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MathGroup Archive 2004

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[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]=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]=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] == 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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