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 Differential Equations Ordinary and Partial Differential Equations Math Forum

 September 13th, 2015, 10:01 AM #1 Member   Joined: Sep 2013 Posts: 31 Thanks: 0 Transport eq. with two initial conditions? Given the usual homogenous transport equation: $\displaystyle \partial_t \rho+ v \partial_x \rho=0$ with initial conditions: $\displaystyle \rho(0,t)=g(t)$ and $\displaystyle \rho(x,0)=0$. How do you find $\displaystyle \rho(x,t)$? September 24th, 2015, 06:11 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Since this is a partial differential equation with two independent variables, of course there are two conditions, one for each variable. (Technically, the condition for x is a "boundary condition", not an "initial condition". Your differential equation is $\displaystyle \frac{\partial \rho}{\partial x}+ v\frac{\partial \rho}{\partial t}= 0$ (are we to assume that v is a constant?) with initial conditions $\displaystyle \rho(0, t)= g(t)$ and $\displaystyle \rho(x, 0)= 0$. It should be easy to see that $\displaystyle \rho(x, t)= F(x- vt)$ is a solution for F any differentiable function. It should also be easy to show that no such function satisfies both conditions. Last edited by Country Boy; September 24th, 2015 at 06:15 AM. September 25th, 2015, 07:54 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,919 Thanks: 2201 The equation given is $\dfrac{\partial \rho}{\partial t}+ v\dfrac{\partial \rho}{\partial x}= 0$, not $\dfrac{\partial \rho}{\partial x}+ v\dfrac{\partial \rho}{\partial t}= 0$. It is satisfied by $\rho(x, t) = F(x - vt)$, where $F$ is differentiable. The condition $\rho(x, 0) = 0 \implies F(x) = 0 \implies \rho(x, t) = 0$, so $\rho(0, t) = g(t)$ can't also be satisfied unless $g(t) = 0$. Maybe the problem was mistyped. Thanks from szz and Country Boy Tags conditions, initial, transport Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post philm Differential Equations 4 March 14th, 2015 09:56 PM JulieK Applied Math 0 June 25th, 2014 10:37 AM Help_me Economics 2 May 13th, 2014 11:05 AM Jhenrique Calculus 7 January 16th, 2014 10:01 PM Anamitra Palit Physics 3 December 6th, 2012 01:20 PM

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