How to solve a nonhomo quasilinear avection equation?
So I need to solve this equation:
$$ u_t + uu_x = -ku^2, k, t > 0, u(x, 0) = 1, x \in \mathbb{R}$$
While using characteristics doesn't really work - that if I try
$$ \dfrac{dt}{1} = \dfrac{dx}{u} = \dfrac{du}{-ku^2}, $$
then I only get $u = \dfrac{1}{kt + 1}$. But doesn't $u$ also depend on $x$? Should I use other methods to find the solution? Then, how to sketch the characteristics for this equation? Constant lines of $t$?
$\endgroup$ 21 Answer
$\begingroup$The solution $$u(x,t) = \frac{1}{kt+1}$$valid for all times $t> -1/k$is correct. Indeed, the initial condition $u(x,0)=1$ is satisfied, and the PDE $u_t + uu_x = -ku^2$ is satisfied too. To plot the characteristics, note that $dx/dt = u$. Thus, the characteristics are curves of the form$$ x(t) = x_0 + \frac1{k}\ln (kt + 1) $$where $x_0$ is a constant such that $x(0) = x_0$.
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