Finite Difference Method

1D Boundary value problem

Differential equation

$$-a(x)u^{\prime \prime }(x)+b(x)u^{\prime }(x)+c(x)u(x)=f(x),x\in (0,1)$$

Boundary conditions

$$\begin{array}{rl}& u(0)=g_D\text{ Dirichlet boundary condition }\\ & u^{\prime }(1)=g_N\text{ Neumann boundary condition }\\ & \end{array}$$

Goal: Apply Finite Differences to discretize the problem.

Step 1: Meshing

$x\in [0,1]$ is divided with $n$ equidistant parts.

$$x_j=jh,j=0,\dots ,n,h=\frac{1}{n}$$

Step 2: Discretization

The derivative can be calculated using three different ways

The Differential equation then can be written

Step 4: Plotting