I. The heat equation is commonly used to model how heat diffuses through a given region,

                                                                        Du       D²u

where k > O is a diffusivity constant, and {a, 6} are specified Dirichlet boundary conditions (i.e. heat sinks). Consider a uniform discretization in space and time,

                                               Ti =iAx, i = O,    , M, Ax = —

                                                   tn n At, n = O, , N,               = — .

1. Express the above system of equations in matrix form so that a linear solve will obtain a solution at time t ⁿ+ ^l
2. Does this system of equations have a unique solution? Why or why not?

2. In a closed system of three components, the following reaction path can occur:

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with governing differential equations

 = -klCA+k2CBCc dt

                                    = kl CA — k2 CB Cc — kg              (1)

dt

                                                                       CA(0) = 1, CB(0) = CC(0) = O.

dö

1. Find the steady state(s) of the system by solving the nonlinear system — dt — — O, where 6 = [CA, CB, CA^T.
2. Simulate the reaction pathway for kl = 0.08 s-1 , k2 = 2 X 10⁴ and kg = 6 x 10⁷ on an appropriate time-scale using an appropriate numerical solver.
3. Linearize (j) about the equilibrium point, and analyze the eigenvalues of the resulting system to draw conclusions about the local behavior close to the equilibrium point. Comment specifically if your numerics match your analysis.

3. Please complete E2.12 [TC p. 91].
4. Please complete E7.9 (TC p. 308].
5. Consider the following first-order PDE,

(t, x) e (O, 00) x (O, DC).

with initial condition

= cos27r:r, x 2 0

and boundary condition u(t, O) = e ¯t , t > O

1. Please solve the system analytically.
2. Please generate both a surface plot and a pcolor plot on a reasonably sized domain that illustrates the behavior of the solution.
3. Generate an approximation to the solution using finite-differencing. Justify your choice of discretizations.
4. Generate a surface plot and a pcolor plot of the error Of your numerical approximation, commenting on what you observe. Remember that when you are trying to visualize multiple scales, it is often helpful to consider a logarithmic scale.

6. Table Ugives the error of a numerical approximation as a function of Ax, A plot of the data reveals that there are likely two different rates of convergence: one when Ax > a second when Ax <

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