Solving a general, constant-coefficient, first-order, two-indep-variable system of PDEs

Question

I have the following system of PDEs that I want to solve as "analytically" as possible:

$$\left(\partial_t + A\partial_x + B\right)\mathbf{u}(t, x) = 0,$$

where $A$ and $B$ are constant, diagonalizable matrices, and $t \ge 0$, $x \ge 0$. For simplicity say we're given $\mathbf{u}(0, x)$ and $\mathbf{u}(t, 0)$. How do I do this?

I first thought to try to solve this with the method of characteristics. This requires removing the $B$ term (right?), so I used the change of variables $\mathbf{u} = e^{-Bt}\mathbf{v}$, so that the PDE becomes

$$\left(\partial_t + e^{Bt}Ae^{-Bt}\partial_x\right)\mathbf{v}(t, x) = 0,$$

where it no longer has constant coefficients. But now I can't diagonalize and decouple the system, right? So I got stuck.

I also tried using the Laplace transform, which allowed me to obtain:

$$\mathbf{U}(t, s) = e^{(sA-B)t}\left[\int_0^\infty\mathbf{u}(0, x)e^{-sx}dx - \int_0^te^{-(sA-B)t'}A\mathbf{u}(t', 0)dt'\right],$$

where $\mathbf{U}$ is the Laplace transform of $\mathbf{u}$ in $x$, and $s$ is the transform variable. I think I may be able to invert this analytically, but wanted to see if I'm on the right track.

Any help would be appreciated!

Answer 1

In general, the method of characteristics is not going to give you anything like the d'Alembertian solution of the wave equation unless $A$ and $B$ are simultaneously diagonalizable.

For example, consider the $2$-by-$2$ system $$ u_t + u_x + v = v_t - v_x + u = 0,\tag 1 $$ which is in the form you have listed, with $A$ and $B$ each being diagonalizable (and having distinct eigenvalues). Note that this system implies that $u$ satisfies the second order scalar equation $$ u_{tt} - u_{xx} - u = 0.\tag 2 $$ Conversely, if $u(x,t)$ satisfies (2), then setting $v = -(u_t+u_x)$ gives you a pair $(u,v)$ that satisfies (1), so the two systems are equivalent.

Now, it is a (perhaps not so well-known) theorem of Sophus Lie that the general solution of (2), unlike the wave equation $u_{tt}-u_{xx} = 0$, cannot be written in the form $$ u(x,t) = F\bigl(x,t,f(x{+}t),g(x{-}t),f'(x{+}t),g'(x{-}t),\ldots, f^{(n)}(x{+}t),g^{(n)}(x{-}t)\bigr) $$ for any function $F$ of $2(n{+}2)$ variables, where $f$ and $g$ are arbitrary functions of a single variable.

Thus, one has to use other techniques to solve the initial value problem for the system (1).

Answer 2

The best option I came up with was to just solve the system numerically with spectral methods, which reduces to solving a single (huge) linear system.