       [DSolve] Re: newbie question DSolve (revisited again)

• To: mathgroup at smc.vnet.net
• Subject: [mg52226] [DSolve] Re: newbie question DSolve (revisited again)
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Mon, 15 Nov 2004 03:17:35 -0500 (EST)
• Organization: The University of Western Australia
• References: <cn1p31\$evf\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <cn1p31\$evf\$1 at smc.vnet.net>, pdesai1 at umbc.edu wrote:

> The Dirac Delta forces a discontinuity in the system, so we actually have
> two parts of the string
> 1. between 0 and d  y1[x]
> 2. between d  and l  y2[x]
>
> so the b.c are
>
> y1=0
> y2[l]=0
>
> at the constraint
>
> y1[d]=y2[d]  Shape is continuous

So, actually,

y1[d] == y2[d] == y[d]

> y1'[d]-y2'[d]=c*lamda*y[d]  the jump discontinuity due to the delta function
>
> I am thinking of an alternate construction without Dirac Delta function
>
> y1''[x]-lamda^2*y1[x]=0  0<x<d
>
> with B.C
>
> y1=0
> y1'[d]=y2'[d]-c*lamda*y[d]
>
> and
>
> y2''[x]-lamda^2*y2[x]=0  0<x<d

you mean d < x < l

> y2'[d]=y1'[d]+c*lamda*y[d]
>
> Please let me know if this is applicable and how I can implement in
> Mathematica. Also can mathematica give you Eigenfunctions-- non-trivial
> solutions to a homogenous BVP

Of course it can -- if you state the problem fully. Clearly, however,
the eigenvalue problem requires, in general, the solution to a
transcendental equation.

Here is the first equation with boundary condition and continuity.

sol1 = First[DSolve[{y1''[x] - lambda^2 y1[x] == 0,
y1 == 0, y1[d] == y[d]}, y1, x]]

Here is the second equation with boundary condition and continuity.

sol2 = First[DSolve[{y2''[x] - lambda^2 y2[x] == 0,
y2[l] == 0, y2[d] == y[d]}, y2, x]]

Now apply the derivative jump discontinuity condition.

FullSimplify[{
y1'[d] == y2'[d] - c lambda y[d],
y2'[d] == y1'[d] + c lambda y[d]} /. sol2 /. sol1]

As stated, however, c is not determined by the differential equations or
boundary conditions.

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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