michael 2007-9-4 11:14
Entering equations
[size=2]Can FlexPDE work with these two equations, how to define them in EQUATIONS section of script? 4@#o$O*S)eS-Y2Or
1. telegraph eq. Uxx = a^2* Utt + b* Ut + c* u .x5rw*i$z}9g-Y)Zg9[
2. pseudo-parabolic eq. Ut = Uxxt + Uxx + f ? [/size]
michael 2007-9-4 11:15
[size=2]In both cases, you need to define auxiliary variables to reduce each equation to second order. LT3Q hv
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In the first equation, define a variable V=dt(U). Then substitute in equation 1, resulting in the system
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U: DT(U) = V
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V: a^2*DT(V) = DXX(U) - b*V + c*U ft/y:Rd5a[/]
The Natural BC for boundary flux must attach to the V equation, since that is now where the DXX() term is.
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See the discussion in "Samples | Time_Dependent | Stress | Vibrate.pde".
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The second equation presents more difficulty, because if you define V=UXX, one of the variables will not be marked by FlexPDE as requiring time integration, and the use of the time derivative of that variable may not work. ._|?`} }Ywjb
One system to try would be
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U: DXX(U) = V
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V: DT(U) = DT(V) + V + F w _:t
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Notice that now the DXX(U) is in the U equation, and that equation must be assigned any NATURAL() boundary conditions.[/size]
michael 2007-9-4 11:15
On second thought, maybe the way to handle problem 2 is the same as problem 1: +vzdp Yd
Define V=DT(U) O8W!j%Cd5T a
Then the system is
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U: DT(U) = V
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V: DXX(V) + DXX(U) + F = V k7|cD_u
The Natural BC for the V equation must now define the outward normal component of DX(V)+DX(U).
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NATURAL(V) = <uflux> + DX(V)