Reference no: EM133029893

Questions -

Q1. Solve the diffusion equation with the initial condition

φ(x) = 1 for |x| < l and φ(x) = 0 for |x| > l.

Write your answer in terms of Erf(x).

Q2. Do the same for φ(x) = 1 for x > 0 and φ(x) = 3 for x < 0.

Q3. Use (8) to solve the diffusion equation if φ(x) = e^3x. (You may also use questions 6 and 7 below.)

Q4. Solve the diffusion equation if φ(x) = e^-x for x > 0 and φ(x) = 0 for x < 0.

Q5. Prove properties (a) to (e) of the diffusion equation (1).

Q6. Compute ₀∫^∞e^{-x^2} dx. (Hint: This is a function that cannot be integrated by formula. So use the following trick. Transform the double integral ₀∫^∞ e^{-x^2} dx ·₀∫^∞e^{-y^2} dy into polar coordinates and you'll end up with a function that can be integrated easily.)

Q7. Use question 6 to show that _-∞∫^∞e^{-p^2} dp = √π. Then substitute p = x/√(4kt) to show that _-∞∫^∞S(x, t) dx = 1.

Q8. Show that for any fixed δ > 0 (no matter how small), max_δ≤|x|<∞ S(x, t) → 0 as t → 0.