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)
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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>
- Re: DSolve for a real function