# Finite Differences Metods for the Laplace's Equation.

My question is not about content, but I would like to ask for references to a topic I have never studied. I need to do research on the following topic: "Finite Difference Methods for the One-Dimensional Laplace's Equation: Stability, Consistency and Convergence".

Already taking advantage of the question, researching PDE references the corresponding Laplace’s equation (i.e, one-dimensional) is $$u''(x) = 0$$ and is trivial: the solution is a linear function connecting the two boundary values. I was a little confused by this, would the content of the topic be applying finite difference methods to the equation $$u''(x) = 0$$?

Indeed, the question seems a bit trivial for the analytic solution is easy to find, namely $$u(x) = a\, x + b$$ with constants $$a, b$$.
But maybe you are supposed to do research on the finite difference methods provided you don't know the analytic solution. Assume, all what you know is that you must satisfy the boundary conditions $$u(x_1) = u_1$$ and $$u(x_2) = u_2$$ for say $$x_1 < x_2$$. How would you proceed?
One well-known method is to take an initial "random" smooth function $$f(x)$$ which satisfies the boundary conditions (but does not satisfy the Laplace equation) and solve a time-dependent diffusion equation $$\partial_t u(t,x) = D\, \Delta u\tag{1}$$ with the initial data $$u(t,x) = f(x)$$ and the above mentioned boundary conditions. It can be shown that the solution will tend to the solution of the Laplace equation as $$t \rightarrow \infty$$.
Now, one can discretize the Laplace operator, e.g. $$\Delta u(x) \rightarrow \frac{u_{n+1} - 2 u_n + u_{n+1}}{\Delta x^2}$$ with $$\Delta x$$ being the space discretization step and solve (1) numerically in finite time-steps $$\Delta t$$. When $$\Delta t$$ is too large relative to $$\Delta x$$ the algorithm will not converge.