# Exact solution for two coupled non-homogeneous transport equations

I want to solve the following system \eqalign{ & {y_t} = -{y_x} + z{\text{ in (0}}{\text{,T)}} \times {\text{(0}}{\text{,1)}} \cr & {z_t} = {z_x} + y{\text{ in (0}}{\text{,T)}} \times {\text{(0}}{\text{,1)}} \cr & y(0,x) = {y_0},{\text{ }}z(0,x) = {z_0},{\text{ }} \cr} I have tried to solve explicitly the first and the second, but the problem is in the coupling, if the coupling is just on one of them there will be no problems. I have tried also to compute the associated semi-group to the system but I didn't get a result. Do you know any method to deal with such a system? Thank you.

• This system rewrites as $q_t + A q_x = S q$ where $q=(y,z)^\top$. In the present case, $A$ and $S$ are not simultaneously diagonalizable. Thus, it seems difficult to find an explicit solution by the method of characteristics. However, the pde is linear, so that Fourier transform can be used (cf. Repost) – Harry49 Dec 11 '18 at 22:46
• @Harry49 Thank you for the answer. I want to use characteristics to solve the system but I don't find the starting point. Can you give me a hint? Thanks. – Gustave Dec 12 '18 at 7:24
• With the method of characteristics, we can solve $q_t + A q_x = 0$ or $q_t = S q$ separately, leading to an approximate resolution by operator splitting. However, since $A$ and $S$ are not simultaneously diagonalizable, it seems difficult to solve $q_t+Aq_x = S q$ exactly with the method of characteristics. By the way, cross-posting a question on multiple sites is not recommended. – Harry49 Dec 12 '18 at 9:14