       Mathematica Question - Using DSolve with Boundary Conditions

• To: mathgroup at smc.vnet.net
• Subject: [mg97477] Mathematica Question - Using DSolve with Boundary Conditions
• From: James Huth <jh288507 at ohio.edu>
• Date: Sat, 14 Mar 2009 05:36:12 -0500 (EST)

```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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