8.1.2. Basic Concept

Given a first-order differential equation of the form:

𝑑𝑦 = 𝑓(𝑥,𝑦), 𝑦(𝑥 )= 𝑦 𝑑𝑥 0 0

This equation represents a first-order ordinary differential equation (ODE), where '𝑓' is a function that depends on two variables, '𝑥' and '𝑦'. In this context, '𝑦' is a function we want to solve for, and '𝑑𝑦/𝑑𝑥' describes how '𝑦' changes with respect to '𝑥'. The initial condition '𝑦(𝑥₀) = 𝑦₀' provides a starting point for solving the equation, meaning at a specific '𝑥₀', '𝑦' takes the value of '𝑦₀'.

Think of this equation like a recipe where '𝑦' is the dish you're trying to prepare and '𝑓(𝑥,𝑦)' tells you how the temperature of the oven (represented by '𝑥') influences how the dish (represented by '𝑦') cooks over time.

Picard’s method rewrites it as an equivalent integral equation using the Fundamental Theorem of Calculus:

𝑥 𝑦(𝑥) = 𝑦 + ∫ 𝑓(𝑡,𝑦(𝑡)) 𝑑𝑡 0

𝑥 0

Picard's method transforms the differential equation into an integral equation. The Fundamental Theorem of Calculus links differentiation and integration, allowing us to express the solution '𝑦(𝑥)' in terms of integrals. Here, '𝑦' is the initial value, and the integral computes the cumulative effect of the change described by '𝑓(𝑡,𝑦(𝑡))' from the starting point up to '𝑥'. The integral captures how '𝑦' accumulates changes due to the function '𝑓' over the interval.

Imagine you're tracking how much water fills a tank over time. The initial amount of water is like the initial value of '𝑦', and the integral represents how you track the flow rate of water into the tank over time, leading to the total amount of water at any point '𝑥'.

This equation is then solved iteratively using successive approximations:

𝑥 𝑦 (𝑥)= 𝑦 + ∫ 𝑓(𝑡,𝑦 (𝑡)) 𝑑𝑡 𝑛+1 0 𝑛

𝑥 0

Picard's method involves creating a series of approximations to refine our estimate of the solution. Starting with an initial guess (often the initial condition itself), we repeatedly apply the integral equation to improve our approximation. Each new function '𝑦ⁿ⁺¹(𝑥)' builds upon the previous one '𝑦ⁿ(𝑡)', indicating that with each iteration, we use the most recent approximation to calculate the next, effectively honing in on the actual solution as we go.

Consider polishing a rough stone. Your first attempt might yield only minor improvements, but with each pass (or iteration), you refine the surface to reveal the stone’s true shine. Each iteration in Picard's method is similar, as you continually refine your estimate of the solution until it's as smooth (accurate) as needed.

1. Start with an initial approximation: Usually, this is taken as a constant function based on the initial value:

𝑦 (𝑥) = 𝑦 0 0

2. Iterate using the integral form: Compute subsequent approximations using:

𝑥 𝑦 (𝑥)= 𝑦 + ∫ 𝑓(𝑡,𝑦 (𝑡)) 𝑑𝑡 𝑛+1 0 𝑛

𝑥 0

3. Repeat until convergence: Continue the iteration until the difference between successive approximations is sufficiently small.

The steps of Picard's method begin with taking our initial guess for '𝑦(𝑥)' as simply '𝑦₀'. In the second step, we calculate the new value using the integral form, incorporating our previous approximation. This process is repeated, refining our approximation with each cycle until the difference between successive approximations becomes negligible (convergence), indicating we've arrived close to the true solution.

Think of adjusting a dial to get a precise temperature on a thermostat. You start by setting it close to what you think is correct ('𝑦₀'). Each time the system observes the actual temperature, you make small adjustments (iterations) until the displayed temperature is the same as the set point (convergence).

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