Solution of differential equations: Picard’s Method

Picard’s method, also known as the Picard iteration or Picard’s successive approximation method, is a technique used to approximate solutions to ordinary differential equations (ODEs) or systems of ODEs. It’s an iterative process where you start with an initial guess and refine it successively until you converge to a solution.

Here’s a general outline of how Picard’s method works:

Formulate the ODE: Start with the given ordinary differential equation or system of ODEs.
Guess an Initial Solution: You need to make an initial guess for the solution. This can be any function that satisfies the initial conditions of the problem.
Iterative Process: Define a sequence of functions or solution approximations using an iterative formula. This formula typically involves integrating the ODE with the previous approximation.
Convergence Check: Check whether the sequence of approximations is converging to a solution. This is usually done by examining the difference between consecutive approximations and ensuring it decreases with each iteration.
Stopping Criterion: Decide on a stopping criterion, such as reaching a desired level of accuracy or convergence.
Solution: If the sequence converges within the desired tolerance, the limit of this sequence is considered the solution to the ODE.

Here’s a more detailed explanation with an example:

Example: Consider the first-order ordinary differential equation:

$\frac{d y}{d x} = x + y,$ with the initial condition

$y (0) = 1$ .

Step 1: Formulate the ODE:

$\frac{d y}{d x} = x + y$

Step 2: Guess an Initial Solution: Choose an initial guess. Let’s start with $y_{0} (x) = 1$ since it satisfies the initial condition.

Step 3: Iterative Process: The Picard iteration formula for this problem is:

$y_{n + 1} (x) = 1 + \int_{0 x} (x + y_{n} (x)) d x$

We start with

$n = 0$ , so

$y_{1} (x) = 1 + \int_{0 x} (x + 1) d x = 1 + \frac{x ^{2}}{2} + x$ .

Step 4: Convergence Check: Compute

$y_{2} (x)$ ,

$y_{3} (x)$ , and so on until the difference between consecutive iterations becomes small enough or until reaching a desired number of iterations.

Step 5: Stopping Criterion: Stop the iteration when the difference between consecutive iterations is less than a specified tolerance.

Step 6: Solution: The limit of the sequence of approximations (if it converges) is the solution to the ODE.

Picard’s method is powerful for theoretical analysis and can be implemented computationally, although it may not always converge or may converge slowly for certain problems. Additionally, it’s important to choose an appropriate initial guess and stopping criterion for practical applications.