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Homework 4 Problem 1 Consider heat conduction in one dimensional bar, ∂2u ∂u = k 2, ∂t ∂x 0 ≤ x ≤ L, with insulated left boundary (∂u/∂x = 0) and fixed temperature (u = 1) right boundary. Assume the initial temperature of the bar is zero. Find the temperature evolution of the bar. Problem 2 Consider predator-prey model with fixed predator and prey population on the boundaries, ∂u ∂2u = 0.01 2 + u − uv − u2 ∂t ∂x ∂v ∂2v = 0.01 2 − v + uv ∂t ∂x (1) with u(0, t) = v(0, t) = u(1, t) = v(1, t) = 1 and u(x, 0) = 1 if 0.8 ≤ x ≤ 0.9, and zero otherwise, and v(x, 0) = 1 if 0.1 ≤ x ≤ 0.2, and zero otherwise. Solve this equation using MATLAB up to time 2. Describe what is the difference between this model and the model with insulated boundary conditions considered in the class. Problem 3 Using MATLAB solve the heat conduction problem with convection ∂u ∂ ∂u ∂u = · (k(x) ) − h ∂t ∂x ∂x ∂x in an inhomogeneous bar. Assume the bar has fixed temperature T = 1 at the ends, and initial temperature T = 0. Assume k = 0.001 if x < 0.9 and k = 1 elsewhere. Solve for h = 0, h = 0.1 and h = 0.5 up to the time 2. Describe briefly the differences of these cases, and explain why. 1