Picard Iterative Integration Method for Solving ODEs

by admin in , , on April 27, 2019

This code uses Picard Iterative Integration Method to Solve single Ordinary Differential Equations (ODE), The method calculated using Matlab Symbolic Toolbox. The method is slow but very accurate even for a few iterations 3 or 4.

Code Outputs:

  • Chart presenting Picard Solution compared with Exact solution (if exist).
  • Error chart (the absolute error between exact and Picard solutions).
  • Printed Maximum absolute error.

Max. Error = 1.1314e-07

Input Requirements:

  • The differential equation intended to be calculated.
  • Integral Terms.
  • Exact solution if exist.
  • Number of iterations (Picard series order)

About the Method:

Picard–Lindelöf theoremPicard’s existence theoremCauchy–Lipschitz theorem, or existence and uniqueness theorem gives a set of conditions under which an initial value problem has a unique solution. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Consider the initial value problem

Suppose f is uniformly Lipschitz continuous in y (meaning the Lipschitz constant can be taken independent of t) and continuous in t, then for some value ε > 0, there exists a unique solution y(t) to the initial value problem on the interval [t0 – epsilon, t0 + epsilon].

Proof Sketch:

The proof relies on transforming the differential equation, and applying fixed-point theory. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation

A simple proof of existence of the solution is obtained by successive approximations. In this context, the method is known as Picard iteration.

Set

and

It can then be shown, by using the Banach fixed point theorem, that the sequence of ‘Picard iterates’ φk is convergent and that the limit is a solution to the problem. An application of Grönwall’s lemma to |φ(t) − ψ(t)|, where φ and ψ are two solutions, shows that φ(t) = ψ(t), thus proving the global uniqueness (the local uniqueness is a consequence of the uniqueness of the Banach fixed point).

 

References:

[1] Coddington, Earl A.; Levinson, Norman (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill..

[2] Lindelöf, E. (1894). ‘Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre’. Comptes rendus hebdomadaires des séances de l’Académie des sciences116: 454–457. (In that article Lindelöf discusses a generalization of an earlier approach by Picard.)

[3] Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.

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Release Information

  • Price
    :

    $4.99

  • Released
    :

    April 27, 2019

  • Last Updated
    :

    May 28, 2019

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