[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