Module

for

Taylor's Method for solving O.D.E.'s

   

     The Taylor  series method is of general applicability and it is the standard to which we can compare the accuracy of the various other numerical methods for solving an I. V. P.  It can be devised to have any specified degree of accuracy.  

 

Theorem  (Taylor Series  Method of Order n)  Assume that  f(t,y)  is continuous and satisfies a Lipschits condition in the variable  y,  and consider the  I. V. P. (initial value problem)

        
[Graphics:Images/TaylorDEMod_gr_1.gif] with [Graphics:Images/TaylorDEMod_gr_2.gif],  over the interval  [Graphics:Images/TaylorDEMod_gr_3.gif].
        
The Taylor series method uses the formulas [Graphics:Images/TaylorDEMod_gr_4.gif],  and  

        [Graphics:Images/TaylorDEMod_gr_5.gif]     for  [Graphics:Images/TaylorDEMod_gr_6.gif]  

where [Graphics:Images/TaylorDEMod_gr_7.gif] is  [Graphics:Images/TaylorDEMod_gr_8.gif]  evaluated at  [Graphics:Images/TaylorDEMod_gr_9.gif],  as an approximate solution to the differential equation using the discrete set of points  [Graphics:Images/TaylorDEMod_gr_10.gif].  

Proof  Taylor Series  Method for O.D.E.'s  Taylor Series  Method for O. D.E.'s  

 

Theorem  (Precision  of Taylor Series  Method of Order n)  Assume that  [Graphics:Images/TaylorDEMod_gr_11.gif]  is the solution to the I.V.P.  [Graphics:Images/TaylorDEMod_gr_12.gif]  with  [Graphics:Images/TaylorDEMod_gr_13.gif].  If  [Graphics:Images/TaylorDEMod_gr_14.gif]  and  [Graphics:Images/TaylorDEMod_gr_15.gif]  is the sequence of approximations generated by the Taylor series method of order  n, then at each step, the local truncation error is of the order  [Graphics:Images/TaylorDEMod_gr_16.gif],  and the overall global truncation error  [Graphics:Images/TaylorDEMod_gr_17.gif] is of the order

        
[Graphics:Images/TaylorDEMod_gr_18.gif],  for  [Graphics:Images/TaylorDEMod_gr_19.gif].  

The error at the right end of the interval is called the final global error  

        
[Graphics:Images/TaylorDEMod_gr_20.gif].  

Proof  Taylor Series  Method for O.D.E.'s  Taylor Series  Method for O. D.E.'s  

 

Animations (Taylor Series  Method for O.D.E.'s  Taylor Series  Method for O.D.E.'s).  Internet hyperlinks to animations.

 

Algorithm (Taylor Series Method). To compute a numerical approximation for the solution of the initial value problem  [Graphics:Images/TaylorDEMod_gr_21.gif] with  [Graphics:Images/TaylorDEMod_gr_22.gif]  over [Graphics:Images/TaylorDEMod_gr_23.gif]  at a discrete set of points using the formulas  

    [Graphics:Images/TaylorDEMod_gr_24.gif],  and  [Graphics:Images/TaylorDEMod_gr_25.gif],  for  [Graphics:Images/TaylorDEMod_gr_26.gif]  
    
where [Graphics:Images/TaylorDEMod_gr_27.gif] is  [Graphics:Images/TaylorDEMod_gr_28.gif]  evaluated at  [Graphics:Images/TaylorDEMod_gr_29.gif].

Computer Programs  Taylor Series  Method for O.D.E.'s Taylor Series  Method for O.D.E.'s  

Mathematica Subroutine (Taylor Series  Method of Order n=4).

[Graphics:Images/TaylorDEMod_gr_30.gif]

Example 1.  Solve the I.V.P.  [Graphics:Images/TaylorDEMod_gr_31.gif].  
Solution  1.

 

Example 2.  Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/TaylorDEMod_gr_60.gif].  
Solution  2.

 

Example 3.  Plot the error for the Taylor  series 's method.
Solution  3.

 

Example 4.  Reduce the step size by  [Graphics:Images/TaylorDEMod_gr_80.gif] and see what happens to the error.
Recalculate points for Taylor  series 's method, and the analytic solution using twice as many subintervals.
Then Plot the error for Taylor  series 's method.
Solution  4.

 

Example 5.   Solve  [Graphics:Images/TaylorDEMod_gr_91.gif]  with  [Graphics:Images/TaylorDEMod_gr_92.gif]  over  [Graphics:Images/TaylorDEMod_gr_93.gif].
Solution  5.

 

Example 6.  Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/TaylorDEMod_gr_122.gif].  
Solution  6.

 

Example 7.  Plot the absolute value of the error for Taylor  series 's method.
Solution  7.

 

Example 8.  Reduce the step size by  [Graphics:Images/TaylorDEMod_gr_140.gif] and see what happens to the error.
Recalculate points for Taylor  series 's method, and the analytic solution using twice as many subintervals.
Then Plot the error for Taylor  series 's method.
Solution  8.

 

Example 9.  Solve the I.V.P.  [Graphics:Images/TaylorDEMod_gr_151.gif].  
Solution  9.

 

Example 10. Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/TaylorDEMod_gr_194.gif].  
Solution  10.

 

Various Scenarios and Animations for the Taylor's Method for solving O.D.E's

[Graphics:Images/TaylorDEMod_gr_214.gif]

Example 11.  Solve the I.V.P.  [Graphics:Images/TaylorDEMod_gr_215.gif].    Compute a Taylor series solution of order n=3 solution to the I.V.P.
Solution  11.

 

Example 12.  Solve the I.V.P.  [Graphics:Images/TaylorDEMod_gr_253.gif].    Compute a Taylor  series solution of order n=4 solution to the I.V.P.
Solution  12.

 

Animations (Taylor's Method  Taylor's Method).  Internet hyperlinks to animations.

 

Old Lab Project (Taylor's Method for O.D.E.'s  Taylor's Method for O.D.E.'s).  
Internet hyperlinks  to an old lab project.  

 

Research Experience for Undergraduates

Taylor Series  Method for ODE's  Taylor Series  Method for ODE's   Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Taylor's Method for solving O.D.E.'s

 

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(c) John H. Mathews 2004