Finding $u$ for $u_t+uu_x=f$

Question

For a river flowing on a slope, the resistive force is $R = av^2$ and gravitational force, $F = bh$. The flow speed adjusts itself to $av^2=bh \implies v^2=\frac{b}{a}h$. Given the mass conservation law,

$$h_t + (hv)_x = r.$$

If $v^2=\frac{b}{a}h$,

$$v^2=\frac{b}{a}h\rightarrow (v^2)_t =\frac{b}{a} h_t\rightarrow h_t =\frac{a}{b}(v^2)_t$$ $$(hv)_x=(\frac{a}{b}v^3)_x=\frac{a}{b}3v^2v_x$$

Thus, $$h_t+(hv)_x=\frac{a}{b}(v^2)_t+\frac{a}{b}3v^2v_x=r$$ $$(v^2)_t+3v^2v_x=r\frac{b}{a}$$

With $u=3\frac{b}{a}h=3v^2$ write, $$\frac{u_t}{3}+u\frac{(u^{\frac{1}{2}})_x}{3}=r\frac{b}{a}$$ $$\frac{u_t}{3}+u\frac{\frac{1}{2}u_x}{3u^{\frac{1}{2}}}=r\frac{b}{a}$$ $$\frac{u_t}{3}+\frac{u^{\frac{1}{2}}u_x}{6}=r\frac{b}{a}$$

I am trying to get the PDE in the form $u_t+uu_x=f$ (inviscid Burger's equation) but am out of tricks.

Answer 1

$$\frac{u_t}{3}+\frac{u^{\frac{1}{2}}u_x}{6}=r\frac{b}{a} \qquad (1)$$ Supposing $a$ , $b$ and $r$ are constants.

Solving thanks to the method of characteristics :

The system of differential equations for the characteristics curves is : $$3dt=6\frac{dx}{ u^{\frac{1}{2} }}=\frac{a}{br}du$$ From $\quad 3dt=\frac{a}{br}du\quad$ the equation of a first family of characteristic cuves is : $$\frac{a}{br}u-3t=c_1$$ From $\quad 6\frac{dx}{u^{\frac{1}{2}}}=\frac{a}{br}du \quad\to\quad 6\frac{br}{a}dx=u^{\frac{1}{2}}du\quad$ the equation of a second family of characteristic cuves is : $$6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}=c_2$$ The general solution of the PDE $(1)$ is obtained on the form of an implicit equation : $$\Phi\left(\frac{a}{br}u-3t \; , \; 6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}\right)=0$$ where $\Phi$ is any differentiable function of two variables.

Other equivalent forms to express the general solution are : $$\frac{a}{br}u-3t =F\left( 6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}\right)$$ $$6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}=G\left( \frac{a}{br}u-3t \right)$$ where $F$ and $G$ are any differentiable functions, but related ( one is the inverse of the other).

The possibility (or not) to reduce the implicit equation to an explicite equation $u(x,t)$ depends on the kind of function $\Phi$ (or $F$ or $G$).

Unfortunately the boundary conditions are missing in the wording of the question. Without them, one cannot determine the function $\Phi$ (or $F$ or $G$). So, we will not go further.