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I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or Neumann. Often it is homogeneous BC like $u(0)=0,u'(R)=0$, which seems to be fine to deal with. One way is to introduce and then eliminate ghost points with the BC.

But if it is inhomogeneous BC (e.g., for a Dirichlet $u(0)=a,u(R)=b$), the nonzero constants have a sting in the tail. Effectively they're put into an inhomogeneous column as shown below, which is also clearly introduced in this note. For instance, the eigenvalue probelm $Lu=\lambda u$ of the operator $L=-\frac{\partial^2}{{\partial x}^2}$ becomes an inhomogeneous eigenvalue problem $Mu+R=\lambda u$ with $$M=\frac{1}{h^2}\begin{bmatrix} 2 &-1 & & & \\ -1 &2 &-1 & & \\ &\ddots &\ddots &\ddots & \\ & & -1 &2 &-1\\ & & & -1 &2 \end{bmatrix}, R=\frac{1}{h^2}\begin{bmatrix} a \\ 0 \\ \vdots \\ 0\\ b \end{bmatrix}.$$

But after this, practically how to proceed to solve the original problem? Any other formulation? Or we simply don't need to think about such problem for some reason?

This may not be a problem for usual DE solving (I naively assume one-sided stencil often suffices.). But for eigenvalue case, I can't find any way out. The inhomogeneous eigenvalue problem doesn't have any standard or general solution as far as I've searched.

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    $\begingroup$ back up for a sec here: does it really make sense to set inhomogeneous BCs to such an eigenvalue problem? after all, you're looking for nontrivial solutions (eigenvalue/eigenfunction pairs) to the corresponding homogeneous problem (with homogeneous BCs of either Dirichlet or Neumann type). have a look at how chebfun's eigs function computes eigenvalues of differential operators - either here, or in chapter 6 of Nick Trefethen's "Exploring ODEs" book $\endgroup$ – GoHokies Mar 28 '18 at 16:50

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