Exercises 1

Question 1. Use d'Alembert's solution to solve to the IVPs

(a) u_tt = u_xx, u(x, 0) = 0, u_t(x, 0) = 1/(1 + x²)

                                       1   |x| ≤ 1
(b) utt = uxx, u(x, 0) =                                u_t(x, 0) = 0.
                                        0   otherwise,

Sketch the solution u(x, t) at t = 0 and several times t > 0.

Question 2. By a similar approach to that used in obtaining d'Alembert's solution, find the solution to

u_tt = c²u_xx, u(x, x) = f (x), u_t(x, x) = g(x),

in which conditions are given on the line t = x.

Question 3. Verify that the two-dimensional Laplacian ? for polar coordinates (r, θ), where x = r cos θ, y = r sin θ, is

? = ∂²/∂r² + 1/r.∂/∂r + 1/r² ∂²/∂θ² .

Question 4. Using the result Γ( 1/2 ) = √π in the series expression for J_1/2 (x), show that J_1/2(x) = √2/πx. sinx and find a corresponding expression for J_-1/2 (x).

Question 5. Consider the series expansion for J_ν (x) in the case that ν is an integer. Prove that for integer index n,

J_-n(x) = Jn(-x) = (-1)ⁿJ_n(x).

Question 6. Prove (a)

d/dx(x^ν J_v (x)) = x^v J_v-1(x),      d/dx(x^-vJ_v(x)) = x^-ν J_ν+1(x)

and (b) deduce that

J'_ν (x) = 1/2 (J_ν-1(x) - J_ν+1(x)),  J_ν (x) = x/2ν (J_ν-1(x) + J_ν+1(x)).

Question 7. Prove that

G(x, t) := e^x/2 (t-t^-1) = ∑^∞_n=-∞ J_n(x)tⁿ.

G(x, t) is a generating function for Bessel functions of integer order.

(a) By using the invariance of G(x, t) under certain transformations of x and t, deduce the results of Q5.

(b) By an appropriate choice of t, deduce from (*) that

cos(x sin θ) = J₀(x) + 2∑_k=1^∞ J_2k(x) cos(2kθ),

and

sin(x sin θ) = 2∑_k=0^∞ J2_k+1(x) sin((2k + 1)θ).

Question 8. Prove by induction that for all non-negative integers n,

J_n(x) = (-x)ⁿ(1/x.d/dx)ⁿ J₀(x).

Question 9. You are given that for ν > -1, all zeros of J_ν are real. Using Rolle's theorem and the identities in Q6 (a), prove that for ν > -1 there is a zero of J_ν between and any two positive zeros of J_ν+1 and vice versa. Deduce that between consecutive positive zeros of Jν there is exactly one zero of J_ν+1. (It is said that the zeros of J_ν and J_ν+1 are interlaced.)

Question 10. Let ν > -1. You are given that xJ'_ν (x) + hJ_ν (x) has infinitely many positive zeros. Show that if λ_n is the nth such zero then

₀∫¹rJ_ν (λ_mr)J_ν (λ_nr) dr = 1/2 δ_mn[1 + (h² - ν²)/λ²_n)] J_ν (λ_n)².

The Fourier-Dini expansion of f on [0, 1] is the expression

f (r) = ∑_n=1^∞A_nJ_ν (λ_nr).

Show that

A_n = 2λ_n²/ (λ_n² + h² - ν²)J_ν (λ_n)² ₀∫¹rf(r)J_ν (λ_nr) dr.

Question 11. A semi-infinite cylinder of conducting material r ≤ a, z ≥ 0 has temperature T such that T = T₁ on z = 0 and T → T₀ at z → ∞. Outside the curved surface the temperature is also T0 and Newton's law of cooling gives the boundary condition

∂T/∂r = -c(T - T₀), on r = a,

where c is a positive constant.

The steady temperature distribution is an axially symmetric (i.e. θ-independent) solution of Laplace's equation. Show that it is given by

T(r, z) = T₀ + 2(T₁ - T₀) Σ^∞_n=1 (λ_nJ₁(λ_n)J₀(λ_nr/a)e^-λ_nz/a)/(λ_n² + a²c²)J₀(λ_n)²)

where λ_n is the nth positive zero of xJ'₀(x) + acJ₀(x).

