# Neumann boundary conditions of a fourth order BVP

To numerically solve the following 2-point fourth order eigenvalue problem $$y''''(t)=\lambda y$$ over $[0,1]$ with boundary conditions $y(0)=y'(0)=y(1)=y'(1)=0.$

I intend to propose four ghost points on either side of the domain, i.e. $u_{-2}<u_{-1}<0=u_0<u_1<\cdots u_{N}=1<u_{N+1}<u_{N+2}$. The Neumann boundary conditions implies $u_{-1}=u_{1}$ and $u_{N+1}=u_{N-1}.$ But how to eliminate $u_{-2}$ and $u_{N+2}$?

Note: four ghost points are necessary for applying the centered second order finite difference approximation scheme for fourth order derivatives at the end points.

• Is the use of centered schemes an absolute requirement? Otherwise, you can try asymmetric approximations of the derivatives at the boundary. They do not require ghost points, but you lose symmetry of your matrix, though. – Carl Christian Aug 3 '17 at 21:22
• That's not a requirement. By 'asymmetric approximation' did you mean one-sided approximations? – user31899 Aug 5 '17 at 11:43
• Yes. I had forgotten the correct English term, i.e., one-sided approximations. – Carl Christian Aug 9 '17 at 20:57