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-users@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 : >>> >>> >>> >>> >>
