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?