Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [characteristics]

The method of characteristics is a way of solving certain partial differential equations by reducing them to ordinary differential equations. It is most often used for 1st order equations. Use with the (pde) tag.

The method of characteristics is a way of solving certain partial differential equations (PDEs) by reducing them to ordinary differential equations (ODEs). Also, the method can be seen as a change of variables to simplify the PDE. It is most often used for 1st order linear or nonlinear PDEs. Use with the tag.

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Shockwaves in traffic flow

I have been struggling on this problem for a while now so here it is: I am looking to work out when the shockwaves occur in the traffic flow model given below: A traffic flow governed by the equations below is observed at $t = 0$ to have a dip in…
juper
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A first order hyperbolic problem using method of characteristics

Consider the initial value problem for the equation $u_t+au_x=f(t,x)$ with $u(0,x)=1$ and $f(t,x)=1$ if $x\ge 0$ and $0$ otherwise. Assume that $a$ is positive. Show that the solution is given by $u(t,x)=0$ if $x\leq 0$, $x/a$ if $x\ge 0$ and…
David Hansen
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Shockwave trajectory

A traffic flow governed by the equations below is observed at $t = 0$ to have a dip in the density distribution at $x = 0$ given by: $$ \rho (x,t=0) = \left\{\begin{aligned} &a &&: x \lt 0\\ &a- \frac {a}{2}(1+x)&&: -1
juper
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First order method of characteristics

I am tasked with solving $$\cos(y)\frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y} = \frac{x}{u}$$ And can't seem to get past the characteristics, namely: $$\frac{dx}{\cos(y)} = \frac{dy}{x}$$ and $$\frac{du}{x/u} =…
Dimitri_896
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Hard Partial Differential Equations - Characteristics

$$ (x-y)\frac{\partial u}{\partial x}+(x+y)\frac{\partial u}{\partial y}=\alpha u $$ where α is a constant, with initial condition $u(x, 0) = x^2$ , $x > 0$. How do I solve this partial differential equation? I have tried to use the method of…
MathematicianP
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