bndnchrs
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So here's the problem:
I'm asked to find the solutions to the 1-D Wave equation
[tex]u_{tt} = u_{xx}[/tex]
subject to
[tex]u(x,0) = g(x), u_t(x,0) = h(x)[/tex]
but also
[tex]u_t(0,t) = A*u_x(0,t)[/tex]
and discuss why A = -1 does not allow valid solutions. I can't figure it out at all. The solutions to the usual problem with initial conditions is just the D'alembert formula... but adding this restriction seems to overdetermine the problem. We can turn that second condition into the full 1-D wave equation but it involves squaring A, which means that if A = +/- 1 it works... and otherwise it is overdetermined. Can anyone else help me?
I'm asked to find the solutions to the 1-D Wave equation
[tex]u_{tt} = u_{xx}[/tex]
subject to
[tex]u(x,0) = g(x), u_t(x,0) = h(x)[/tex]
but also
[tex]u_t(0,t) = A*u_x(0,t)[/tex]
and discuss why A = -1 does not allow valid solutions. I can't figure it out at all. The solutions to the usual problem with initial conditions is just the D'alembert formula... but adding this restriction seems to overdetermine the problem. We can turn that second condition into the full 1-D wave equation but it involves squaring A, which means that if A = +/- 1 it works... and otherwise it is overdetermined. Can anyone else help me?