We Helped With This Differential Equations Homework: Have A Similar One?

Exercise

We consider a model with diffusion and convection for some concentration C:

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with initial condition C(t = 0,x) = 1 for 2 ≤ x ≤ 3 and C(t = 0,x) = 0 otherwise. We use boundary conditions C(t,x = 0) = C(t,x = 6) = 0, but C(t,x = 0) = 1 for 4 ≤ t ≤ 5. This is a partial differential equation where C depends on both time t and space x. It cannot be solved easily analytically. Let us make a numerical solution with forward-Euler. For this, we are going to discretize this system as follows. We compute the solution C at discrete points x_i in space only. These points lie at a distance h, so x_i = ih with i = 0,1,2,...,N with N = 6/h. With this notation we can write C_i = C(t,x_i). Then we approximate the spatial derivatives with the following formulas

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•    Write down the differential equations for C₁,C₂,C₃,C₄,C₅ if h = 1. Do this in a matrix form C⁰ = AC + b so that C = (C₁....C_N−1). That is, give the matrix A and vector b. Also specify the initial condition C(t = 0).

•    Use dt = 0.25 to simulate the system for 0 ≤ t ≤ 10. Make a contourplot of your solution with t and x on the axes. Also plot the solution for t = 1,5,10.

•    Now use h = 0.25 and simulate the system for 0 ≤ t ≤ 10. You cannot use the same dt, but need a smaller value. Experiment with smaller values until your simulation makes sense. Hint: you may for-loops to initialize the matrix A once. Make the same plots as before.

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