Mathematics > EXAM > MAE3456+MEC3456-S1-2018-Exam-Solution-FINAL-ALL ANSWERS CORRECT (All)

MAE3456+MEC3456-S1-2018-Exam-Solution-FINAL-ALL ANSWERS CORRECT

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Question 1 [25 marks Total] General understanding Part A) [5 marks] Please indicate if each statement below is TRUE or FALSE. i.) The false-position method and Secant method each require two initial guesses. TRUE [1.25 mark] ii.) Gaussian quadrature can be used to do numerical integration for tabulated data values. FALSE [1.25 mark] iii.) Richardson extrapolation can be used to combine two ODE solutions to obtain one with higher order of accuracy. TRUE [1.25 mark] iv.) The central difference scheme for the first derivative has an order of accuracy of O(Δx). FALSE [1.25 mark] Part B) [20 marks] i.) Briefly explain the similarities and differences between the false-position method and the Secant method for root finding [2 marks] The similarity is that a new guess for the root is determined by drawing the secant (chord) between the last 2 guesses (1 mark) The difference is that the two guesses must bracket the root with false-position, but do not need to in the secant method (1 mark) You need 2 starting guesses (0 marks) ii.) Show how you can combine two estimates of an integral using Simpson’s 1/3 rule (each with a different step size) to obtain an estimate of the integral which has an accuracy of order O(h6). (Show your working) [4 marks]Page 2 of 19 ANSWER: Also allowable is dividing 1st eqn by 16, or using a different step size to h/2 (e.g. h/3) – the numbers will of course be different iii.) Explain each of the steps required in order to use the shooting method to find the solution y(x) to the first-order boundary value problem below. ?? ?? = ?(?, ?), ?(?)) = ?), ?* ≤ ? ≤ ?) Use illustrations to help with your explanation. [6 marks] ANSWER: First we need to find the value of y(x0) that gives y(xn)= yn The steps to do this are 1. Make a series of guess for y0=y(x0) (1 mark) 2. Integrate each guess from x0 to xn using a suitable ODE integrator (e.g. RK4) and estimate y(xn) (1 mark) 3. If the calculated y(xn)= yn then stop, else choose a new guess for y0 (1 mark) 4. The choice of a new y0 should be based on a root finding method, where the equation is like f(y0)= y(xn) - yn e.g. Secant Method, bisection, false position (NOT Newton-Raphson) (1 mark) Once we have found correct y(x0), integrate one more time to calculate the solution over the entire domain (1 mark)

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