UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Thursday, 5 June, 2014 9:00 am to 11:00 am PAPER 69 THE UNIFIED METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS AND MEDICAL IMAGING 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 Let u(x, t) satisfy the PDE Ut = Wx + aux, u(x, 0) = u0(x), 0 << 0, u(0,t) = g0(t), 0<t < T, 0 < < < 0, 0 < < < T, where a and T are given positive constants, uo(x), go(t) are given functions with sufficient smoothness, uo has sufficient decay as x -> 00, and uo(0) = go(0). (i) Rewrite the above PDE as a one parameter family of divergence forms and derive the associated global relation. (ii) Derive an integral representation for u(x, t) in the complex Fourier plane involving appropriate transforms of u0, g0 and ux(0,t) (the Jordan lemma can be used without proof). (iii) Use the global relation to eliminate the transform of the unknown function ux (0, t), and hence obtain a solution of the above initial boundary value problem. (iv) Prove that the expression for u(x, t) obtained in (iii) satisfies the above PDE and the given initial and boundary conditions. Can the above problem be solved via the classical sine transform? Part III, Paper 69
UNIVERSITY OF CAMBRIDGE 3 2 Assuming the validity of the Poincare lemma Z F = z dF, (1) where F is a one-form and D is anappropriate domain with regular boundary OD, implement the following: (i) Derive the Pompeiu formula. (ii) By employing the spectral analysis of the equation az OF(z, z, k, k) -kF(z,z,k,k) = q(z, z), z E C, k E C, (2) derive the two-dimensional Fourier transform pair. (iii) Sketch, without giving all the mathematical details, the steps needed in order to derive the attenuated Radon transform, via the spectral analysis of 1 -0 -0 1 OF(@ ?; t2, k) 1 + -1 1 OF(xf;522, k) 2 f(x1kr2), (x1, x2) ER2, k & C. -0 -p(x1, @2)F(x1, x2, k) ket. Part III, Paper 69 [TURN OVER