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## Symbolic generating of system equations for 2D regular grid for solving the Laplace equation using ﬁnite diﬀerence method

Math 228A, UC Davis, Fall 2010   Compiled on May 21, 2020 at 2:30am

$\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial x^2} = f(x,y)$

When solving the above equation using ﬁnite diﬀerence method on square, in order to make it easy to see the internal structure of the A matrix using the standard 5 points Laplacian scheme, the following is a small function which generates the symbolic format of these equations for a given $$N$$, the number of grid points on one edge. At the end of this note, the system equations are generated for $$N=4,5,6,7,8$$. One can see the form of the $$A$$ matrix with the dominant diagonal.