Re: Re: Tricky differential equation
- To: mathgroup at smc.vnet.net
- Subject: [mg41567] Re: [mg41553] Re: Tricky differential equation
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 27 May 2003 01:47:28 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
A rather more elegant way is the one that was just recently presented
in this list by Paul Abbott:
In[1]:=
w[x_] := x*x*Derivative[2][u][x] + x*Derivative[1][u][x] +
(x*x - 1)*Sin[u[x]]
In[2]:=
Solve[w[x] + O[x]^6 == 0, Table[Derivative[i][u][0],
{i, 0, 5}]]
From In[2]:=
Solve::ifun:Inverse functions are being used by Solve, so some
solutions may \
not be found.
From In[2]:=
Solve::svars:Equations may not give solutions for all "solve" variables.
Out[2]=
(5) 1 2 4
{{u [0] -> -- u'[0] (20 + 40 u'[0] + 3 u'[0] ),
32
(4) (3) 1 2
u [0] -> 0, u [0] -> -(-) u'[0] (6 + u'[0] ),
8
u''[0] -> 0, u[0] -> 0},
(5) (4) (3)
{u [0] -> 0, u [0] -> 0, u [0] -> 0, u''[0] -> 0,
u'[0] -> 0, u[0] -> 0}}
On Monday, May 26, 2003, at 06:46 pm, Dr. Wolfgang Hintze wrote:
> Luiz,
>
> I have tried to find a appoximate solutions to your differential
> equation in terms of polynomials of degree m for the special boundary
> conditions u[0] = 0, u'[0] != 0. The method can be generalized.
>
> This is probably not a very elegant method in Mathematica but it seems
> to work. Perhaps someone in the group can improve the method. Here it
> is
> (notice that in the text below comments are not depicted explicitly;
> that's because I copied it from a notebook in which I used Ctrl+7
> format
> for comments)
>
> In[362]:=
> ClearAll["Global`*"]
>
> Differential equation
>
> In[363]:=
> w[x_] := x*x u''[x] + x u'[x] + (x*x - 1) Sin[u[x]]
>
> Series ansatz
>
> In[364]:=
> u[x_] := a[0] + Sum[a[k] x^k, {k, 1, m}]
>
> Constructing a solution (e.g. for m=5; you can try other values of m
> and
> look what happens) by substituting the ansatz for u[x] into the
> differential eqaution and requiring that the coefficients of all powers
> of x vanish
>
> In[365]:=
> m = 5;
>
> In[366]:=
> cond = {}; sol = {};
> For [i = 0, i <= m,
> AppendTo[cond, (D[w[t], {t, i}] /. t -> 0) == 0];
> AppendTo[sol, a[i]];
> i++];
>
> You can display cond and sol if you like deleting the comment marks
> below
> (*
> cond
> sol
> *)
>
> In[370]:=
> asl = Solve[cond, sol]
>
> From In[370]:=
> Solve::"ifun": "Inverse functions are being used by \!\(Solve\), so
> some \
> solutions may not be found."
>
> From In[370]:=
> Solve::"svars": "Equations may not give solutions for all \"solve\" \
> variables."
>
> Out[370]=
> \!\({{a[5] -> \(a[1]\ \((20 + 40\ a[1]\^2 + 3\ a[1]\^4)\)\)\/3840, a[4]
> -> 0,
> a[3] -> \(-\(1\/48\)\)\ a[1]\ \((6 + a[1]\^2)\), a[2] -> 0,
> a[0] -> 0}, {a[5] -> 0, a[4] -> 0, a[3] -> 0, a[2] -> 0, a[1]
> -> 0,
> a[0] -> 0}}\)
>
>
>
> Substituting the coefficients into the ansatz
>
> In[371]:=
> h[x_] := u[x] /. asl
>
> Displaying the solution
>
> In[372]:=
> Clear[a]
>
> In[359]:=
> h[x]
>
> Out[359]=
> \!\({x\ a[1] -
> 1\/48\ x\^3\ a[
> 1]\ \((6 +
> a[1]\^2)\) + \(x\^5\ a[1]\ \((20 + 40\ a[1]\^2 + 3\ \
> a[1]\^4)\)\)\/3840, 0}\)
>
> Plotting the solution for a special value of a[1]:
>
> a[1] = 0.5;
> Plot[h[x][[1]], {x, 0, \[Pi]}, PlotRange -> {{0, \[Pi]}, {-1, 1}}];
>
> or, for different a[1]
>
> For[z = 0.1, z < \[Pi], a[1] = z;
> Plot[h[x][[1]]/\[Pi], {x, 0, 2\[Pi]}, PlotRange -> {{0, \[Pi]},
> {-1, 1}}];
> z = z + 0.2];
>
> Hope this hepls.
>
> Wolfgang
>
>
>
>
> Luiz Melo wrote:
>
>> Hello everyone,
>>
>> I'm trying to find the numerical solution of the following
>> differential equation (r is the independent variable):
>>
>> x''[r] + 1/r x'[r] + (p - 1/r^2)*Sin[x[r]]*Cos[x[r]] == 0 ,
>>
>> with boundary conditions: x'[1] == 0 , and x[0] -> "has to be finite",
>>
>> but I'm having at least two problems:
>>
>> 1) I don't know how to submit the BC "finite" to Mathematica;
>> 2) The coefficient p is about 10^4. For this reason, it seems
>> that the Runge-Kutta method usually used for numerical
>> integration of ordinary differential equations turns out
>> to be unsuccessfull in our case. Do we need a special method
>> to solve this?
>>
>> The solution of this equation gives the internal magnetic structure
>> of a cylinder. The function x[r] is the angle between the
>> magnetization and the axial direction, and it depends on the radial
>> direction, r.
>>
>> I would like to plot the Cossine of the result as a function of r
>> (which varies from 0 to 1), for several values of p.
>>
>> Any help will be very appreciated!
>> Thank you
>>
>> Luiz Melo
>>
>> Ecole Polytechnique de Montreal,
>> Montreal, Quebec
>> luiz.melo at polymtl.ca
>>
>>
>>
>>
>
>
>
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/