UNIVERSITY OF CAMBRIDGE Part III MATHEMATICAL TRIPOS Monday, 6 June, 2016 9:00 am to 11:00 am PAPER 328 BOUNDARY VALUE PROBLEMS FOR LINEAR PDES Attempt no more than TWO questions. There are THREE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 1 Consider the following initial-boundary value problem: Ut = Uxx + aux, 0 <<< L, 0<t<T, u(x, 0) = uo(x), O << L, u(0, t) = go(t), u(L,t) = h(t), 0 < < T, where uo(x), go(t), h(t) are given smooth functions and a, L, T, are given positive constants. (i) Obtain an integral representation for the solution u(x,t) in terms of u0(x), g0(t) and h(t). (ii) Discuss how the solution of this problem can be evaluated numerically for the case that the relevant integral transforms of u0, g0 and h can be computed explicitly. 2 The temperature u(x, t) of a semi-infinite solid rod in contact at x = 0 with a tank filled with a certain liquid satisfies the following initial-boundary value problem: Ut = Wxx, 0<x<oo, t > 0, u(x, 0) = u0(x), 0 < x < 80, ut (0,t) + aux(0, t) + Bu(0,t) = f(t), t > 0, where a and B depend on the specific heat and conductivities of the rod and the liquid and f(t) is the uniform temperature of the surrounding of the tank. Obtain an integral representation of the solution in the particular case that B = 2. a2 4 Part III, Paper 328
UNIVERSITY OF CAMBRIDGE 3 3 Let u(x, y) denote a solution of the modified Helmholtz equation, namely of the equation Uxx + Uyy - k2u = 0, k > 0. (i) Write a family of divergence forms involving a solution v of the adjoint equation. (ii) Write a particular solution of v parameterised in terms of an arbitrary complex parameter A. (iii) Derive two global relations and show that if u is real, then the second global relation can be obtained from the first via complex conjugation followed by the transformation > >> >. (iv) Consider the Dirichlet problem for the modified Helmholtz equation in the interior of the square with corners at the points -1+i, -1-i, 1-i, 1+i. By expanding the boundary values in terms of appropriate basis functions obtain two approximate global relations. (v) Determine four suitable sets of collocation points, each associated with each side. Show that these sets will yield a diagonaly dominant system of algebraic equations for the unknown coefficients in the expansions of (iv). END OF PAPER Part III, Paper 328