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04-BS-5: December 2013

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

  1. Question 1 Power-Series Solution of a Legendre-Type ODE
  2. Question 2 Fourier Series of a Piecewise Function
  3. Question 3 Two-Sided Exponential and Its Fourier Transform
  4. Question 4 Newton's Divided Differences and Finite-Difference Derivatives
  5. Question 5 Newton's Method, Bisection, and Fixed-Point Iteration
  6. Question 6 Romberg Integration of Tabulated Data
  7. Question 7 LU Decomposition and Solving a Linear System

Start with Question 1 →

National Exams — December 2013 — 04-BS-5 Advanced Mathematics. Three-hour, closed-book exam (one double-sided 8.5"×11" aid sheet permitted; approved Casio/Sharp calculator allowed). Format: seven questions (Q1 20, Q2 20, Q3 20, Q4 20, Q5 20, Q6 20, Q7 20 marks, per the printed marking scheme); any five constitute a complete paper and only the first five appearing in the answer book are marked. All seven are solved below for completeness.

Reference texts: Kreyszig, Advanced Engineering Mathematics (10th ed., Wiley) — power-series solutions of ODEs about an ordinary point (Ch. 5), Fourier series and the Fourier transform (Ch. 11), numerics in general: interpolation, root-finding, Romberg integration (Ch. 19), numeric linear algebra: LU factorization (Ch. 20); Burden & Faires, Numerical Analysis (9th ed., Cengage) — Newton divided-difference interpolation, finite-difference derivative stencils, Newton–Raphson/bisection/fixed-point convergence theory, Romberg extrapolation, Doolittle LU factorization.