Non-Linear BVP | Sahithyan's Notes

A boundary value problem is nonlinear iff its differential equation cannot be written as

y'' = p(x)y' + q(x)y + r(x)

for functions $p$ , $q$ , and $r$ .

Nonlinear Shooting Method

Finds the initial slope $y'(a)$ that makes an IVP solution hit $y(b) = \beta$ .

For a guess $t$ , let $y(x,t)$ solve

y'' = f(x,y,y'), \quad y(a) = \alpha,\ y'(a) = t

Find $t$ such that $y(b,t) - \beta = 0$ with Newton’s method.

Define $z(x,t) = \dfrac{\partial y(x,t)}{\partial t}$ . Differentiating the IVP gives

z'' = f_y(x,y,y')z + f_{y'}(x,y,y')z', \quad z(a) = 0,\ z'(a) = 1

The Newton update is

t_k = t_{k-1} - \frac{y(b,t_{k-1}) - \beta}{z(b,t_{k-1})}

Choose an initial guess $t_0$
For $k = 0, 1, 2, \ldots$ , solve the IVPs for $y_k(x) = y(x,t_k)$ and $z_k(x) = z(x,t_k)$
Stop when $|y_k(b) - \beta|$ is sufficiently small
Update $t_{k+1} = t_k - \dfrac{y_k(b)-\beta}{z_k(b)}$

Replace derivatives with centered differences and solve the resulting nonlinear system with Newton’s method.

A unique solution exists iff

$f$ , $f_y$ , and $f_{y'}$ are continuous for $x \in [a,b]$ and $y, y' \in (-\infty, \infty)$
$f_y(x,y,y') \geq \delta$ for some $\delta > 0$
A constant $k$ exists with $k = \max\big(|f_y(x,y,y')|,\ |f_{y'}(x,y,y')|\big)$

For mesh points $x_0, \ldots, x_{N+1}$ , with $y(x_i) \approx w_i$ ,

w_0 = \alpha, \quad w_{N+1} = \beta, \quad -\frac{w_{i+1}-2w_i+w_{i-1}}{h^2} + f\left(x_i,w_i,\frac{w_{i+1}-w_{i-1}}{2h}\right) = 0

for $i = 1, \ldots, N$ , with truncation error $O(h^2)$ .

The $N$ equations are

2w_1-w_2+h^2f\left(x_1,w_1,\frac{w_2-\alpha}{2h}\right)-\alpha=0 \\[8pt] -w_{i-1}+2w_i-w_{i+1}+h^2f\left(x_i,w_i,\frac{w_{i+1}-w_{i-1}}{2h}\right)=0, \quad i=2,\ldots,N-1 \\[8pt] -w_{N-1}+2w_N+h^2f\left(x_N,w_N,\frac{\beta-w_{N-1}}{2h}\right)-\beta=0

Solve with Newton’s method. A unique solution exists whenever $h < \dfrac{2}{L}$ .