Understanding Picard Iteration: A Guide for Teens

[mathelord]

[SOLVED] picard iteration

I tried reading beyond my level,but i am having some problems.i do not understand the picard iteration,who can explain this to a boy of 16.

 

[AKG]

http://www.tmt.ugal.ro/crios/Support/ANPT/Curs/math/s1/s1pic/s1pic.html explains it. If you have a DE

y' = f(x,y)

then y is that function which satisfies:

$$\mathbf{y}(x) = y(x_0) + \int _{x_0} ^x f(\xi , \mathbf{y}(\xi ))d\xi$$

Notice how I boldfaced the "y"s. Pick functions f and w "randomly," and plug them into the right side of the equation. Compute it, and you'll get function of x, call it z(x). Is this z(x) the same as your w(x)? Probably not. Picard iteration gives you a technique so that even if you're starting out with some function y_0 that doesn't satisfy the differential equation, you can iteratively get new functions, and approach the actual function in the limit. Suppose you're looking for the smallest non-negative number, and you start with 2. That's not small enough, so you divide by 2 and get 1. Still not small enough, so you keep dividing by 2, and in the limit you will approach the answer, 0. Picard's technique is a similar idea. It gives you an iterative procedure to find a sequence of functions whose limit is the function that satisfies the DE. This process works because the functions in question form a complete metric space, so you can prove that this sequence has a limit, and that the limit is a fixed point if you try to apply the iterative function to it. Our function w was not a fixed point, because we put it in the right side, and got a different function, z, coming out. But remember how I put "y" in bold. You can see that y is a fixed point. You put y in the right side, and you get y back.

 

[M98Ranger]

Sure, I would be happy to explain Picard Iteration to you. Picard Iteration is a mathematical method used to solve differential equations. It is named after the French mathematician Charles Émile Picard.

To understand Picard Iteration, you first need to know what a differential equation is. A differential equation is an equation that contains a function and its derivatives. For example, dy/dx = x is a simple differential equation.

Now, let's say we have a more complicated differential equation like dy/dx = x^2 + y. This equation cannot be solved directly, so we use Picard Iteration to approximate the solution.

Picard Iteration works by breaking down the original equation into smaller, simpler equations. We start with an initial guess for the solution, let's say y = 0. Then, we use this guess to find a better approximation for the solution.

Using our initial guess, we can plug in y = 0 into the equation dy/dx = x^2 + y, which gives us dy/dx = x^2. This is a simpler equation that we can solve easily. We find the solution to this equation, let's say y = x^3/3.

Now, we take this new solution and plug it back into the original equation, dy/dx = x^2 + y. This gives us a new equation dy/dx = x^2 + x^3/3. We solve this equation to get a better approximation for the solution.

We keep repeating this process, each time using our new approximation to find an even better one. The more times we repeat this process, the closer we get to the actual solution of the original differential equation.

I hope this explanation helps you understand Picard Iteration better. It may seem complicated at first, but with practice, you will be able to solve more complex differential equations using this method. Keep learning and don't give up!