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Solving eigenvalue problems for ode's
*To*: mathgroup at yoda.physics.unc.edu
*Subject*: Solving eigenvalue problems for ode's
*From*: news at murdoch.acc.virginia.edu
*Date*: Thu, 4 Jun 92 16:25:49 EDT
I posted this problem earlier this week but I got a few responses
asking for more detail, so here it is.
This is the an aerospace problem for supersonic flow over a simply
supported flat plate. I have the following governing pde and
boundary conditions:
d*W""""(x,t) + m*W''(x,t) + a*W"(x,t) - b*W'(x,t) = 0
(1)
W(0,t)=W(L,t)=W""(0,t)=W""(L,t) = 0, 0<= x <= L, t>0
where
" - denotes differentiation w.r.t. x
' - denotes differentiation w.r.t. t
W - transverse motion of plate
and a, b, d, m are positive constants.
Since this is a linear pde, I would like to find the normal modes
of (1). I can then expand W(x,t) and then apply Galerkin's method
to reduce the problem to a set of equations.
Applying the separation of variables technique, we assume that
W(x,t)=F(x)*G(t). Subbing into (1) gives
d*F'''' a*F' m*G'' b*g'
------- + ---- = - ----- + ---- = v
F F G G
where and ' denotes differentiation and v is a constant. The sign of
v can be less than, equal to, or greater than zero. The normal modes
are represented by F(x). Rearranging gives the classic eigenvalue
problem;
d*F''' + a*F' - v*F = 0
(2)
F(0)=F(L)=F''(0)=F''(L) = 0
and
m*G'' - b*G' + v*G = 0. (3)
The constants a and b are related to the the air speed. If a=b=0,
then we have a simply supported flat plate and solving (2) gives
the normal modes as K*sin(n*pi*x/L), K is a constant. The eigenvalues
are v=(n*pi/L)^4, n=0,1,-1,2,-2,...
(This problem was been solved many times.)
Howevwer, for a and b nonzero, the solution is not as easy. I figure
let Mathematica handle it. Use DSolve to solve (2) for any 3 of the
4 boundary conditions. Solve the resulting expression for v subject
to the 4th boundary condition. This fails because applying either of
the conditions F''(0)=0 or F''(L)=0 to (2) gives the trivial solution
of zero.
For v>0, The Mathematica command stream looks like:
DSolve[{d F'''' + a F' - v F == 0, F[0] == 0, F[L] == 0,
F''[0] == 0}, F, x]
or
DSolve[{d F'''' + a F' - v F == 0, F[0] == 0, F[L] == 0,
F''[L] == 0}, F, x]
which both return
{}.
But if I evaluate
DSolve[{d F'''' + a F' - v F == 0, F[0] == 0, F[L] == 0}, F, x]
I get a expression as expected but how do I apply the 2 remaining
boundary conditions.
The constant v can be negative or zero also. Of course, I must
consider the possibility that the solution may not exist at all.
I hope this clarifies the problem. Please e-mail response to me.
Keith
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