Enter the initial value, radius of convergence, and number of iterations into the calculator to determine the result according to Picard's Theorem. This theorem is used in differential equations to find the number of times a function's values will iterate over a certain point.
Picard's Theorem Formula
The following iterative method is used to calculate the result according to Picard's Theorem.
y_{n+1} = y_0 + int_{t_0}^{t} f(s, y_n(s)) dsVariables:
- y_0 is the initial value
- r is the radius of convergence
- n is the number of iterations
- y_n is the result after n iterations
To calculate the result according to Picard's Theorem, an iterative method is used where the function's values are computed over and over again, each time using the previous iteration's result. The specific function f(s, y) and the limits of integration would depend on the differential equation being solved.
What is Picard's Theorem?
Picard's Theorem is a fundamental result in the theory of ordinary differential equations. It states that if a function satisfies certain conditions, then there exists a unique solution to the differential equation in the neighborhood of a given point. The theorem provides a method for constructing a sequence of approximate solutions that converge to the actual solution.
How to Calculate Using Picard's Theorem?
The following steps outline how to calculate using Picard's Theorem.
- First, determine the initial value (y_0).
- Next, determine the radius of convergence (r).
- Next, determine the number of iterations (n).
- Finally, apply the iterative method to calculate the result after n iterations (y_n).
- After inserting the variables and performing the iterations, check your answer with the calculator above.
Example Problem :
Use the following variables as an example problem to test your knowledge.
Initial Value (y_0) = 1
Radius of Convergence (r) = 2
Number of Iterations (n) = 3