Exercises 2

Consider Bessel's equation with ν = 1/2,

y'' + 1/x.y' + (1 - 1/4x²) y = 0. (*)

You should find the general solution in two ways:

Question 1. (Easier) Find and solve the second order ODE satisfied by new dependent variable z = √x.y. Hence find the general solution of (*).

Question 2. (Harder) Use the method of Frobenius to find a series solution ( Σ^∞_n=0 a_nx^n-1/2) in which the first two coefficients (a₀ and a₁) are arbitrary. Find a closed form expression for this solution (i.e. in terms of standard functions not infinite series).

Compare the two answers you get. (They should be the same of course.)

Exercises 3

Consider a semi-infinite conducting cylinder of radius a with the closed end at z = 0. The temperature at z = 0 is maintained at T₁ and the curved sides are insulated. The temperature T → T₀ as z → ∞. The steady state temperature T in the interior of the cylinder satisfies Laplace's equation. Assuming that. T does not depend on the polar angle θ, write down the boundary value problem that determines T.

Show that T(r, z) = T₀ + ∑^∞_m=1 A_me^-λm^z/a J_o(λ_mr/a),

where λ_m is the mth zero of J'₀ and A_m are constants. Find an expression for A_m in terms of T₀, T₁, λ_m and J₀(λ_m) and J₁ (λ_m).

Exercises 4

Question 1. Use the base cases P₀(x) = 1, P₁(x) = x and the recurrence formula

(n + 1)P_n+1(x) = (2n + 1)xP_n(x) - nP_n-1(x)

for n ≥ 1, to derive the Legendre polynomials P_n(x) for 2 ≤ n ≤ 4.

Question 2. Determine all non-zero associated Legendre functions P n m (x) for m ≤ n ≤ 3.

Question 3. By considering the generating function G(x, t) = (1-2xt+t²)^-1/2 = Σ^∞_n=0 P_n(x)tⁿ, prove that for all n ≥ 0, P_n(-x) = (-1)ⁿP_n(x) and P_n(1) = 1.

Question 4. For all integers n ≥ 0 the Legendre polynomial P_n(x) satisfies Legendre's equation (1 - x²)y'' - 2xy' + n(n + 1)y = 0. Use this fact to prove that for any n ≥ 0, P'_n(1) = 1/2n(n + 1) and P'_n(-1) = (-1)ⁿ1/2n(n + 1).

Exercises 5

Prove that (guided by the steps indicated below)

(n + 1)P_n+1 - (2n + 1)xP_n(x) + nP_n-1 = 0, (*)
for n ≥ 0.

Step 1 The polynomial xP_n(x) has degree n + 1. Explain why it may expressed as a linear combination of Legendre polynomials P_m(x) where m is odd/even when n is even/odd.

Step 2 Further, show that xP_n(x) is a linear combination of just two Legendre polynomials aP_n+1(x) + bP_n-1(x) where a and b are constants to be determined.

Step 3 Evaluate xP_n(x) and its derivative at x = 1 and solve the resulting simultaneous equations for a and b.

Step 4 Obtain the relation (*).

Exercises 6

Consider the IVP

u_t = (σx²u), u(x,0) = {x   0 < x ≤ 1

                                 {0   x > 1 
where σ > 0 is a constant and x > 0. Find the changes of variable which transform this into an IVP for the heat equation. Hence solve the IVP for u(x, t).

Exercises 7

Find the solutions of the Cauchy problems with u(x, 0) = f (x) and u_t(x, 0) = g(x) where

Question (i) f(x) = 1/(1+ x² + y² + z²), g(x) = 0,

Question (ii) f(x) = 0, g(x) = 1/(1 + x² + y² + z²)

Question (iii) f(x) and g(x) are arbitrary but depend only on r.

Exercises 8

Question 1) Find the solution of the problem for u(x^{^}, 0) = 0 and u_t(x^{^}, 0 ) = xy. The solution must be xyt as computed via the 2D formula.

Question 2) Likewise, compare the solution for u(x^{^}, 0) = z² u_t(x^{^}, 0) = 0 and verify that it needs u(x^{^}, t ) = z² + c²t²

Question 3) assume u(x^{^}, 0) = x² + y² + z² and compare the solution of obtained by means of the 3D solution with that computed radial symmetry

U(n, t) = 1/2h ((h-ct) f(h - ct) + (h + cr)f(h + ct)) as g = 0.