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We are asked to solve the Laplace equation numerically using the standard five-point formula on the square (0 \le x \le 1, 0 \le y \le 1) with a mixed boundary condition. Let’s solve it step by step. --- Step 1: Set up the grid Given (h = k = 1/2), the grid points along (x) and (y) are: [ x_0 = 0, ; x_1 = 1/2, ; x_2 = 1 ] [ y_0 = 0, ; y_1 = 1/2, ; y_2 = 1 ] So the interior point is ((x_1, y_1) = (1/2, 1/2)). --- Step 2: Standard five-point formula For interior points, the five-point formula for Laplace’s equation is: [ \frac{u_{i+1,j} - 2 u_{i,j} + u_{i-1,j}}{h^2} + \frac{u_{i,j+1} - 2 u_{i,j} + __________ ________ _____ __________ __________.
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