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Re: DSolve for a real function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127946] Re: DSolve for a real function
  • From: Bob Hanlon <hanlonr357 at gmail.com>
  • Date: Tue, 4 Sep 2012 05:47:26 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • Delivered-to: l-mathgroup@wolfram.com
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  • References: <k1k8np$90f$1@smc.vnet.net>

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^-21I,
   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'Arm=E9e
<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
>>>
>



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