# Linear Difference equation

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Posted by Christian Oehreneder on August 13, 1998 at 12:10:08:

I need to solve a problem of the following kind:

d^2 U(x,y)/dx^2 + d^2 U(x,y)/dy^2 - U(x,y)*f(x,y) = - g(x,y)

or in discretized form

U[m-1,n] + U[m+1,n] + U[m,n-1] + U[m,n+1] - 4*U[m,n] - U[m,n]*f[m,n] = -g[n,m]

f >= 0

The problem is to be solved on a square domain.
At the boundary I use a symmetric continuation of U
to give the above equation meaning.

Is this a "known" problem. If yes, under what name is
it referenced in the literature?

For the 1D case it involves the solution of a tridiagonal
symmetric Matrix with subdiagonal elements all the same.

I solved with some special solver for tridiagonal Matrizes
which works fine.

For the 2D case everything seems more complicated. In view
of the very regular structure of the equation I thought
there might be a special purpose solver for that type
of problems. It seems to be in close relation to other
finite difference problems.

Can anyone give me an advise?
Many Thanks
Christian