PaperIII_66 (2)

# PaperIII_66 (2) - MATHEMATICAL TRIPOS Part III Thursday 29...

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MATHEMATICAL TRIPOS Part III Thursday, 29 May, 2014 9:00 am to 12:00 pm PAPER 66 NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS Attempt no more than THREE questions from Section A and ONE from Section B. There are SEVEN questions in total. The questions in Section B carry twice the weight of those in Section A. Questions within each Section carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 SECTION A 1 Let κ be a constant. We consider the partial differential equation ∂u x [0, 1], t ≥ 0, x2 + κ ∂x , given with initial conditions at t = 0 and zero Dirichlet boundary conditions at x = 0 and x = 1. (a) Prove that the equation is well posed. (b) The equation is semi-discretised by the ODE system 1 u′m = m = 1, . . . , M, Δx (um+1 − um−1), where Δx = 1/(M + 1) and um (t) ≈ u(mΔx, t). Carefully justifying your analysis, prove that the method is stable.

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