Preface


This section demonstrates Picard's iteration scheme applicable to boundary value problems.

Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to the main page for the course APMA0330
Return to the main page for the course APMA0340
Return to Part VII of the course APMA0330

Picard Iterations


As early as 1893, Émile Picard constructively treated the second order differential equation subject to Dirichlet boundary conditions

\[ y'' (x) = f\left( x, y, y' \right) , \qquad y(a) = A , \quad y(b) = B , \]
where f(x,y,p) is a continuous functions of three independent variables.

A differential equation (as above) involves a derivative operator that is unbounded operator. This means that the derivative operator can map a bounded function into unbounded one. Therefore, iterations with noncontinuous operators may be problematic to deal with. This is avoided upon applying the (left) inverse operator \( \texttt{D}^{-2} \) to obtain a fixed point equation with bounded operator:

\[ y(x) = \texttt{D}^{-2}f\left( x, y, y' \right) . \]
The Picard iteration scheme is based on its equivalent integral reformulation (upon application of \( \texttt{D}^{-2} \) ):
\[ y(x) = A + \left( x-a \right) \frac{B-A}{b-a} + \frac{1}{2} \int_0^x \left( x-t \right) f(t)\,{\text d}t - \frac{1}{2} \int_x^b \left( x-t \right) f(t)\,{\text d}t + \frac{1}{2} \,\frac{x-b}{a-b} \int_a^b \left( a-t \right) f(t) \,{\text d}t - \frac{1}{2} \,\frac{x-a}{b-a} \int_a^b \left( b-t \right) f(t) \,{\text d}t. \]

Example: For our first example, we consider a standard "benchmark" problem:

\[ y'' + \pi^2 y =x, \qquad y(0) =1, \quad y \left( \frac{1}{2} \right) = -1. \]
Application of Picard's iteration gives
\[ y_{m+1} (x) = 1-4x - \frac{\pi^2}{2} \int_0^x \left( x-t \right) y_m (t)\,{\text d}t + \frac{\pi^2}{2} \int_x^{1/2} \left( x-t \right) y_m (t)\,{\text d}t + \pi^2 x \int_0^{1/2} \left( \frac{1}{2} -t \right) y_m (t) \,{\text d}t - \left( x- \frac{1}{2} \right) \int_0^{1/2} t\, y_m (t) \,{\text d}t, \quad m=0,1,2,\ldots . \]
As an initial approximation, we choose the linear function that satisfies the given bundary conditions:
\[ y_0 = -2x . \]
The truue solution is
\[ y(x) = \cos (\pi x) - \frac{1 + 2\,\pi^2}{2\,\pi^2}\,\sin (\pi x) + \frac{x}{2\pi^2} . \]
   ■

Example: Consider the boundary value problem

\[ y'' = 6\, y^2 , \qquad y(0) = \frac{1}{4} , \quad y(1) = \frac{1}{9} . \]
Its exact solution is
\[ y(x) = \left( x+2 \right)^{-2} . \]
Using the initial approximation,
\[ y_0 (x) = \frac{1}{4} - \frac{5}{36}\, x , \]
we apply the formula for the inverse derivative operator \( \texttt{D}^{-2} \) to obtain
\[ y_{m+1} (x) = \frac{1}{4} - \frac{5}{36}\,x + x \int_0^x \left( x-t \right) y_m^2 (t)\,{\text d}t - 3 \int_x^{1} \left( x-t \right) y_m^2 (t)\,{\text d}t + 3 \left( x-1 \right) \int_0^{1} t \, y_m^2 (t) \,{\text d}t - 3\,x \int_0^{1} t\, y_m^2 (t) \,{\text d}t, \quad m=0,1,2,\ldots . \]
   ■

Example:    ■

 

  1. Agarwal, R.P. and Loi, S.L., On approximate Picard's iterates for multipoint boundary value problems, Nonlinear Analysis: Theory, Methods & Applications, 1984, Vol. 8, Issue 4, pp. 381--391.
  2. Lai, M. and Moffatt, D., Picard;s successive approximation for non-linear two-point boundary value problems, Journal of Computational and Applied Mathematics, 1982, Vol. 8, No 4, pp. 233--236.
  3. Robin, W.A., Solving differential equations using modified Picard iteration, International Journal of Mathematical Education in Science and Technology, 2010, Vol. 41, No. 5, pp.649--665; https://doi.org/10.1080/00207391003675182

 

Return to Mathematica page
Return to the main page (APMA0330)
Return to the Part 1 (Plotting)
Return to the Part 2 (First Order ODEs)
Return to the Part 3 (Numerical Methods)
Return to the Part 4 (Second and Higher Order ODEs)
Return to the Part 5 (Series and Recurrences)
Return to the Part 6 (Laplace Transform)
Return to the Part 7 (Boundary Value Problems)