Reference no: EM133029914

Questions -

Q1. Solve the diffusion equation u_t = ku_xx with the initial condition u(x, 0) = x² by the following special method. First show that u_xxx satisfies the diffusion equation with zero initial condition. Therefore, by uniqueness, u_xxx ≡ 0. Integrating this result thrice, obtain u(x, t) = A(t)x² + B(t)x +C(t). Finally, it's easy to solve for A, B, and C by plugging into the original problem.

Q10. (a) Solve Question 1 using the general formula discussed in the text. This expresses u(x, t) as a certain integral. Substitute p = (x - y)/√(4kt) in this integral.

(b) Since the solution is unique, the resulting formula must agree with the answer to question 1. Deduce the value of _-∞∫^∞p²e^{-p^2} dp.

Q3. (a) Consider the diffusion equation on the whole line with the usual initial condition u(x, 0) = φ(x). If φ(x) is an odd function, show that the solution u(x, t) is also an odd function of x. (Hint: Consider u(-x, t) + u(x, t) and use the uniqueness.)

(b) Show that the same is true if "odd" is replaced by "even."

(c) Show that the analogous statements are true for the wave equation.

Q4. The purpose of this exercise is to calculate Q(x, t) approximately for large t. Recall that Q(x, t) is the temperature of an infinite rod that is initially at temperature 1 for x > 0, and 0 for x < 0.

(a) Express Q(x, t) in terms of Erf.

(b) Find the Taylor series of Erf(x) around x = 0. (Hint: Expand e^z, substitute z = -y², and integrate term by term.)

(c) Use the first two non-zero terms in this Taylor expansion to find an approximate formula for Q(x, t).

(d) Why is this formula a good approximation for x fixed and t large?