PLS, some one help me find the 1st 4 iterates using Picard's iteration.

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In summary, the conversation discusses using Picard's iteration to solve the problem dx/dt = -tx, x(0) = 1 and provides a link for further information. It also reminds the asker to provide relevant equations and concepts when seeking homework help. The first 4 iterates of solutions are then shown using the given information.
  • #1
manttiz
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Question

Use Picard's iteration to find the first 4 iterates of solutions of the problem dx/dt = -tx, x(0) = 1.

If anyone can help me solve this, I'll be so grateful. I wait anxiously to get the solution.
 
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  • #2
Welcome to the PF. We do not provide solutions to your homework. You must show us the relevant equations and concepts that apply to your question, and you must show us your work so far so that we can provide some tutorial help. In the future, please completely fill out the Homework Posting Template that is provided to you when you start a new Homework Help thread.
 
  • #3
I'm finding it impossible to write the equations here, though the clue can be gotten at http://www.cse.uiuc.edu/eot/modules/ode/picard/
 
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  • #4
If you have the differential equation dx/dt= f(x,t) with x(t0)= x0 then "Picard's interation", used in his proof of the existence of solutions of the initial value problem, is
[tex]x(t)= \int_{t_0}^t f(x,t)dt+ x_0[/tex]
starting with x(t)= x0 and then using the resulting x(t) in the next iteration.

Here, f(x,t)= -tx and x(0)= x0= 1 so the first "iteration" is
[tex]\int_0^t -tdt+ 1[/itex]
Surely you can continue from there.
 

1. What is Picard's iteration?

Picard's iteration is a mathematical technique used to solve iterative equations. It was developed by the French mathematician Charles Émile Picard and is commonly used in the field of numerical analysis.

2. How does Picard's iteration work?

Picard's iteration involves repeatedly applying a given function to an initial guess or estimate. The resulting sequence of values converges to the solution of the equation, assuming certain conditions are met.

3. What are the steps to find the first four iterates using Picard's iteration?

The steps to find the first four iterates using Picard's iteration are as follows:
1. Choose an initial guess or estimate for the solution.
2. Apply the given function to the initial guess to get the first iterate.
3. Use the first iterate as the new guess and apply the function again to get the second iterate.
4. Repeat this process for the third and fourth iterates.
5. The fourth iterate is the solution to the equation.

4. What are the conditions for Picard's iteration to converge?

For Picard's iteration to converge, the given function must be a contraction mapping, meaning it shrinks the distance between any two points. Additionally, the initial guess must be close enough to the solution and the equation must have a unique solution.

5. What are some applications of Picard's iteration in science?

Picard's iteration has applications in various fields of science, including physics, engineering, and economics. It can be used to solve differential equations, compute fixed points of functions, and find numerical solutions to various problems. It is also commonly used in machine learning and optimization algorithms.

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