Non-Linear BVP - ODE - using finite differences

Question

I have the following ODE

$EI\frac{{{d^2}y}}{{d{x^2}}} = [\frac{{ - qLx}}{2} + \frac{{q{x^2}}}{2}] \cdot {[1 + {(\frac{{dy}}{{dx}})^2}]^{\frac{3}{2}}}$

$\begin{array}{l} y(0) = 0\\ y(L) = 0 \end{array}$

Using a standard second order discretization of y′′(x) on a grid with uniform spacing the linear system of equations to solve becomes:

\begin{array}{l} EI\frac{{y(i + 1) - 2 \cdot y(i) + y(i - 1)}}{{{h^2}}} = [\frac{{ - qLx}}{2} + \frac{{q{x^2}}}{2}] \cdot {[1 + {(\frac{{y(i + 1) - y(i - 1)}}{{{h^2}}})^2}]^{\frac{3}{2}}} \end{array}

I' am not sure how to use Gauss-Seidel method or other method for solving this non-linear ODE using finite differences.

Thanks :)

Answer 1

Consider $y$ an approximation and $y+\delta y$ a closer approximation. Then in first order \begin{align} EI(y''+δy'')&=f(x)[1+y'^2+2y'δy'+δy'^2]^{3/2} \\ &=f(x)\left([1+y'^2]^{3/2}+3[1+y'^2]^{1/2}y'δy'+O(δy'^2)\right) \end{align} Ignoring the higher order terms this gives a linear DE for $δy$. Solving this and updating $y$ with the solution constitutes a Newton step, so one can expect rapid convergence (if there is any convergence at all).