Re: NDSolve with Dirichlet boundary condition

• To: mathgroup at smc.vnet.net
• Subject: [mg107320] Re: [mg107228] NDSolve with Dirichlet boundary condition
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Tue, 9 Feb 2010 02:44:46 -0500 (EST)
• References: <201002060823.DAA14151@smc.vnet.net>

```I suspect you'll get better responses, now, after posting the actual
problem.

Bobby

On Mon, 08 Feb 2010 12:15:26 -0600, Frank Breitling <fbreitling at aip.de>
wrote:

> Hi Bobby,
>
> My real equation is
>
> D[r^2 k0 T[r]^(5/2) T'[r], r] == 3/2 kB T'[r]-(kB T[r])/n[r] n'[r]
>
> where kB and k0 are constants and n[r] is a monotonically decreasing
> (non analytic) function with n[r]->0 for r->infinity.
>
> I think I can't apply your transformation here.
> But anyways thanks a lot for your thoughts!
>
> Frank
>
>
> On 2010-02-08 18:43, DrMajorBob wrote:
>>> Therefore my question is whether it is possible to solve my simplified
>>> example using NDSolve or any other non analytic method of Mathematica.
>>
>> I solved this with DSolve in my post, but NDSolve also works:
>>
>> NDSolve[{y''[r] == 0, y[0] == 50, y[1] == 1/2}, y, {r, 0, 1}]
>>
>> {{y->InterpolatingFunction[{{0.,1.}},<>]}}
>>
>> If your simplified example is like the real problem, there should be a
>> way to transform, as I did, and solve.
>>
>> Bobby
>>
>> On Mon, 08 Feb 2010 03:46:38 -0600, Frank Breitling <fbreitling at aip.de>
>> wrote:
>>
>>> Dear Bobby,
>>>
>>> Unfortunately my original problem doesn't allow for an analytic
>>> solution, since the equation is more complex and involves interpolating
>>> functions.
>>> Therefore my question is whether it is possible to solve my simplified
>>> example using NDSolve or any other non analytic method of Mathematica.
>>>
>>> Frank
>>>
>>>
>>> On 2010-02-06 22:42, DrMajorBob wrote:
>>>> Define y as follows and compute its derivative:
>>>>
>>>> Clear[x,y,r]
>>>> y[r_]=x[r]^2/2;
>>>> y'[r]
>>>>
>>>> x[r] (x^\[Prime])[r]
>>>>
>>>> Hence your equations are equivalent to
>>>>
>>>> {y''[r]==0, y[0] == 50, y[1] == 1/2}
>>>>
>>>> The first equation says that y is linear. Specifically,
>>>>
>>>> y[r_] = InterpolatingPolynomial[{{0, 50}, {1, 1/2}}, r]
>>>>
>>>> 50 - (99 r)/2
>>>>
>>>> and hence,
>>>>
>>>> x[r_] = Sqrt[2 y[r]]
>>>>
>>>> Sqrt[2] Sqrt[50 - (99 r)/2]
>>>>
>>>> Solving the same thing in Mathematica, we get:
>>>>
>>>> Clear[y]
>>>> DSolve[{y''[r]==0,y[0]==50,y[1]==1/2},y,r]
>>>> {{y->Function[{r},1/2 (100-99 r)]}}
>>>>
>>>> Or, for the original problem:
>>>>
>>>> Clear[x, r]
>>>> DSolve[{D[x[r] x'[r], r] == 0, x[0] == 10, x[1] == 1}, x, r]
>>>>
>>>> {{x -> Function[{r}, -I Sqrt[-100 + 99 r]]}}
>>>>
>>>> That's the same as the earlier (real-valued) solution, even though it
>>>> appears to be Complex.
>>>>
>>>> Simplify[-I Sqrt[-100 + 99 r] - Sqrt[2] Sqrt[50 - (99 r)/2],
>>>>  r < 100/99]
>>>>
>>>> 0
>>>>
>>>> Bobby
>>>>
>>>> On Sat, 06 Feb 2010 02:23:21 -0600, Frank Breitling
>>>> <fbreitling at aip.de>
>>>> wrote:
>>>>
>>>>> Hello,
>>>>>
>>>>> I was not able to solve the following differential equation with
>>>>> Mathematica 7.01.0 using:
>>>>>
>>>>> NDSolve[{D[x[r]x'[r],r]==0, x[0]==10, x[1]==1}, x, {r,0,1}]
>>>>>
>>>>> Since my original problem is inhomogeneous and involves interpolating
>>>>> functions DSolve is not an option.
>>>>>
>>>>> Is there a way to solve this problem using Mathematica?
>>>>> Any help is highly appreciated.
>>>>>
>>>>> Best regards
>>>>>
>>>>> Frank
>>>>>
>>>>
>>>>
>>>
>>
>>
>

--
DrMajorBob at yahoo.com

```

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