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} + u_{i,j-1}}{k^2} = 0 ] Since (h = k = 1/2), we have (h^2 = k^2 = 1/4). Multiply both sides by (1/4) to simplify: [ u_{i-1,j} + u_{i+1,j} + u_{i,j-1} + u_{i,j+1} - 4 u_{i,j} = 0 ] Or equivalently: [ u_{i,j} = \frac{1}{4} \left( u_{i-1,j} + u_{i+1,j} + u_{i,j-1} + u_{i,j+1} \right) ] --- Step 3: Boundary values Boundary conditions are: 1. (x=0) : (u(0, y) = x^2 - y^2 = 0 - y^2 = -y^2) [ u(0,0) = 0, \quad u(0,1/2) = -1/4, \quad u(0,1) = -1 ] 2. (y=0) : (u(x,0) = x^2 - 0 = x^2) [ u(0,0)=0, \quad u(1/2,0) = 1/4, \quad u(1,0)=1 ] 3. (y=1) : (u(x,1) = x^2 - 1) [ u(0,1)=-1, \quad u(1/2,1)= -3/4, \quad u(1,1)= 0 ] 4. (x=1) : Robin condition (3u + 2 u_x = x^2 + y^2). We approximate (u_x \approx \frac{u_{2,j} - u_{1,j}}{h} = \frac{u_{2,j} - u_{1,j}}{1/2} = 2(u_{2,j}-u_{1,j})). So the Robin condition becomes: [ 3 u_{2,j} + 2 \cdot 2 (u_{2,j} - u_{1,j}) =