Re: Re: Re: function of a function

*To*: mathgroup at smc.vnet.net*Subject*: [mg62694] Re: [mg62667] Re: [mg62650] Re: function of a function*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 30 Nov 2005 22:09:19 -0500 (EST)*References*: <200511301040.FAA07278@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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? >>> | >> >