I'm in trouble finding the solution of this system of 2 PDEs:
\begin{equation}
\frac{\partial u\_1}{\partial t} + a\_1 \frac{\partial u\_1}{\partial x} = b (u\_1-u\_2)\\
\frac{\partial u\_2}{\partial t} + a\_2 \frac{\partial u\_2}{\partial x} = -b (u\_1-u\_2)
\end{equation}


with:
\begin{equation}
u\_1 = u\_1(t,x)\\
u\_2 = u\_2(t,x)\\
a\_1,a\_2,b = constant
\end{equation}


initial condition is given for t=0:
with:
\begin{equation}
u\_1(0,x) = f(x)\\
u\_2(0,x) = g(x)\\
\end{equation}


I'm really frustrated cause I'm not able to find any valid method to approach it. Is it possible to find an analytical solution to obtain explicit functions u1, u2?



System of first order linear coupled PDEs
You might try a Laplace transform in $t$, obtaining a parametrized system of linear constant-coefficient ODE's in $x$.


