Boundary Value Problems, Relxation Methods

• Derive the second order central difference approximation for $\frac{d^2y}{dx^2}$ on a uniform grid with grid spacing $\Delta x$.
• For the ODE $$y^{\prime\prime} + Py^\prime + Qy = F$$with constant $P$, $Q$, and $F$, derive the finite difference approximation on a grid with 5 interior points.
  • Assume Dirichlet boundaries with values $y_L$ and $y_R$ (for left and right)
  • Write the five equations for each interior point.
  • Convert this to a matrix $Ay=F$. You can use symbols for $A$, but define all values.
  • Be careful to treat the boundary conditions appropriately.
• What is the form of the linear system that is obtained?
  • What is the name of the algorithm used to solve this system?
• Consider the little details. Like, do you include the boundaries in your solution array, or do you insert them after?