This question is followed up from this question How to solve a certain coupled first order PDE system
Here I consider the non-homogeneous advection system
\begin{equation} \displaystyle\left\{\begin{array}{l} \frac{\partial U}{\partial t}+b_1\frac{\partial U}{\partial x}=(r+l_1)U-l_1V, \\ \frac{\partial V}{\partial t}+b_2\frac{\partial V}{\partial x}=(r+l_2)V-l_2 U, \\ \end{array}\right. t\in [s,T] \end{equation}
with the following boundary conditions (where $W$ could be $U$ and $V$)
\begin{equation} \left\{\begin{array}{l} W(x,t)=0 \ \ \text{as} \ \ x\to-\infty,\\ W(x,t)\sim S_0e^x \ \ \text{as} \ \ x\to\infty,\\ W(x,T)=\max \{ S_0e^x-100,0\} \\ \end{array}\right. \end{equation}
First I tried it with mathematica using DSolve
DSolve[{
D[u[x, t], t] + b1*D[u[x, t], x] - (r + l1)*u[x, t] + l1*v[x, t] ==0,
D[v[x, t], t] + b2*D[u[x, t], x] - (r + l2)*v[x, t] + l2*u[x, t]==0
}, {u[x, t], v[x, t]}, {x, t} ];
No things come out. I then tried it with Maple 12
sys := [diff(u(x,t),t)+b1*diff(u(x,t),x)=(r+l1)*u(x,t)-l1*v(x,t),
diff(v(x,t),t)+b2*diff(v(x,t),x)=(r+l2)*v(x,t)-l2*u(x,t)]:
sol:=pdsolve(sys,[u,v]) assuming b1>b2;
here what I got
It seems for me that Maple 12 can find the solution but I don't know what does the symbol $_c_{1}$ in the output mean ?
My questions here are :
1) Can we tell mathematica does the same job as Maple 12 does ?
One of my friends recommended the following approach: we assume that solution is of the form $u=u_0e^{d_1 t+d_2x}, v=v_0e^{d_1 t +d_2x}$, then I find $u_t, v_t, u_x, v_x$,---> plug them into the system---> collecting terms in terms of $(u_0, v_0)$, we have a linear system whose determinant must be zeros in order to have a solution for the advection system. At the end, you can find infinitely many $(d_1,d_2)$, hence the solution can be represented in term of an infinite series ( which is not likable :))) )
My next questions are :
1) How can he guess such a form of the solution? (I am not saying friend is incorrect but I am not sure whether we can assume such a form )
2) Using the asymptotic limit: $U, V\sim S_0 e^x$, can I assume that $d_2\equiv 1$?
3) Is there any other simple way to solve the above system ?
Thank you so much for your time . I really appreciate it.