Last modified: December 22, 2024

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Picard's Iteration Method

Picard's method, alternatively known as the method of successive approximations, is a tool primarily used for solving initial-value problems for first-order ordinary differential equations (ODEs). The approach hinges on an iterative process that approximates the solution of an ODE. Though this method is notably simple and direct, it can sometimes converge slowly, depending on the specifics of the problem at hand.

Mathematical Formulation of Picard's Method

Picard's method originates from the idea of expressing the solution to an ODE as an infinite series. We then approximate this solution by truncating the series after a certain number of terms. Given a first-order ODE of the general form $y' = f(x, y)$, complemented by the initial condition $y(x_0) = y_0$, we can express the iterative formula for Picard's method as follows:

$$y_{n+1}(x) = y_0 + \int_{x_0}^{x} f(t, y_n(t)) dt$$

Derivation

Picard's method of successive approximations is based on the principle of transforming the differential equation into an equivalent integral equation, which can then be solved iteratively. Let's take a look at how it's derived.

Consider a first-order ordinary differential equation (ODE):

$$ y'(t) = f(t, y(t)),$$

with an initial condition:

$$ y(t_0) = y_0.$$

The main idea of Picard's method is to rewrite the ODE as an equivalent integral equation by integrating both sides with respect to $t$ over the interval $[t_0, t]$:

$$\int_{t_0}^t y'(t) dt = \int_{t_0}^t f(t, y(t)) dt.$$

The left-hand side of the equation becomes $y(t) - y(t_0)$ by the fundamental theorem of calculus, which we can rewrite using our initial condition to obtain:

$$y(t) = y_0 + \int_{t_0}^t f(s, y(s)) ds.$$

Now we can see the integral equation form of the initial value problem.

To solve this equation using Picard's method, we generate a sequence of functions ${y_n(t)}$ iteratively as:

$$y_{n+1}(t) = y_0 + \int_{t_0}^t f(s, y_n(s)) ds,$$

where $y_0(t)$ is the initial guess (often taken as the constant function $y_0(t) = y_0$).

Each approximation $y_{n+1}(t)$ is defined in terms of the previous approximation $y_n(t)$, creating an iterative process which can be repeated until a desired level of accuracy is achieved.

Algorithm steps

  1. Start with an initial approximation $y_0(x)$.
  2. Calculate the next approximation $y_{n+1}(x) = y_0 + \int_{x_0}^{x} f(t, y_n(t)) dt$.
  3. Repeat step 2 for a pre-determined number of iterations $n$ or until the approximation of the solution meets the required accuracy.

Example Application of Picard's Method

Suppose we are given the ODE $y' = x + y$ with the initial condition $y(0) = 1$.

Starting with an initial approximation $y_0(x) = 1$, we follow these steps:

  1. Compute $y_1(x) = 1 + \int_0^x (t + y_0(t)) dt = 1 + \int_0^x (t + 1) dt = 1 + [\frac{1}{2}t^2 + t]_0^x = 1 + \frac{1}{2}x^2 + x$.
  2. Compute $y_2(x) = 1 + \int_0^x (t + y_1(t)) dt = 1 + \int_0^x (t + 1 + \frac{1}{2}t^2 + t) dt = 1 + [\frac{1}{6}t^3 + t^2 + t]_0^x = 1 + \frac{1}{6}x^3 + x^2 + x$.
  3. Continue the iterations until a satisfactory approximation of the solution is reached or the maximum allowable iterations is met.

In this example, the successive approximations for the solution of the ODE $y' = x + y$ become $y(x) = 1 + \frac{1}{2}x^2 + x$ and $y(x) = 1 + \frac{1}{6}x^3 + x^2 + x$.

Advantages of Using Picard's Method

Limitations of Picard's Method

Table of Contents

  1. Picard's Iteration Method
  2. Mathematical Formulation of Picard's Method
  3. Derivation
  4. Algorithm steps
  5. Example Application of Picard's Method
  6. Advantages of Using Picard's Method
  7. Limitations of Picard's Method