Approximate the solutions to the following elliptic partial differential equations

Chapter 12, Problem 12.1.3

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Approximate the solutions to the following elliptic partial differential equations, using Algorithm 12.1: a. 2u x 2 + 2u y2 = 0, 0 < x < 1, 0 < y < 1; u(x, 0) = 0, u(x, 1) = x, 0 x 1; u(0, y) = 0, u(1, y) = y, 0 y 1. Use h = k = 0.2, and compare the results to the actual solution u(x, y) = xy. b. 2u x 2 + 2u y2 = (cos(x + y) + cos(x y)), 0 < x < , 0 < y < 2 ; u(0, y) = cos y, u(, y) = cos y, 0 y 2 , u(x, 0) = cos x, u x, 2 = 0, 0 x . Use h = /5 and k = /10, and compare the results to the actual solution u(x, y) = cos x cos y. c. 2u x 2 + 2u y2 = (x 2 + y2)exy , 0 < x < 2, 0 < y < 1; u(0, y) = 1, u(2, y) = e2y , 0 y 1; u(x, 0) = 1, u(x, 1) = ex , 0 x 2. Use h = 0.2 and k = 0.1, and compare the results to the actual solution u(x, y) = exy . d. 2u x 2 + 2u y2 = x y + y x , 1 < x < 2, 1 < y < 2; u(x, 1) = x ln x, u(x, 2) = x ln(4x 2 ), 1 x 2; u(1, y) = y ln y, u(2, y) = 2y ln(2y), 1 y 2. Use h = k = 0.1, and compare the results to the actual solution u(x, y) = xy ln xy.