Elliptic1D Pure Neumann Boundary Conditions#
Solves the 1D Poisson boundary value problem with Neumann boundary conditions.
\[
-\nabla^2 u(x) = x - \frac{1}{2}
\]
with \(x\in[0,1]\).
The boundary conditions are given by
\[
au + b\frac{du}{dx} = g
\]
with the left hand side boundary condition (Neumann) satisfying
\[
0u(0) + 1\frac{du(0)}{dx} = 0
\]
and the right hand boundary condition (Neumann) satisfying
\[
0u(1) + 1\frac{du(1)}{dx} = 0
\]
This corresponds to the call to addScalarBC1D of addScalarBC1D(A,b,k,m,dx,dc,nc,v), where dc, nc, and vc are vectors which hold the coefficients for \(a\), \(b\), and \(g\) in the above system of equations. \(a=[0,0]\), \(b=[1,1]\) and \(g=[0,0]\).
Substituting these values in gives:
\[
\frac{du(0)}{dx} = 0
\]
\[
\frac{du(1)}{dx} = 0
\]
The exact solution ( with constant \(C\) ) is:
\[
u(x) = C + \frac{x^2}{4} - \frac{x^3}{6}
\]
This example is implemented in:
Additional MATLAB/ OCTAVE variants of this example with different boundary conditions: