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Re: Mathematica Question - Using DSolve with Boundary Conditions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg97532] Re: Mathematica Question - Using DSolve with Boundary Conditions
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Sat, 14 Mar 2009 18:15:47 -0500 (EST)
  • References: <gpg1aj$cft$1@smc.vnet.net>

Hi,

write a function, that solve your problem and rename it to
DSolve[], because DSolve[] build in Mathematica
can't solve this kind of problems.

Regards
   Jens

James Huth wrote:
> Dear Mathgroup:
> 
> I am trying to reproduce the analytic solution to a pair of partial 
> differential equations subject to boundary conditions. The analytic 
> solution has been published in an older journal article - I am trying to 
> reproduce/ verify/ understand the solution.
> 
> I believe I have set up DSolve correctly.  However, when the line is 
> executed DSolve simply returns the equations and boundary conditions rather 
> than solving the system.
> 
> Notes:
> * There are two partial differential equations, two unknown variables, and 
> three boundary conditions.
> * The second and third boundary conditions are supposed to hold when: WN, 
> WA and the partial derivatives of WN , WA with respect to t are evaluated 
> at the value t == ts, where ts is an arbitrary constant.
> * Arbitrary constants include: a, b, d, e, k, r, s
> 
> I would like to solve for WN and WA which are both functions of [t, m].
> 
> Here is the code I tried:
> 
> DSolve[{-t m + (b e - d m) D[WN[t, m], m] + a t D[WN[t, m], t] + 1/2 s^2 
> t^2 D[WN[t, m], t, t] == r WN[t, m],
> -t m - d m D[WA[t, m], m] + a t D[WA[t, m], t] + 1/2 s^2 t^2 D[WA[t, m], t, 
> t] == r WA[t, m],
> WN[0, m] == 0, WN[ts, m] == WA[ts, m] - k, D[WN[ts,m],t] == D[WA[ts,m],t]}, 
> {WN[t, m], WA[t, m]}, {t, m}]
> 
> OR...
> 
> DSolve[{-t m + (b e - d m) D[WN[t, m], m] + a t D[WN[t, m], t] + 1/2 s^2 
> t^2 D[WN[t, m], t, t] == r WN[t, m],
> -t m - d m D[WA[t, m], m] + a t D[WA[t, m], t] + 1/2 s^2 t^2 D[WA[t, m], t, 
> t] == r WA[t, m],
> WN[0, m] == 0, WN[ts, m] == WA[ts, m] - k, (D[WN[t,m],t]/.t->ts) == 
> (D[WA[t,m],t]/.t->ts)}, {WN[t, m], WA[t, m]}, {t, m}]
> 
> 
> The published solution is of the form (where A and y are positive constants 
> to be determined):
> 
> WN[t, m] = A t^y - t m / (r + d - a) - b e t / ((r - a) (r + d - a))
> 
> WA[t, m] = - t m / (r + d - a)
> 
> Can anyone advise how to use DSolve to yield solutions for WN and WA?
> 
> Sincerely,
> 
> James Huth
> jh288507 at ohio.edu
> 
> 


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