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.
2026-03-25 06:10:47.1774419047
Exact solution for two coupled non-homogeneous transport equations
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