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This is the Past Exam Paper of Math Tripos which includes Restriction and Kakeya Phenomena, Minkowski Dimensions, Real Numbers, Besicovitch Set, Besicovitch Subset, Absolute Constant, Measurable Function, Discrete Fourier Analysis etc. Key important points are: Numerical Solution of Differential Equations, Implicit Midpoint Rule, Runge–Kutta Method, Symmetric Matrix, Order of Method, Diffusion Equation, Boundary Conditions, Boundary Conditions, Order of Magnitude
Typology: Exams
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Friday 30 May 2003 1.30 to 4.
Attempt THREE questions from Section A and attempt ONE question from Section B.
Each question from Section B carries twice the weight of a question from Section A.
Section A
1 The implicit midpoint rule for the ODE system y′^ = f (t, y), t > 0, y(0) = y 0 , is
yn+1 = yn + hf (tn + 12 h, 12 (yn + yn+1)), n > 0.
a. Formulate the implicit midpoint rule as a Runge–Kutta method and determine its order. Is the method A-stable?
b. Suppose that it is known that, for every initial condition y 0 , the solution of the ODE posesses the invariant y>(t)Sy(t) ≡ y> 0 Sy 0 , t > 0, where S is a given symmetric matrix. Prove that y> n Syn ≡ y> 0 Sy 0 , n > 0.
2 Consider the two-step ODE method
yn+2 − (^) 2+^4 α yn+1 + (^2) 2+−αα yn = (^) 2+hα (fn+2 + 2αfn+1 − fn),
where fm = f (tm, ym), while α 6 = −2 is a parameter.
a. Determine the range of α for which the method is convergent. For every such α compute the order of the method.
b. Prove that no α exists so that the method is both convergent and A-stable.
3 The diffusion equation
∂u ∂t
∂x
a(x)
∂u ∂x
where a is a positive function, is given for 0 6 x 6 1, t > 0, with initial conditions at t = 0 and zero Dirichlet boundary conditions at x = 0, 1. It is solved by the fully-discretized finite difference method
un m+1+1 = unm + μ[am− 1 / 2 unm− 1 − (am− 1 / 2 + am+1/ 2 )unm + am+1/ 2 unm+1],
where μ = ∆t/(∆x)^2 and aγ = a(γ∆x).
a. Derive the order of magnitude of the local error.
b. Determine the range of μ > 0 for which the method is stable for every function a such that 0 < a− 6 a(x) 6 a+ < ∞.
Paper 67
