Appendix A2. Picard Iteration
The following first order initial value problem
y'(t) = f( t , y(t) ) , y(0) = 0
can be reformulated as the an integral equation
y(t) =
See Ledder, Appendix A.2.
Assume that f(t, y) and its y partial are continuous on a rectangle containing the point (0,0). Then starting with an initial "guess" for a solution
the recursive relation
generates a sequence of functions converging to the solution. These functions are called the Picard iterates generated by g(t).
Mathematica can generate some of these iterates, use an iteratively defined process.
The following example is taken from Ledder, see Chapter A.2, Exercise 2.
Example Obtain the first three Picard iterates for the following IVP, generated by the function g(t) = 0.
y'(t) = t +
Sketch their graphs and the graph of the exact solution.
Start with the definition of the function f.
In[1]:=
f[t_,y_] := t + t^2*y
Use it to define the differential equation.
In[2]:=
DE = y'[t] == f[t,y[t]]
Out[2]=
Define the expression Y[0,t] to be 0. Then define Y[n,t] recursively as shown below.
In[3]:=
Y[0,t_] := 0
Y[n_,t_] := Integrate[f[s,Y[n-1,s]], {s,0,t}]
Here are the first three Picard iterates.
In[5]:=
Table[Y[n,t], {n,1,3} ]
Out[5]=
The solution formula is obtained next. It is not pretty.
In[6]:=
soln = DSolve[ {DE, y[0]==0}, y[t], t ]
Out[6]=
The plot of the solution, red and thick, and the three iterates, blue, green, black:
In[7]:=
Show[ Plot[ y[t]/.soln, {t,0,2},
PlotStyle->{RGBColor[1,0,0],Thickness[0.02]} ],
Plot[ Y[1,t], {t,0,2}, PlotStyle->{RGBColor[0,0,1]} ],
Plot[ Y[2,t], {t,0,2}, PlotStyle->{RGBColor[0,1,0]} ],
Plot[ Y[3,t], {t,0,2} ], PlotRange->{0,5} ]
The next output reveals what appears to be a bug in Mathematica. The Picard iterates are the partial sums for the Taylor series for the solution yet Series generates a series approximation to the solution that does not converge to the correct solution curve.
In[21]:=
Series[ y[t]/.soln[[1]], {t,0,8} ];
Normal[%]
Out[22]=
In[23]:=
Plot[ %, {t,0,2}, PlotRange->{0,5} ]
Created by Mathematica (December 9, 2004)