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Consider the following elliptic PDE: au z2 = x2 1, = [0, 1]; y= [0, 1] ay? Using finite difference methd with step size h = 0.25 for both x-and y-domains, determine u(x, y) at all nodes of the solution mesh with the following boundary conditions: (a) u (2, 0) = u(x, 1) = 1; u(0, y) = u(1, y) = cos(2ny) (b) u (x, 0) = u (x, 1) = 1; (0, y) = (1, y) = 1 Derive the governing finite difference equation for numerical solution of the following PDE[a is a constant.]: 2 au ar2 at = a Use step size h for t-domain and k for x-domain. Use indices i and for t-and x-domains, respectively. Present your derivations in handwritten form. Include the solution for this problem in your report. Consider the following elliptic PDE: au z2 = x2 1, = [0, 1]; y= [0, 1] ay? Using finite difference methd with step size h = 0.25 for both x-and y-domains, determine u(x, y) at all nodes of the solution mesh with the following boundary conditions: (a) u (2, 0) = u(x, 1) = 1; u(0, y) = u(1, y) = cos(2ny) (b) u (x, 0) = u (x, 1) = 1; (0, y) = (1, y) = 1 Derive the governing finite difference equation for numerical solution of the following PDE[a is a constant.]: 2 au ar2 at = a Use step size h for t-domain and k for x-domain. Use indices i and for t-and x-domains, respectively. Present your derivations in handwritten form. Include the solution for this problem in your report