Re: Re: Re: function of a function

*To*: mathgroup at smc.vnet.net*Subject*: [mg62696] Re: [mg62667] Re: [mg62650] Re: function of a function*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 1 Dec 2005 00:45:59 -0500 (EST)*References*: <200511301040.FAA07278@smc.vnet.net> <2D884FA6-65F6-4ECB-B489-D2FDB07F04FF@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

I forgot to include the definition of f[0]: f[0][x_]:=Cos[x] Andrzej On 1 Dec 2005, at 09:26, Andrzej Kozlowski wrote: > Actually, first equation is also solvable, but you have to use a > "functional method" (which indeed corresponds to "solving > infinitely many differential euations"). Here is a rather simple > approach; more sophisticated ones are possible. > > We want to solve the system: > > > Clear[x, f, EQ]; > EQ = {f'[x] == f[Cos[x]], f[0] == 1}; > > > We are going to try to approximate the entire function f over the > range [0,1]. This is actually a case when it can be done.We need a > starting function that satisfies the initial condition f[0]=1. The > functions Cos[x] itself will do. We now do the following: > > > Do[f[n] = f[n] /. > First[NDSolve[{D[ > f[n][x], x] == f[n - 1][Cos[ > x]], f[n][0] == 1}, f[n], {x, 0, 1}]], {n, 1, 100}] > > The solutions is > > Plot[f[100][x], {x, 0, 1}] > > By plotting > > Plot[{f[100]'[x], , f[100][Cos[x]]}, {x, 0, 1}, PlotStyle -> {Red, > Green}] > > we see that the two curves coincide perfectly. > > Obviously you could not expect NDSOlve to do this by itself! > > Andrzej Kozlowski > > > > On 30 Nov 2005, at 19:40, Andrzej Kozlowski wrote: > >> >> On 30 Nov 2005, at 14:07, Narasimham wrote: >> >>> Jens-Peer Kuska wrote: >>> >>>> it can't work because f [0] ==1 given in your differential equation >>>> f ' [0]==f [1] and NDSolve[] can't find the value for >>>> f[1] until it has integrated the equation. >>> >>> ??? >>> >>>> The nested dependence is equivalent to an infinite >>>> system of ordinary differential equations and it seems to be >>>> hard to do this by a finte computer. >>> >>> I cannot understand this. In the following two examples the first >>> one >>> works, not the second. >>> >>> Clear[x,f,EQ]; >>> EQ={f'[x] == f[Cos[x]],f[0]== 1}; >>> NDSolve[EQ,f,{x,0,4}]; >>> f[x_]=f[x]/.First[%]; >>> Plot[f[x],{x,0,4}]; >>> >>> Clear[x,f,EQ]; >>> EQ={f'[x] == Cos[f[x]],f[0]== 1}; >>> NDSolve[EQ,f,{x,0,4}]; >>> f[x_]=f[x]/.First[%]; >>> Plot[f[x],{x,0,4}]; >> >> >> Surely, you mean the second one works, the first one does not!? Also, >> I think I agree with Jens. These cases are quite different and the >> problem he mentione does not arise in the second case. Ine the second >> case the derivative at a point x is defined only in terms of the >> value of the function at x. Thus values of the function, it's >> derivative, function etc, can be computed sequentially. In the first >> case, however, in order to compute the derivative at x you need to >> know the value of the function at Cos[x], which in general will not >> be known yet. This is, I think, what Jens meant and it seems to me >> clearly right. >> >>> >>> It appears (to me) the power of programming with functions in >>> Mathematica has not been used to the full. >>> >>> >> >> What do you mean? Can you suggest an approximation scheme for this >> sort of problem? >> >> Andrzej Kozlowski >> >> >> >>> Jens-Peer Kuska wrote: >>>> Hi, >>>> >>>> it can't work because f[0]==1 give in your >>>> differential equation >>>> f'[0]==f[1] and NDSolve[] can't find the value for >>>> f[1] until it >>>> has integrated the equation. >>>> The neted dependence is equvalent to a infinite >>>> system of >>>> ordinary differential equations and it seems to be >>>> hard to do >>>> this by a finte computer. >>>> >>>> Regards >>>> Jens >>>> >>>> "Narasimham" <mathma18 at hotmail.com> schrieb im >>>> Newsbeitrag news:dmha20$932$1 at smc.vnet.net... >>>> | Tried to solve numerically: >>>> | >>>> | >>>> http://groups.google.com/group/sci.math/browse_frm/thread/ >>>> 248f76d024c1ac57/0bba983777a07bc9#0bba983777a07bc9 >>>> | >>>> | thus: >>>> | >>>> | EQ= { f'[x] == f[f[x]], f[0]== 1} ; >>>> NDSolve[EQ,f,{x,0,2}]; >>>> | >>>> | But gives an error. NDSolve::ndnum: >>>> Differential equation does not >>>> | evaluate to a number at x = 0. >>>> | >>>> | Also does not work even with other f[0] values. >>>> Any way to do that? >>>> | >>> >> >

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