We start with a problem where we need to find u such that
$$\frac { { d }^{ 2 }u }{ dx^{ 2 } } +\lambda { e }^{ u }=0$$
with 0 < x < 1, u(0)=u(1)=0 and $\lambda$ is a known constant.
We discretize the problem by using central finite-differences. We obtain the following non-linear set of equations $$\frac { u_{ i+1 }-2u_{ i }+u_{ i-1 } }{ h^{ 2 } } +\lambda { e }^{ u_{ i } }=0,\ i=1...n$$
where $u_i$ approximates $u(x_i)$ and $x_i$ = ih for i = 1,...,n with h = 1/(1+n). We also denote U $\in R^n$ the vector with entries $u_1$,...,$u_n$
We write the set of equations for $\{u_i\}_{i=1}^{n}$ as a non-linear equation f(U) = 0 with f:$R^n \to R^n$. What is the Jacobian of f?