Solving Picard Iteration: Step-by-Step Guide

[AkilMAI]

I'm doing some revision and I'm having some trouble understanding the principle behind picard's iteration.I tried to do some practice by tackling the following problem but I got nowhere as I don't know with what should I substitute f(s,u(s)) in the integral.
The problem states the following:
u'= v and v'= −u
with initial conditions u(0) = 1 and v(0) = 0, find an approximate solution by
performing 4 steps of Picard iteration and compare the results with the actual solution.
If someone has the time and patience to enlighten me with a step by step explanation that will be incredible.
Thanks at least for taking the time to read this

 

[lippyka]

.Picard's iteration is a way of finding approximate solutions to nonlinear differential equations by using an iterative process. The idea is to start with an initial guess for the solution, then use this guess to calculate a better approximation of the solution. This is repeated until the desired accuracy is reached. In this problem, the differential equation is u' = v and v' = -u, with initial conditions u(0) = 1 and v(0) = 0. The first step of Picard's Iteration is to make an initial guess for the solution. Since we know the initial conditions, we can make an educated guess for the solution: u(s) = 1 and v(s) = 0. Now we use this initial guess to calculate a better approximation of the solution. This is done by substituting the initial guess into the equation and solving for the next step of the iteration:u'(s) = v(s) = 0 v'(s) = -u(s) = -1We can now use these equations to calculate our next approximation of the solution. We do this by integrating each equation with respect to s and using the initial conditions to find the constants of integration:u(s) = 1 + sv(s) = -sNow that we have a better approximation of the solution, we can repeat this process until we reach the desired accuracy. We do this by substituting our current guess for the solution back into the equation and solving for the next step:u'(s) = v(s) = -s v'(s) = -u(s) = -1 - sWe then integrate each equation with respect to s and use the initial conditions to find the constants of integration:u(s) = 1 + s - s^2/2v(s) = -s + s^2/2We can continue to repeat this process until we reach the desired accuracy. After four steps of Picard iteration, we get the following approximation of the solution:u(s) = 1 + s - s^2/2 + s^3/6 - s^4/24v(s) = -s + s^2/2 - s^3/6 + s^4/