Re: Difficulty with NDSolve (and DSolve)

*To*: mathgroup at smc.vnet.net*Subject*: [mg106167] Re: Difficulty with NDSolve (and DSolve)*From*: Mark McClure <mcmcclur at unca.edu>*Date*: Sun, 3 Jan 2010 03:43:31 -0500 (EST)*References*: <201001021004.FAA07409@smc.vnet.net>

On Sat, Jan 2, 2010 at 5:04 AM, KK <kknatarajan at yahoo.com> wrote: > I am trying to numerically solve a differential equation. However, I > am encountering difficulty with Mathematica in generating valid > numerical solutions even for a special case of that equation. > > The differential equation for the special case is: > F'[x] == - (2-F[x])^2/(1-2 x + x F[x]) and > F[1]==1. > These equations are defined for x in (0,1). Moreover, for my context, > I am only interested in solutions with F[x] in the range <1. Of course, the basic problem is that the solution is not one-to-one in a neighborhood of your initial condition. Thus, let's set up an initial condition y0 at x0=1/2 and try to choose y0 so that the solution passes through (1,1) - kind of like the shooting method for solving boundary value problems. We can set up a function that tells us the value of the solution at x=1 as a function of the initial condition at x0=1/2 like as follows: In[1]:= valAt1[y0_?NumericQ] := Quiet[Check[(F[x] /. NDSolve[{F'[x] == -(2 - F[x])^2/(1 - 2 x + x F[x]), F[1/2] == y0}, F[x], {x, 1/2, 1}][[1, 1]]) /. x -> 1, bad]]; In[2]:= {valAt1[-4], valAt1[-3]} Out[2]= {0.208967, bad} Note that we use Check to check for error messages in case that x=1 is outside the range of the solution. The computation shows that this singularity between y0=-4 and y0=-3. We can use a bisection method to find more precisely where this happens like so: In[3]:= a = -4; In[4]:= b = -3; In[5]:= Do[ c = (a + b)/2; If[valAt1[c] === bad, b = c, a = c], {50}]; In[6]:= N[a] Out[6]= -3.35669 Now, we find the solution taking y0=a. In[7]:= Clear[F]; In[8]:= F[x_] = F[x] /. NDSolve[{F'[x] == -(2 - F[x])^2/(1 - 2 x + x F[x]), F[1/2] == a}, F[x], {x, 0.001, 1}][[1]]; In[9]:= F[1] Out[9]:= 1. The computation shows that it worked. Hope that helps, Mark McClure

**References**:**Difficulty with NDSolve (and DSolve)***From:*KK <kknatarajan@yahoo.com>