Analytical fixed point iteration method

Question

I have a system of nonlinear ordinary diffferential equations and i want to use analytical fixed point iteration method. Unfortunately i could only see computational resources but i couldnt find anything on this method. Can somebody please explain me how one obtains analytical solution to nonlinear ODE's using fixed point analytical method.

Answer 1

I imagine that the iteration gives an analytic solution of nonlinear ODE only in very exceptional cases. Here is one, extremely simple example: $$y'=y^2, \quad y(0)=1$$ The integral form is $$y_{n+1}=1+\int_0^x y_n^2(t)\,dt$$ Starting with $y_0=1$, we get $y_1=1+x$, $y_2=1+x+x^2+x^3/3$, and so forth. All are polynomials.

Claim: for $0\le k\le n$, the coefficient of $x^k $ in $y_n $ is equal to $1$.

Proof: by indiction on $n$. By assumption, $$y_{n-1}=1+x+\dots +x^{n-1} + \text{(powers above $n-1$)}$$ Hence, the coefficient of $x^{k-1}$ in $y_{n-1}^2$ is the same as its coefficient in $$(1+x+\dots+x^{k-1} )((1+x+\dots+x^{k-1})$$ which is $k$. The integration of $ kx^{k-1}$ produces $x^k$, proving the claim.

From the above, one can conclude that the iterations (which are already known to converge) converge to $\sum_{k=0}^\infty x^k$, which is $\dfrac{1}{1-x}$, the solution of the ODE.