Reference no: EM132580817

Case Study Questions -

Q1. By taking the Laplace transform of the diffusion equation and boundary conditions, write down an ordinary differential equation for w(x; s) and solve that ordinary differential equation to show that

w(x; s) = e^-x√(s/D)/s,

for 0 < x < ∞.

Q2. By taking an appropriate inverse Laplace transform show that

u(x, t) = erfc(x/2√(Dt)),

for 0 < x < ∞ and t > 0.

Q3. Calculate an expression J(x, t), simplifying where possible.

Q4. Write a short MATLAB function that produces two different kinds of plot. The first plot should show u(x, t) for a particular choice of D and t as a function of x. The second plot should plot J(x, t) for a particular choice of D and t as a function of x.

Write your function so that you are able to superimpose the solution for various sensible choices of t so that your plots clearly show the time evolution of u(x, t) in one subplot and the time evolution of J(x, t) in another subplot. Label your solutions appropriately so that the variation with t is clear.

Submit your carefully labeled plots together with your commented MATLAB code. Provide/describe sufficient evidence to support the choices you make in plotting the results.

Q5. Compare plots of u(x, t) and J(x, t) in Question 4 for some particular choice of D and an appropriate range of t. Do your plots of the solution make sense in terms of the underlying physical model? Write a few sentences to explain what you see and your interpretation of what you see in terms of the underlying physical model.

Q6. Repeat Questions 1-4 (above) to develop solutions of Equation (3) for the following initial conditions and boundary conditions:

Initial condition at t = 0: u(x, 0) = A for 0 < x < ∞,

Boundary condition at x = 0: u(0, t) = 1 for t > 0, and

Boundary condition as x → ∞: u(x, t) = A for t > 0,

for some constant A > 0. Using your solutions and plots explore solutions for different choices of A, in particular explore A > 1, A < 1 and A = 1. Write a few sentences to explain what you see and your interpretation of what you see in terms of the under-lying physical model. Answering this question will require the generation of carefully commented MATLAB code that will be submitted together with your mathematical derivation and written interpretation.

Attachment:- Assignment File - Case Study.rar