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.

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Inhomogenous nonlinear transport equation $u_t+uu_x = -Du$

We have the following setup: $$u_t+uu_x = -Du \\ u(x,0)=\sin x.$$ The question is to find the time $T_s$ of a first shock formation. So basically, I need to solve the equation using method of ...
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Cauchy Problem for inviscid Burgers' equation

Consider the Cauchy Problem of finding $u(x,t)$ such that $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0,x\in\mathbb{R},t>0$$ $$u(x,0) = u_0(x), x\in\mathbb{R}$$ Which choices ...
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Why is the solution single-valued?

I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-...
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Rarefaction and shock waves colliding in Burgers' equation

I'm confident I've solved all but the last segment of this problem, to which I have an answer that just doesn't seem right. The problem is to solve the inviscid Burgers' equation $$u_{t}+uu_{x}=0$$ ...
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Finding shock curves, Burgers equation

I'm working on this problem: $u_{t}+uu_{x}=0\\ u(0,x)=-x \mathbb{1}_{[a,b]}$. For the case when $a<b=0$ I want to find a shock curve starting from point $(t,x)=(1,0)$. On the field without ...
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How to Solve This PDE and Deal with Boundary Value Using Method of Characteristics. Work Shown.

The function $u(x,y)$ satisfies $u_y + u_x = 0$ in $x > 0$, $y > 0$ together with the initial condition $u(x, 0) = \sin(x)$, $x > 0$ and the boundary condition $u(0, y) = \sin(y)$, $y > 0$....
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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 ...
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Entropy solution - Burgers' equation

Why doesn't the following problem have a solution for $t\ge1$? $u_{t}+uu_{x}=0\\ u(0,x)=-x$. The characteristics don't intersect and they cover the whole space above t=1.