Btw I fixed a bug so now
norm((exp(u)-exp(-u))-2diff(u)./(x+1)-diff(u,2))
returns 0.0 (not sure why not machine eps…)
On 21 Jun 2014, at 10:58 pm, 'Stéphane Laurent' via julia-users
<julia-users@googlegroups.com> wrote:
> Sorry but I really don't understand you first sentence.
> About the code, I will possibly experiment it during the next days or weeks,
> and I'll send you some feedback.
> Thank you again for this library and your help.
> - Stéphane
>
> Le samedi 21 juin 2014 14:24:03 UTC+2, Sheehan Olver a écrit :
> Thanks!
>
> I'm interested to know if the ultraspherical spectral approach has any clear
> advantages over spectral collocation for nonlinear odes. (For linear odes the
> advantages are clear.)
>
> So if you find the code better than expected please let me know. It's a bit
> buggy and unoptimized, so chances are you won't.
>
> Sent from my iPad
>
> On 21 Jun 2014, at 10:15 pm, 'Stéphane Laurent' via julia-users
> <julia...@googlegroups.com> wrote:
>
>> Ok, I understand. It works and it is really awesome. Thank you !
>>
>>
>> Le samedi 21 juin 2014 12:17:22 UTC+2, Sheehan Olver a écrit :
>> Hi,
>>
>>
>> You didn’t quite get the Newton iteration right: if you want to solve
>>
>> B u + [1,0] = 0
>> L u + g(u) = 0
>>
>> then Newton iteration becomes
>>
>> u = u - [B, L + g’(u)]\[B u + [1,0], L u + g(u)]
>>
>>
>> i.e., your bc right hand side is not right. Below is the corrected code.
>>
>> Cheers,
>>
>> Sheehan
>>
>>
>>
>> x=Fun(identity, Interval(0.,1.))
>> d=x.domain
>> B=neumann(d)
>> D=diff(d)
>> # Solves Lu + g(u)-1==0
>> L = D^2 + 2/(x.+1)*D
>> g = u -> -(exp(u)-exp(-u)); gp = u -> -(exp(u)+exp(-u))
>>
>> u=-0.3*x+0.5 #initial guess
>>
>> for k=1:5 # this crashes if Ii increase the number of iterations
>> u=u-[B,L+gp(u)]\[diff(u)[0.]+1.,diff(u)[1.],L*u+g(u)];
>> end
>>
>> plot(u)
>>
>> On 21 Jun 2014, at 6:54 pm, 'Stéphane Laurent' via julia-users
>> <julia...@googlegroups.com> wrote:
>>
>>> Hello,
>>>
>>> I'd like to solve this equation with Neumann boundary conditions. My code
>>> below does not work. Am I doing something bad or is it a failure of the
>>> Newton algorithm ?
>>>
>>> # solves u" = (exp(u)-exp(-u))-2u'/(x+1) , u'(0)=-1, u(1)=0
>>> x=Fun(identity, Interval(0.,1.))
>>> d=x.domain
>>> B=neumann(d)
>>> D=diff(d)
>>> # Solves Lu + g(u)-1==0
>>> L = D^2 + 2/(x.+1)*D
>>> g = u -> -(exp(u)-exp(-u)); gp = u -> -(exp(u)+exp(-u))
>>>
>>> u=-0.3*x+0.5 #initial guess
>>>
>>> for k=1:5 # this crashes if Ii increase the number of iterations
>>> u=u-[B,L+gp(u)]\[-1.,0.,L*u+g(u)];
>>> end
>>>
>>>
>>> The solution should look like that :
>>>
>>>
>>>
>>>
>>
