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]=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]=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'[0] 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[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". >> >> >>Thanks again for your reply again Dr Hintze, >> >> >> >>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[1]:= >>>s = DSolve[{-y[x] + Derivative[2][y][x] == DiracDelta[d - x]*y[d], y[0] >>>== 0, y[L] == 0}, y[x], x] >>> >>>From In[1]:= >>>DSolve::"nvld" : "The description of the equations appears to be >>>ambiguous or \ >>>invalid." >>> >>>Out[1]= >>>{{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[2]:= >>>DSolve::"nvld" : "The description of the equations appears to be >>>ambiguous or \ >>>invalid." >>> >>>Extracting the solution to u[x] >>> >>>In[4]:= >>>u[x_] = y[x] /. s[[1]] >>> >>>you can Plot it, after assigning numeric values to all relevant >>>quantities: >>> >>>In[6]:= >>>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]==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 >>>> >>>>Please advise and thanks in advance, >>>> >>>> >>>>Pratik Desai >>>> >>>> >>>>ps: This is my third attempt at posting my query, I hope this time it >>>>makes it to the list :) >>>> >>>> >>>> >>>> >>>> >>>> >> >> >> >