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I am trying to use NDSolve to analyze transients in transmission lines. I started with this simple one:

s = NDSolve[{∂_(t,t) u[t,x] == 2*∂_(x,x) u[t,x], u[0,x] == 1, 
              u^((1,0))[0,x] == 0, u[t,0] == Sin[t]},
             u, { t, 0, 40}, { x, -10, 10}];

So: voltage is initially constant (at t = 0). One end of the line is shorted.

But, this is not approved. Mathematica says:

NDSolve::ndnco: The number of constraints (1) (initial conditions) is not equal to 
the total differential order of the system plus the number of discrete variables (2). >>

How could NDSolve be used for transmission line transient analysis?

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Can you please post copiable Mathematica code? Select your code, right click, copy as -> plain text. –  Szabolcs Jul 10 at 13:46
    
The u[t,0] == Sin[t] is not defined on the boundary of the domain (x == 10 or x == -10). At least if I try u[t, -10] == Sin[t + 10]`, I get a solution. I'm not sure why the particular error message is the one we get. –  Michael E2 Jul 10 at 14:18
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