= Problem 3. Consider the diffusion equation with proportional growth Ut = Duzx + Cu u(x,0) = sin? I x 0 5 x 5L u(0, t)

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Problem 3 Consider The Diffusion Equation With Proportional Growth Ut Duzx Cu U X 0 Sin I X 0 5 X 5l U 0 T 1 (84.42 KiB) Viewed 19 times

= Problem 3. Consider the diffusion equation with proportional growth Ut = Duzx + Cu u(x,0) = sin? I x 0 5 x 5L u(0, t) = 0 t> 0 u(L, t) = 0 t> 0 The population density at time t and position x is denoted u(x,t). The Dirichlet boundary conditions represents the assumption that the population cannot live outside the patch 0 < x <L. (a) Derive the Crank-Nicolson method for this equation. (6) Set D = 1. Use the Crank-Nicolson Method to approximate the solution and find the smallest C for which the population equation, on the patch (0,10] (that is, take L 10), survives in the long run. (c) Try to confirm that your results do not depend on the step size choices (that is, the method is uncon- ditionally stable). Hint: Find the code crank.m in the course folder. Hint: Your answer for the smallest C should be close to 72/100. =