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Re: DSolve for a real function

• To: mathgroup at smc.vnet.net
• Subject: [mg127948] Re: DSolve for a real function
• From: Andreas Talmon l'Armée at smc.vnet.net
• Date: Tue, 4 Sep 2012 05:48:06 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• Delivered-to: l-mathgroup@wolfram.com
• Delivered-to: mathgroup-newout@smc.vnet.net
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• References: <k1k8np$90f$1@smc.vnet.net> <20120901062746.B3D7F687B@smc.vnet.net> <504462A6.5070404@fsm.tu-darmstadt.de> <CAEtRDSfhfr=mca7_3XsfGrF_5MbAuvAF5f8oVo7y+M9=_C68Ow@mail.gmail.com>

You are completely right but I think there should be no real part in the 
solution.

I did the solution of the differential equation on paper and got the 
term for y1[x].

Then I determined the constants C[5]...C[8] with the boundary conditions 
with Solve[] in  Mathematica. There I got an unusual behaviour.

When using Simplify on the sol1 could generate a soltuion with only a 
real part.
When using FullSimplify I get the imaginary part which is very small for 
small x but for greater x also the imaginary part starts to oscillate 
with a great amplitude.

I do not understand how a solution can be Real or Complex depending on 
the Simplifications scheme I use.

Shouldn't there be one solution to a problem with specified boundary 
conditions.?

The notebook where I did all this is on:
http://dl.dropbox.com/u/4920002/DGL_4th_Order_with_own_solution.nb


Thanks for your great help,

Andreas.



