Table 1 Solution for the Lighthill-Whitham-Richards model for different g
For | Solution to \(\bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} \mathrm {d}u =-(1-2u)\cdot u_{x} \,\mathrm {d}t,\\ u(x,0) = g(x),\quad x\in [0,1].\end{array}}\) |
|---|---|
g(x)=x | \(u(x,t)=\frac{x-t}{1-2t}\), \(t\neq \frac{1}{2}\) |
\(g(x)=x-x^{2}\) | \(u(x,t)=\frac{\sqrt{-4t^{2}+t(8x-4)+1}+2t-1}{4t}-\frac{(\sqrt{-4t^{2}+t(8x-4)+1}+2t-1)^{2}}{16t^{2}}\), t ≠ 0 |