@Carl Love 

I delete the last bc condition and accept your comment, setting exp(-10000)=0.

Still, the maple says... the system is too complex.

As you can see the equation I want to solve is the hyperbolic wave equation.

To see the system is working, I simply put the boundary conditions R(30)=R(-30)=0 and get the result.

Maple worked smoothly and gave me the plot as I expected....

The problem is... the wave do not evolve to the boundary... It simple stays at the same region while its amplitude is growing so fast... omg...

I believe it's an issue of bcs...

Do you have any idea why maple cannot accept the first two bcs I gave in the original question?

@Preben Alsholm 

Thank you for your comment.

I do know how to plot several plots in different time t on the same plot.

What I meant earlier is that I want to plot u(x,t) at a fixed x, for example u(100,t) with time range 30<t<100.

Do you have any idea might help me?

@Carl Love 

I'm woking on the wave equation R(x,t) with source u(x,t)

v(x,t) is a time derivative of R(x,t).

If there were second derivatives of u or v, the main equation R(x,t) becomes third order PDE which is not what I wanted.

The reason why the bcs has derivative form is to follow the purely outgoing condition at both boundaries.

when I enter the aboves...

the maples says... cannot relate the Bcs and variables because the system is too complex.

When I put an arbitrary value as BCs, the PDE system is solved.

Those Bcs, ((30^2+1^2)/30^2)^(1/4)*v(-30, t) = -((30^2+1^2)/30^2)^(1/2)*(D[1](R))(-30, t) is important because it implies the wave propagates only outward at the boundary.

Can you help me?