Explicit solution for transport equation with source given by ODE

Question

Consider the transport equation on $(0,\infty) \times (0,1)$ where the source solves an ODE: $$ \begin{cases} u_t(t,x)+au_x(t,x)+f(t,x)= 0\\ f_t(t,x) = u(t,x) \\ u(t,0) = g(t) & t>0 \\ u(0,x) = u_0(x) & x \in (0,1) \end{cases}$$ Can we give an explicit solution $u$ for this equation with the method of characteristics?

Answer 1

In the present form, the linear system of first order PDEs $$ \begin{bmatrix} u\\ f \end{bmatrix}_t + \begin{bmatrix} a&0\\ 0&0 \end{bmatrix}\begin{bmatrix} u\\ f \end{bmatrix}_x = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix}\begin{bmatrix} u\\ f \end{bmatrix} $$ of the form ${\bf u}_t + {\bf A}{\bf u}_x = {\bf B}{\bf u}$ hardly decouples, since $\bf A$, $\bf B$ are not co-diagonalisable. Therefore the method of characteristics is of little use here if applied directly (except for the case $a=0$). To solve the problem, one option would be to differentiate the first line w.r.t. $t$ which gives us the second-order hyperbolic equation $$ u_{tt} + au_{xt} +u =0 $$ to be solved the standard way (change of variables, reduction to canonical form). Note that the proposed boundary conditions may not be sufficient for this problem to be well-posed.