       newbie question DSolve (revisited again)

• To: mathgroup at smc.vnet.net
• Subject: [mg52153] newbie question DSolve (revisited again)
• From: pdesai1 at umbc.edu
• Date: Fri, 12 Nov 2004 02:14:18 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Thanks for your response Yehuda,

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
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
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

Thanks again

Pratik Desai

> I tried to use LaplaceTransform for your problem and the result agree
> with the message given by DSolve.
> Using LaplaceTransform you you finally get a result that y' depends
> on y[d] (or the opposite), so you really need another external
> constraint for that matter (the DiracDelta function "samples" y at its
> singular point).
> do you have any such information ?
> yehuda
>
> Pratik Desai wrote:
>
>>Thank you Dr. Hintze for your response,
>>
>>I tried your suggestion unfortunately, Mathematica gives me another error
>> it
>>is as follows
>>
>>In:=
>>DSolve[{y''[x] - lamda^2*y[x] == DiracDelta[x - d]*y[d], y == 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".
>>
>>
>>
>>
>>
>>Pratik Desai
>>
>>
>>
>>----- Original Message -----
>>
>>From: "Dr. Wolfgang Hintze" <weh at snafu.de>
To: mathgroup at smc.vnet.net
>>To: mathgroup at smc.vnet.net
>>Subject: [mg52153] [mg52118] [mg52090] Re: newbie question DSolve
>>
>>
>>
>>
>>>If you replace DiracDelta[d - x]*y[x] by the equivalent DiracDelta[d -
>>>x]*y[d] then your equation can be solved as follows (with just one minor
>>>error message appearing twice, which can be ignored)
>>>
>>>In:=
>>>s = DSolve[{-y[x] + Derivative[y][x] == DiracDelta[d - x]*y[d], y
>>>== 0, y[L] == 0}, y[x], x]
>>>
>>>From In:=
>>>DSolve::"nvld" : "The description of the equations appears to be
>>>ambiguous or \
>>>invalid."
>>>
>>>Out=
>>>{{y[x] -> 1/2*E^(-d - x)*(-((E^(2*x)*((-1 + E^(2*d))*UnitStep[-d] -
>>>(E^(2*d) - E^(2*L))*UnitStep[-d + L])*y[d])/
>>>        (-1 + E^(2*L))) + ((E^(2*L)*(-1 + E^(2*d))*UnitStep[-d] -
>>>(E^(2*d) - E^(2*L))*UnitStep[-d + L])*y[d])/
>>>       (-1 + E^(2*L)) - E^(2*d)*UnitStep[-d + x]*y[d] +
>>>E^(2*x)*UnitStep[-d + x]*y[d])}}
>>>
>>>From In:=
>>>DSolve::"nvld" : "The description of the equations appears to be
>>>ambiguous or \
>>>invalid."
>>>
>>>Extracting the solution to u[x]
>>>
>>>In:=
>>>u[x_] = y[x] /. s[]
>>>
>>>you can Plot it, after assigning numeric values to all relevant
>>>quantities:
>>>
>>>In:=
>>>L = 1; d = 0.5; y[d] = 1;
>>>Plot[u[x], {x, -1, 4}, PlotRange -> {{-1, 5}, {-2, 1}}];
>>>
>>>Hope this hepls
>>>Wolfgang
>>>
>>>
>>>Pratik Desai wrote:
>>>
>>>
>>>
>>>>Hello all
>>>>
>>>>I am trying to use DSolve to solve a ode with discontinuity in it (wave
>>>>equation with a viscous damper injected at a location d)
>>>>
>>>>This is what i am using
>>>>
>>>>DSolve[{y''[x]-lamda^2*y[x]==DiracDelta[x-d]*y[x],y==0,y[L]==
>>>>=0},y[x],x]
>>>>
>>>>the problem I am facing is that
>>>>
>>>>y[x] on the right hand side (next the delta function) varies w.r.t to
>>>>the location
>>>>
>>>>y[x]==y[x]&& 0<=x<=d
>>>>y[x]==y[L-x]&&d<=x<=L
>>>>
>>>>I can solve the above equation without the y[x] coupled to the delta
>>>>function
>>>>
>>>>
>>>>
>>>>Pratik Desai
>>>>
>>>>
>>>>ps: This is my third attempt at posting my query, I hope this time it
>>>>makes it to the list :)
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>
>>
>>
>

```

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