We all have experienced the result of someone spraying perfume in the corner of a large room. Within a fairly short amount of time, most people in the room can smell the perfume. In this case, we say that the perfume has diffused from a region of the room in which it was highly concentrated to a region of the room in which its concentration was initially low. Diffusion is the process by which the molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. This process can be investigated using a diffusion model (also known as a heat equation): ди at อใน əx², u(0, t) = 20, u(30, t) = 50, u(x, 0) = 60 - 2x, t> 0, 0 < x < 30, (1) with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t, along a one-dimensional spatial domain, x. a) Calculate the steady-state concentration of perfume particles. b) Formulate the initial-boundary value problem that determines the transient distribution of perfume particles. c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1).

Consider the discrete-time dynamical system modeling the concentration of a chemical in a lung. (Note: round all values at the end of the calculations and use 4 decimal places.)ct+1 = (1-p)ct + pβLet V = 2 L, W = 1 L, and β = 6 mmol/LIf c0 = 7 mmol/L, iterate to find the following values:c1 = ____mmol/Lc2 = ____mmol/Lc3 = ____mmol/Lc4 = ____mmol/LFind the equilibrium of this system:c* = ____mmol/L

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