Painleve Property

Definition (Pole).  A singular point
is called a pole if
has a series expansion about
which includes only a finite number of negative powers
with  
,  i.e.  if  
has a representation of the form


   

      
  valid for  
.  

The leading coefficient must be non-zero,  
,  and we say that
has a pole of order  
  at  
.  When  
  we say it has a simple pole at  
.

Remark.  When you look at the graph  
, a pole at  
  is a vertical asymptote at  
.     

Theorem (Poles and Zeros).  If  
  has a pole at  
  then the function  
  has a removable singularity at  
.  If we define  
   then the equation  
  will have a root at  
.

Definition (Logarithmic Singularity).  A logarithmic singularity involves a logarithmic branch point in the complex plane.

For example, the function  
has a logarithmic singularitie at the point  
.  

Definition (Algebraic Branch Point).  A algebraic branch point is a singular point associated with a fractional power.

For example, the "multivalued function"  
has algebraic branch point at  
.  

Restriction

    The Painlev� property excludes the occurance of logarighmic branch points and algebraic branch points.  The underlying solution must be analytic except at isolated points where it has poles. It is not necessary to dwell on the above definitions, but it is important to know that we are restricting the type of singularities we want to allow.

Definition (Movable Singularity).  If the singularities of a differential equation depend on the initial conditions then they are said to be movable singularities.

Definition 1. (Painlev� Property)  The second-order ordinary differential equation  
has the Painlev� property if  all movable singularities of all solutions are poles.

Remark.  We will take the liberty to extend this concept to first order equations.  

Definition 2. (Painlev� Property)   The first-order ordinary differential equation  
has the Painlev� property if  all movable singularities of all solutions are poles.

Remark.  Movable singularities depend on initial conditions and in general it is difficult to predict their location.  The following examples have been chosen because the analytic solution can be found.

Computed Solution Curves for Differential Equations

    An important problem in numerical analysis is to compute approximate solutions of the differential equation  

(1)        
.  

Under modest (and well known) assumptions on f, the "general solution" of (1) consists of an infinite family of functions, each of which may be distinguished by selection of an initial point
.  Starting from this initial point, numerical methods attempt to approximate the solution  
  on some specified interval  
.   Continuity of  
  does not ensure the continuity of  
.

    Suppose that  
  has an infinite discontinuity at  
,  that is
.  Then the reciprocal  
  tends to zero as  
,  and  
  will have a removable singularity at  
  provided that we define  
.   We can use the change of variable

(2)        
.  

Now differentiate each side of (2) and get

        
  

Then substitute  
  from (1) and obtain

(3)        
  

Differential equation (3) is equivalent to (1) in this sense:  Given a neighborhood N of  
  and a number  
,  equation (1) has a solution with  
  and  
  for all x in N if and only if equation (3) has a solution with  
  and  
.

    We call equation (3) the companion differential equation and write it as

(4)        
.    

    Numerical methods "track" a specific solution curve through the starting point  
.  The success of using (4) for tracking the solution  
  near a singularity is the fact that  
  as  
  if and only if  
  as  
.  A numerical solution  
  to (4) can be computed over a small interval containing  
,  then (2) is used to determine a solution curve for (1) that lies on both sides of the vertical asymptote  
.  

    A procedure such as the Runge-Kutta method, uses a fixed step size  
  and for each  
  an approximation  
  is computed for  
.  If  
  as  
  then the numerical method fails to follow the true solution accurately because of the inherent numerical instability of computing a "rise" as the product of a very large slope and very small "run" (a computation which magnifies the error present in the value
).   One way to reduce this error is to select a bound B and change computational strategy as soon as a value  
  is computed for which  
, that is, as soon as the possibility of a singularity is "sensed."   Then we stop using (1) and start with the point  
  as an initial value to the differential equation (4).  Then proceed to track the reciprocal  
,  which will not suffer from the difficulties created by steep slopes.

    The following strategy can be employed to extend any single-step numerical method.  We use equation (1) and the initial value  

        
  and compute   


where    
   for  
  and  
.  

    Then switch equations and use (4) with the initial value  

        
  and compute   
  

where  
for  
  and  
.  

Continue in a similar fashion and alternate between formula (1) and formula (4) until  
  

    The decision process, for the "extended" Runge-Kutta method is:

    IF
THEN
    
        Perform one Runge-Kutta step using   
  to compute  
,

    ELSE
    
        Set  
  and perform one Runge-Kutta step using   
   to compute  
,  
        and keep track of   
.
        
    ENDIF

Before (4) is used for numerical computations, the formula for  
  should be simplified in advance so that the  "
"  or  "
"  computational problems do not occur.