From PDE to the linear difference equations system

• Differentiation:

  u(xi+½, yj) = (ui+1,j – ui,j) / h ui,j = (ui+1,j – 2ui,j + ui–1,j) / h2
  
  ui,j = ( ui+½,j – ui–½,j) / h ui,j = (ui,j+1 – 2ui,j + ui,j–1) / h2

• –?u(x,y) = 0

? – ui–1,j – ui,j–1 + 4ui,j – ui,j+1 – ui+1,j = 0 for i=0,…,m-1, j=0,…,n-1

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*) corners are unused

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