Background

    If you are tired of your Runge-Kutta solution "blowing up" at a singularity, then this module could help remedy the situation.

Definition (Singularity).  A singularity of    is a point    at which     "blows up" or is not defined.

Example 1.  The functions    have singularities at the point  .  
The functions    have singularities at the point  .  
Solution 1.

Definition (Removable Singularity).  A singular point is called a removable singularity if has a Taylor series expansion about , i.e.  if   has a representation of the form

               

              valid for  .  

A common situation is that is not defined and we only need to define   in order to "remove the singularity."

Example 2.  The function     has a removable singularity at the point  .  
Solution 2.

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  .

Example 3 (a).  The functions     have poles at the point  .  
3 (b).  The functions     have poles at the point  .  
Solution 3 (a).
Solution 3 (b).

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 Painlevé 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. (Painlevé Property)  The second-order ordinary differential equation   has the Painlevé 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. (Painlevé Property)   The first-order ordinary differential equation   has the Painlevé 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.

Example 4.  Investigate the initial value problem    with  .   
Solution 4.

Example 5.   Investigate the initial value problem    with  .   
Solution 5.

Example 6.   Investigate the initial value problem    with  .   
Solution 6.

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.

Proof  Painlevé Property

Computer Programs The Runge-Kutta Method  

Computer Programs Painlevé Property  

Mathematica Subroutine (PoleVault Runge-Kutta Method of Order 4).

Example 7.  Use the extended Runge-Kutta method to compute a numerical approximation for  

               with      over the interval  .  

Solution 7.

Example 8.  Use the extended Runge-Kutta method to compute a numerical approximation for  

               with      over the interval  .  

Solution 8.

Example 9.  Use the extended Runge-Kutta method to compute a numerical approximation for  

               with     over the interval  .  

Solution 9.

Example 10.   Use the extended Runge-Kutta method to compute a numerical approximation for  

               with      over the interval  .  

Solution 10.

Example 11.  Use the extended Runge-Kutta method to compute a numerical approximation for  

               with     over the interval  .  

Solution 11.

Example 12.  Use the extended Runge-Kutta method to compute a numerical approximation for  

               with     over the interval  .  

Solution 12.

References

1.  John H. Mathews, Computed Solution Curves for Differential Equations, The AMATYC Review, Vol. 11, No. 1, (Part I), Fall 1989, pp. 30-33.  
2.  George F. Corliss, Integrating ODE's in the Complex Plane-Pole Vaulting, Mathematics of Computation, Vol. 35, No. 152 (Oct., 1980), pp. 1181-1189.   
3.  Kurtis Fink;  John Mathews, Numerical Methods Using Matlab, 4th Edition, ISBN 0-13-065248-2, Prentice-Hall Pub. Inc., Upper Saddle River, NJ, 2004.  

Research Experience for Undergraduates

Painlevé Property  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Painleve Property

(c) John H. Mathews 2005