Re: NDSolve with Dirichlet boundary condition

• To: mathgroup at smc.vnet.net
• Subject: [mg107313] Re: [mg107228] NDSolve with Dirichlet boundary condition
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Tue, 9 Feb 2010 02:43:30 -0500 (EST)
• References: <201002060823.DAA14151@smc.vnet.net>

```> 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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