Re: Re: function of a function

*To*: mathgroup at smc.vnet.net*Subject*: [mg62682] Re: [mg62650] Re: function of a function*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Wed, 30 Nov 2005 22:08:45 -0500 (EST)*References*: <dmha20$932$1@smc.vnet.net><dmhfhd$bit$1@smc.vnet.net> <200511300507.AAA00115@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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}]; > > It appears (to me) the power of programming with functions in > Mathematica has not been used to the full. > > Regards > Narasimham > [...] The second one gives a result, the first does not. This is a consequence of mathematics, not Mathematica. A first order ODE is of the form dy/dx=F(y,x) where F is some "reasonable" function e.g. piecewise analytic. Your second equation conforms to this description whereas your first does not. My guess is one might approach your first using ansatz from integral equations methods. One such might be to expand f as a power series with unknown coefficients and try to find constraints to give linear equations in those coefficients. Another method might be to form a sequence of functions where f'[n,x] == f[n-1,Cos[x]]. Make f[0,x] something reasonable (perhaps constant) and iterate the process a few times in hope of getting the f[n,x] to converge for all x of interest. The code below may give an idea of what I have in mind. f[0][x_] := 1 deq[n_] := {f[n]'[x]==f[n-1][Cos[x]],f[n][0]==1}; sol[n_] := (f[n][x] = f[n][x]/.First[NDSolve[deq[n],f[n][x],{x,0,4}]]; f[n][t_] := f[n][x]/.x->t) For example, if I now do Do[sol[j], {j, 100}] then I get a bunch of messages telling me extrapolation was needed to go outside the domain. But then I evaluate Table[f[j][1.8], {j, 100}] I see that it is converging to about 10.5214, and similarly f[j][2.8] appears to be converging to about 5.08. Let's check the differential condition. Max[Abs[Table[f[100]'[j] - f[100][Cos[j]], {j, 0, 4, .1}]]] also gives a bunch of extrapolation messages, and a value of around .035, which I think is not too bad. Going to 500 iterations brings the residual to .0039. So this appears to be a viable approach. It may also be a bit naive as I'm never certain when such a method is really justified. Probably requires a bit of analysis to decide if we have some sort of contraction principle to tell us to expect a fixed point. Also those extrapolations might be problematic, and evaluations of f[n][2.8] do not seem to converge terribly well, so it may be that the range needs to be expanded or contracted in order to get reliable results. Daniel Lichtblau Wolfram Research