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ozkan12
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İn some articles...I see something... Let ${x}_{n}=T{x}^{n-1}$ be a Picard Operator...Then ${f}^{n}x \subseteq {f}^{n+1}x$...What is the meaning of this ? Can you help me ?
Picard Iteration is a method used to numerically approximate the solution of a differential equation. It involves repeating a process of solving for the next iteration based on the previous one, until the desired level of accuracy is achieved.
First, an initial guess for the solution is made. Then, this guess is substituted into the differential equation to solve for the next iteration. This process is repeated until the desired level of accuracy is achieved.
This inequality represents the relationship between the solutions at two consecutive iterations in the Picard Iteration process. It means that the solution at the (n+1)th iteration will always be contained within the solution at the nth iteration.
Picard Iteration is useful because it allows for the numerical approximation of solutions to differential equations that cannot be solved analytically. It is also a relatively simple and straightforward method to implement.
One limitation of Picard Iteration is that it may not always converge to the actual solution. This can occur if the initial guess is too far from the true solution or if the differential equation is ill-behaved. Additionally, the method may require a large number of iterations to achieve the desired level of accuracy.