numerically solving differential equations $\frac{d^2 \theta}{dx^2} (1 + \beta \theta) + \beta \left(\frac{d \theta}{d x}\right)^2  - m^2 \theta  = 0$
Boundary Conditions
$\theta=100$ at $x = 0$, $\frac{d\theta}{dx} = 0$ at $x = 2$
$\beta$ and $m$ are constants.
Please help me solve this numerically (using finite difference).
The squared term is really complicating things!
Thank You!
 A: Choose an integer $N$, let $h=2/N$ and let $\theta_k$ be the approximation given by the finite difference method to the exact value $\theta(k\,h)$, $0\le k\le N$. We get the system of $N-1$ equations
$$
\frac{\theta_{k+1}-2\,\theta_k+\theta_{k-1}}{h^2}(1+\beta\,\theta_k)+\beta\,\Bigl(\frac{\theta_k-\theta_{k-1}}{h}\Bigr)^2-m^2\,\theta_k=0,\quad 1\le k\le N-1\tag1
$$
complemented with two more coming from the boundary conditions:
$$
\theta_0=100,\quad \theta_N-\theta_{N-1}=0.
$$
I doubt that this nonlinear system can be solved explicitly.
I suggest two ways of proceeding. The first is to solve the system numerically. The other is to apply a shooting method to the equation.
Choose a starting value $\theta_N=a$. The system (1) can be solved recursively, obtaining at the end a value $\theta_0=\theta_0(a)$. If $\theta_0(a)=100$, you are done. If not, change the value of $a$ and repeat the process. Your first goal is to find two values $a_1$ and $a_2$ such that $\theta_0(a_1)<100<\theta_0(a_2)$. Then use the bissection method to approximate a value of $a$ such that $\theta_0(a)=100$.