On 09/03/2012 03:22 PM, Bob Hanlon wrote:
> Clear[sol];
>
> $Assumptions = Element[{a, c, ic0, ic1}, Reals];
>
> sol[a_, x_] =
>   y[x] /. DSolve[{y''''[x] + a y[x] == 0, y''[-c] == ic0, y''[c] == ic0,
>        y'''[-c] == ic1, y'''[c] == -ic1}, y[x], x][[1]] // FullSimplify
>
> ((1 + I)*a^(1/4)*
>            (I*E^(I*Sqrt[2]*a^(1/4)*c) -
>               I*E^(Sqrt[2]*a^(1/4)*c) +
>               E^(Sqrt[2]*a^(1/4)*
>                    ((1 + I)*c + I*x)) -
>               E^(I*Sqrt[2]*a^(1/4)*x) -
>               E^(Sqrt[2]*a^(1/4)*x) +
>               E^(Sqrt[2]*a^(1/4)*
>                    ((1 + I)*c + x)) +
>               I*E^(Sqrt[2]*a^(1/4)*
>                      (I*c + (1 + I)*x)) -
>               I*E^(Sqrt[2]*a^(1/4)*
>                      (c + (1 + I)*x)))*ic0 +
>          Sqrt[2]*(E^(I*Sqrt[2]*a^(1/4)*
>                    c) + E^(Sqrt[2]*a^(1/4)*
>                    c) + E^(Sqrt[2]*a^(1/4)*
>                    ((1 + I)*c + I*x)) +
>               E^(I*Sqrt[2]*a^(1/4)*x) +
>               E^(Sqrt[2]*a^(1/4)*x) +
>               E^(Sqrt[2]*a^(1/4)*
>                    ((1 + I)*c + x)) +
>               E^(Sqrt[2]*a^(1/4)*
>                    (I*c + (1 + I)*x)) +
>               E^(Sqrt[2]*a^(1/4)*
>                    (c + (1 + I)*x)))*ic1)/
>       E^((-1)^(1/4)*a^(1/4)*(c + x))/
>     (4*a^(3/4)*(Sin[Sqrt[2]*a^(1/4)*
>               c] + Sinh[Sqrt[2]*a^(1/4)*
>               c]))
>
> Table[sol[a, x] /. {
>     c -> RandomReal[{0, 10}, WorkingPrecision -> 20],
>     ic0 -> RandomReal[{0, 10}, WorkingPrecision -> 20],
>     ic1 -> RandomReal[{0, 10}, WorkingPrecision -> 20]},
>   {a, 1, 5}, {x, 0, 4}]
>
> {{0.01540650723811880994 + 0.*10^-21 I, -0.0003703423761496348 +
>     0.*10^-20 I, -0.1278269715054133646 + 0.*10^-20 I,
>    0.0392723836429585004 + 0.*10^-20 I,
>    13.10841701158527505 + 0.*10^-18 I}, {-0.00286328430472954469 +
>     0.*10^-21 I, -1.985775360248369253 + 0.*10^-19 I, -0.311655087881410281 +
>     0.*10^-19 I, -0.718055887186593765 + 0.*10^-19 I,
>    21.42291506291193811 + 0.*10^-18 I}, {0.419394450195258310 + 0.*10^-19 I,
>    0.01666534839706064001 + 0.*10^-21 I,
>    0.528881084742103412 + 0.*10^-19 I, -0.060937975960727530 + 0.*10^-19 I,
>    0.0086995512380544778 + 0.*10^-20 I}, {0.01847749159520439918 + 0.*10^-21 I,
>     0.00517593237021928535 + 0.*10^-21 I,
>    0.0134376958759010127 + 0.*10^-20 I, -0.0679397801010012506 +
>     0.*10^-20 I, -5.21491473427110550 +
>     0.*10^-18 I}, {-0.00168718272676310080 +
>     0.*10^-21 I, -0.00137069152420374693 + 0.*10^-21 I,
>    3.051457492528748582 + 0.*10^-19 I,
>    0.00026536991493989622 + 0.*10^-21 I, -59.38147365750750826 + 0.*10^-18 I}}
>
> The imaginary parts cancel out; the residual imaginary parts are just
> numerical noise. This can be removed with Chop
>
> % // Chop
>
> {{0.01540650723811880994, -0.0003703423761496348, -0.1278269715054133646,
>    0.0392723836429585004,
>    13.10841701158527505}, {-0.00286328430472954469, -1.985775360248369253, \
> -0.311655087881410281, -0.718055887186593765,
>    21.42291506291193811}, {0.419394450195258310, 0.01666534839706064001,
>    0.528881084742103412, -0.060937975960727530,
>    0.0086995512380544778}, {0.01847749159520439918, 0.00517593237021928535,
>    0.0134376958759010127, -0.0679397801010012506, -5.21491473427110550}, \
> {-0.00168718272676310080, -0.00137069152420374693, 3.051457492528748582,
>    0.00026536991493989622, -59.38147365750750826}}
>
> ?Chop
>
> Chop[expr] replaces approximate real numbers in expr that are close to
> zero by the exact integer 0.  >>
>
>
> Bob Hanlon
>
>
> On Mon, Sep 3, 2012 at 3:56 AM, Andreas Talmon l'Armée
> <talmon at fsm.tu-darmstadt.de> wrote:
>> Hi,
>>
>> My initial conditions are the following and I am pretty sure that my
>> solution consists only of a real part.
>> The Solution has four eigenvalues and they are complex conjugated. With all
>> variables and all parameters real numbers I must be able to retrieve a real
>> solution. But being a mathematica newbie,  I do not understand how to do it
>> with mathematca.
>>
>> y''[-c] == ic0, y''[c] == ic0, y'''[-c] == ic1, y'''[c] == -ic1
>>
>> My Notebook is also ready for download at dropbox.com:
>> http://dl.dropbox.com/u/4920002/DGL_4th_Order.nb
>>
>> Clear[sol]
>>
>> $Assumptions = {a \[Element] Reals, ic0 \[Element] Reals,
>>     ic1 \[Element] Reals, c \[Element] Reals};
>>
>> sol[a_, x_] =
>>   y[x] /. DSolve[{y''''[x] + a y[x] == 0, y''[-c] == ic0,
>>        y''[c] == ic0, y'''[-c] == ic1, y'''[c] == -ic1}, y[x],
>>       x][[1]] // FullSimplify
>>
>> Reduce[Element[sol[a, x], Reals], a, Reals]
>>
>>
>>
>> Thanks for your help,
>>
>> Andreas
>>
>>
>>
>>
>>
>>
>>
>> On 09/01/2012 08:27 AM, Bob Hanlon wrote:
>>> What are your initial conditions?
>>>
>>> Clear[sol]
>>>
>>> sol[a_, x_] = y[x] /. DSolve[
>>>        {y''''[x] + a y[x] == 0,
>>>         y[0] == ic0, y'[0] == ic1,
>>>         y''[0] == ic2, y'''[0] == ic3},
>>>        y[x], x][[1]] // FullSimplify
>>>
>>> (1/(2*a^(3/4)))*
>>>      (Cosh[(a^(1/4)*x)/Sqrt[2]]*
>>>           (2*a^(3/4)*ic0*Cos[(a^(1/4)*x)/
>>>                    Sqrt[2]] + Sqrt[2]*
>>>                (Sqrt[a]*ic1 + ic3)*
>>>                Sin[(a^(1/4)*x)/Sqrt[2]]) +
>>>         (Sqrt[2]*(Sqrt[a]*ic1 - ic3)*
>>>                Cos[(a^(1/4)*x)/Sqrt[2]] +
>>>              2*a^(1/4)*ic2*Sin[(a^(1/4)*x)/
>>>                    Sqrt[2]])*
>>>           Sinh[(a^(1/4)*x)/Sqrt[2]])
>>>
>>> Reduce[Element[sol[a, x], Reals], a, Reals]
>>>
>>> a > 0
>>>
>>>
>>> Bob Hanlon
>>>
>>>
>>> On Fri, Aug 31, 2012 at 3:59 AM,  <"Andreas Talmon
>>> l'Arm=E9e"@smc.vnet.net> wrote:
>>>> Hi All
>>>>
>>>> Is there a way to tell mathematica to solve only for real solutions. My
>>>> differential equation is of the kind
>>>>
>>>> y''''[x]+a y[x]==0
>>>>
>>>> a= constant coefficient
>>>>
>>>> I know that I get 4 komplex eigenvalues which are complex conjungated.
>>>> But y[x] is a real function.
>>>> Solving this equation with DSolve always gets a complex function y[x].
>>>>
>>>> Any Ideas.
>>>>
>>>> Thanks, Andreas
>>>>

• References:
  • Re: DSolve for a real function
    • From: Bob Hanlon <hanlonr357@gmail.com>