Determine the solution of ∂ρ/∂t = ρ which satisfies ρ(x,t) = 1 + sin x along x =-2t.

Relevant equations: ∂ρ/∂t + ∂/∂x(q(ρ)) = 0 or ∂ρ/∂t + ∂/∂x(ρu(ρ)) = 0 and q = ρu.

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 Post subject: Re: Partial Differential EquationsPosted: Fri, 28 Oct 2011 07:51:54 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
glebovg wrote:
Can anyone help with these problems? I have no idea where to start. What is the general approach?

Determine the solution of ∂ρ/∂t = (sin x)ρ which satisfies ρ(x,0) = cos x.
Determine the solution of ∂ρ/∂t = ρ which satisfies ρ(x,t) = 1 + sin x along x =-2t.

Relevant equations: ∂ρ/∂t + ∂/∂x(q(ρ)) = 0 or ∂ρ/∂t + ∂/∂x(ρu(ρ)) = 0 and q = ρu.

In this case note that x is constant as far as varying t is concerned, so it is just a collection of first-order linear ODE where you introduce a parameter x.

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 Post subject: Re: Partial Differential EquationsPosted: Fri, 28 Oct 2011 14:34:38 UTC
 S.O.S. Oldtimer

Joined: Mon, 28 Dec 2009 00:16:28 UTC
Posts: 228
So how would I solve them? Which method should I use?

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