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[DSolve] Re: newbie question DSolve (revisited)
*To*: mathgroup at smc.vnet.net
*Subject*: [mg52224] [DSolve] Re: newbie question DSolve (revisited)
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Mon, 15 Nov 2004 03:17:32 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <cmppui$mll$1@smc.vnet.net> <200411100945.EAA11259@smc.vnet.net> <cmve9e$shc$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <cmve9e$shc$1 at smc.vnet.net>,
"Pratik Desai" <pdesai1 at umbc.edu> wrote:
> I tried your suggestion unfortunately, Mathematica gives me another error it
> is as follows
>
> In[10]:=
> DSolve[{y''[x] - lamda^2*y[x] == DiracDelta[x - d]*y[d], y[0] == 0,
> y[l] == 0}, y[x], x]
>
> (DSolve::"litarg"), "To avoid possible
> ambiguity, the arguments of the dependent variable in (the equation)
> should literally match the independent variables".
As others have suggested, try solving
y''[x] - lamda^2 y[x] == DiracDelta[x - d] yd
and then determining a self-consistent solution at x = d. My analysis of
this problem find only the trivial solution y[x] = 0 for 0 <= x <= l for
real lambda. However, for imaginary lambda, there are nontrivial
solutions (i.e., eigenvalues).
[1] Solve the differential equation:
sol = First[DSolve[{y''[x] - lambda^2 y[x] == DiracDelta[x - d] yd,
y[0] == 0, y[l] == 0}, y, x]];
[2] Determine the self-consistent solutions.
consistent = Simplify[y[d] == (y[d] /. sol /. yd -> y[d]), 0 < d <= l]];
[3] Not suprisingly, Solve only yields the trivial solution.
Solve[consistent, y[d]]
[4] To be an eigenvalue the following expression must vanish:
eval = Simplify[Factor[Subtract @@ consistent]/y[d]]
[5] A plot of eval for, say, l -> 1 and d -> 1/2, shows that it never
vanishes. However, putting lambda -> I lambda and simplifying,
complexeval = Simplify /@ ComplexExpand[eval /. lambda -> I lambda]
one _can_ find eigenvalues by numerically solving the resulting
transcendental equation. For example, again with l -> 1 and d -> 1/2,
Simplify[complexeval /. {l -> 1, d -> 1/2}]
FindRoot[%, {lambda, 4}]
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
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