Section A Attempt all questions from this section..

Section A Attempt all questions from this section.

Al. Use the method of characteristics to find the solution of the initial value problem

ut – uur =—

u(x, 0) {1 x < 0 —1 x > 0, for t > 0. Sketch the characteristics and the solution at a typical time t > 0. 10

Consider the kinematic wave equation pt + qx = 0. In the context of modelling traffic flow, define the densitys(x, t) and flux q(x,t). State the connection bet-w-een p(x,?), q(x, t) and the traffic flow speed v(r,1). The flow speed is assumed to depend linearly on density and satisfies the conditions v = 0 when p = R and v = V when p = 0. Derive the corresponding traffic flow model in terms of the density.

A3. Let J,i(x) be the Bessel function of the first kind with integer index n. Using the series definition of J,2(x), or otherwise, prove that J_„(x) = (-1)nJn(x).

A4. Solve the initial value problem for the heat equation

{1 0 < x < 1 = 0(x, 0) = 0 x < 0 or x > 1.

Verify that the solution, *(x, t), you obtain tends to the given initial condition as t —+ 0 and sketch the solution for a typical t > 0.

Section B

Attempt three of the four questions from this section.

B1. (i) Consider a weak solution of the kinematic wave equation pt + qz = 0 with a discontinuity in p and q, a shock, at x = s(t). Show that

= P1 – P2

q1 —q2

where subscripts 1 and 2 denote the values before and after the discontinuity. Deduce that if q = Q(p) is any quadratic expression in p then = 2 (c(Pi) + c(P)) where c(p) = Q'(p) is the kinematic wave speed.

(ii) You are given that the initial value problem

u, + 2uux = 0, u(x, 0) =

{1 1 2,

has multi-valued solution u(x,t) =1 x — 1 1+2t 2. 2t x-2 2t 0

Sketch this solution at a typical small time and at a typical large time in each case indicating the approximate location where any shock or shocks would be placed in the corresponding weak solution. By using the final result from (i), or otherwise, obtain formulae for the location of the shock or shocks at all times t > 0.

B2. (i) Prove the Bessel function identity —d(eJ,(x)) = xy .J,,_1(x) and use this to dx evaluate

fo rJo(Ar) dr. (ii) A uniform conducting cylinder C = (r, 0, z) : r < 1, 0 < z e h is described in cylindrical polar coordinates. Its temperature is maintained at To on the curved surface and on the base z = 0, and at T1 on the top z = h. You are given that the steady state temperature T(r, z) in the interior of the cylinder satisfies the axially symmetric (0-independent) Laplace equation. By assuming a separable solution, show that the most general solution satisfying the first two boundary conditions is

CO

T(r, z) = To + E An„Jo(A„,r)sinh(A„z), ni=o where A,,, are arbitrary constants, stating the condition that defines A. Evaluate the A,,, by imposing the final boundary condition. Using (i), or otherwise, express the solution T(r, z) in a form free of integrals.

B3. (i) Let P„(x) be the nth Legendre polynomial. Show that

1 1 + t2 — 2tx E E pm — (x)p„(x)t–0 n=0

00 CO

Then, using the orthogonality condition Pm(x)Pn(x) dx = 0 for m n, or otherwise, deduce that P„(x)2 dx t2n = t-5 (log(1 + t) — log(1 — t)). n=0 -1

Hence show that

2 P„(x)2 dx 2n + 1

(ii) Starting from Laplace’s equation in spherical polar coordinates with axial symmetry, show that if u = R(r)e(0) then ( -12 R 2 7772- + T72 dRr — A = 0 and d -(vi (1 — )de Tt) + Ag = 0, where /2 = cos 9. Briefly explain why for most applications one must choose A = n(n + 1), where n E Z+. Deduce that the most general solution that is non-singular in 0 is a linear combination of axially symmetric spherical harmonics (A„r” + B„r-n-1)P„(cos 0).

B4. (i) Starting from the general solution u = F(x — ct) + G(x + of the one dimensional wave equation utt = C2Uxr, show that d’Alembert’s solution to the Cauchy problem with u(x, 0) = f (x), ut(x, 0) = g(x) is 1 2c x f , u(x, t) = —2 (f (x — ct) + f (x + ct)) + — g(y) dy.

(ii) Show that spherically symmetric solutions of the three dimensional wave equa-tion zit, = c2Au take the form

1 u(x, = —rF(r — ct),

where r = lxi and F is an arbitrary function. Show that when F is taken to be Dirac’s &function then u(x, 0) = 0 and ut(x, 0) = 47rc o(x). Hence derive the Green’s function G(x, t; e) for the three dimensional wave equation, the solution satisfying initial conditions u(x, 0) = 0 and ut(x, 0) = (5.(x —

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