From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Why nobody answer this question? Date: 24 Oct 2000 08:05:32 GMT Newsgroups: sci.math Summary: [missing] I don't have a full answer to the original question, but I find this to be an interesting example, as I will explain below. Originally, in article , Zakir F. Seidov wrote: >Given ODE: > >(x^2 y')' = -x^2 y^2, > >with initial values: > >y(0)=1, and y'(0)=0. > >Series expansion (get by Mathematica) gives, for the convergence radius, >the numerical value 3.9626. What is the meaning of this value? In response, in article <8sqfit$qi1$1@nntp.itservices.ubc.ca>, Robert Israel wrote (correctly, of course): > >The solution in the complex plane has some sort of singularity >on the circle of radius 3.9626 centred at the origin, if that's the >radius. Actually the series, according to Maple, is > > 2 4 11 6 8 97 10 > y(x) = 1 - 1/6 x + 1/60 x - ---- x + 1/8505 x - -------- x + > 7560 10692000 [and more, deleted by djr] > >and assuming this pattern of signs continues indefinitely the >closest singularities to the origin will be on the imaginary axis. Taking a cue from this expansion we try replacing the (complex) variable x with u=-x^2; positive real values of u correspond to the points x on the imaginary axis. Using the chain rule we find the relationship between y and u to be / 2 \ /d \ |d | 2 6 |-- y(u)| + 4 u |--- y(u)| = y(u) \du / | 2 | \du / and the initial conditions y(0)=1, y'(0)=1/6. This ODE appears to admit the series solution y(u) = 1 + (1/6) u + (1/60) u^2 + ... consistent with Robert Israel's calculation. Furthermore, the ODE may be solved numerically without much trouble for small u (after using, say, the series solution to estimate y(eps) and y'(eps) for some small non-zero eps so as to avoid a division-by-zero error). For example, I find the solution curve has a point near (u, y) = (10, 10.2231422136984752) (where dy/du = 3.81661668823352369 or so). Now, it is proposed that there is a singularity of some kind near u = 3.9626^2 = 15.702, so we try to continue the numerical solution to larger u. This proves difficult, as indeed it appears there is a pole near this value of u, making both y and dy/du large. At this point it becomes preferable to study the curve on its side, that is, to think of u as a function of y, let y increase to infinity, and study the limiting value of u. In an attempt to study rates of convergence of u(y), I noticed my estimates were better expressed in terms of powers of z = y^(-1/2), which is more convenient anyway, since letting y approach infinity is equivalent to dropping z to the finite value 0. So let me do this _first_ (that is, before inverting the roles of u and the dependent variable). In the previous ODE substitute y = 1/z^2; then z = z(u) satisfies this equation: / 2 \ /d \ /d \2 |d | -12 z(u) |-- z(u)| + 24 u |-- z(u)| - 8 z(u) u |--- z(u)| = 1 \du / \du / | 2 | \du / Now the initial conditions are z(0)=1, z'(0)=-1/12. The previously found point on the curve now has coordinates (u, z) = (10, 0.312757547443019738) or thereabouts, and the tangent line has dz/du = -.0583810559410273223. We want to follow the curve to the point where z = 0 and expect to find there a point with u-coordinate near 15.7 . (One approach is to solve this ODE in series: z(u) = 1 - u/12 + ... and then solve the equation: [this approximation] = 0. I won't do it this way.) As I said before, it seems preferable to me to treat z as the independent variable, since the point we seek has a known z-coordinate (=0). To invert the equation, use the inverse function theorem: (dz/du) = 1 / (du/dz) and (d^2z/du^2) = - (d^2u/dz^2) / (du/dz)^3 . If I have done this correctly, the ODE is now: / 2 \ /d \2 /d \ |d | /d \3 -12 z |-- u(z)| + 24 u(z) |-- u(z)| + 8 z u(z) |--- u(z)| - |-- u(z)| \dz / \dz / | 2 | \dz / \dz / My numerical computations show that if we proceed from the point where (z, u) = (0.31... , 10) and du/dz = 1/(-.058...) , we terminate at z=0 with u = 15.7179392534512642 (and du/dz = -19.4224236922900625) (all approximate, of course). (Indeed, one may simply invert formally the previous series solution to get u = -12(z-1) + ... ; the coefficients of this series appear to decrease fairly rapidly so we may evaluate this series at (z-1) = -1 . Fifty terms gives the same value for u to within the magnitude of the last summand, about 10^(-12). ) Now, here is where I find the situation very interesting. The last ODE I wrote down now appears to have a solution which we find has u(0)=15.7... and u'(0)=-19.4... . These two numbers (nearly) satisfy the relation u(0) = u'(0)^2 / 24, which is convenient, since the ODE can be solved for u''(z) and we find a fraction with a numerical value at z=0, plus a term ( (u')^2 - 24 u ) / z which would mean u is not twice differentiable at z=0 unless u(0) = u'(0)^2 / 24 precisely. If we begin with that premise, we can attempt to find a series solution for the ODE near z = 0: for any nonzero constant c we find u(z) = 3 5 6 2 2 72 z 132192 z 12939264 z 7 1/24 c + c z + 18/5 z - -- ---- + ------ ---- - -------- ---- + O( z ) 25 c 625 3 3125 4 c c is a solution to the differential equation ... HOWEVER ... this series cannot be continued! Irrespective of any multiples of z^7, z^8, ... added to the terms shown, we find that when the series is substituted for u(z) in the ODE, there is a nonvanishing term beginning ( - 155271168/3125 / c^3 ) z^6 + ... I guess this shows there is no analytic solution u = u(z) to the differential equation near z=0, and further suppose this means that there is _not_ a simple pole but rather an essential singularity in the solution of the initial ODE, but I am rusty enough on my complex analysis that I would not put a lot of confidence in my conclusions here. Anyway, my computations lead me to believe the radius of convergence is closer to 3.9645856345211241 than to the 3.9626 mentioned by the original poster. I know of no significance to this number. Nothing nearby seemed appropriate in the tables at http://www.cecm.sfu.ca/projects/ISC/ISCmain.html I must say the original differential equation turned out to be more interesting than it first appeared. dave PS - This is almost, but not quite, an example of a Sturm-Liouville equation (p(x) y')' + q(x) y = 0. I know nothing about them, really, but perhaps some of the tools in S-L theory can be applied here too.