Anna University Question Papers – Civil Engineering – IV SEM

MODEL PAPER

B.E. DEGREE EXAMINATION.

Fourth Semester

Civil Engineering

MA 038 — NUMERICAL METHODS

(Common to Mechanical, Instrumentation and Control Engineering, Aeronautical, Automobile, Production, Instrumentation and Mechatronics Engineering)

Time : Three hours                                                                        Maximum : 100 marks

Answer ALL questions.

PART A — (10 ´ 2 = 20 marks)

  1. What is the order of convergence of Newton–Raphson method?
  2. Compare Gaussian elimination and Gauss–Jordon methods in solving the linear system .
  3. Given  , use  to show that .
  4. What is the order of interpolating polynomial could be constructed, if n sets of  are given?
  5. Find , using Newton’s forward difference interpolation.
  6. What is the geometrical meaning of trapezoidal rule?
  7. What is single step method? Give examples.
  8. How do you apply Runge–Kutta method of order form to solve ,  and ?
  9. What is the order of convergence of Crank–Nicolson method for solving parabolic partial differential equation  subject to , and ?
  10. Write down the finite difference scheme for solving the Poisson equation  on  with  for  where  denotes the boundary of .

PART B — (5 ´ 16 = 80 marks)

  1. (i)      Using iterative method, find the root of  in [1, 2].     (6)

             (ii)     Solve the following system by applying first two iterations by Gauss–Jacobi method and continue using the Gauss Seidel method.                                               (10)

.

  1. (a)     (i)      Approximate  using  the following data and the Newton’s forward difference formula :
 : 0.0 0.2 0.4 0.6 0.8
() : 1.0000 1.22140 1.49182 1.82212 2.22554

                       (ii)    Use the Newton’s backward difference formula to approximate (0.65).

                       (iii)   Use Stirlings formula to approximate (0.43).

Or

             (b)     (i)      Derive the Lagrange’s interpolation for unequal intervals.          (6)

                       (ii)    Find an approximate polynomial using Hermite’s interpolation. (10)

 () ()
0.8 0.22363 2.16918
1.0 0.658091 2.04669
  1. (a)     (i)      Given the following table of values of  and  :
 : 1.0 1.05 1.10 1.15 1.20 1.25 1.30
 : 1.0000 1.0247 1.0488 1.0723 1.0954 1.1180 1.1401

                                find  and  at  = 1.00, 1.25 and 1.15.                                 (10)

                       (ii)    Estimate the length of arc of the curve 3 =  from (0, 0) to  using Simpson’s   rule taking 4 subintervals.      (6)                                                                              

Or

             (b)     (i)      For the following values of  and , find the first derivative at
= 4.                                                                                                     (6)

 : 1 2 4 8 10
 : 0 1 5 21 27

                       (ii)    Evaluate  by trapezoidal rule with  =  = 0.5. (6)

                       (iii)   Evaluate  by two point Gaussian quadrature.             (4)

  1. (a)     Solve  with

                       (i)      Use Taylor series at x = 0.2 and x = 0.4                                            (4)

                       (ii)    Use Runge–Kutta method of order 4 at x = 0.6                               (6)

                       (iii)   Use Adam–Bashforth predictor–corrector method at x = 0.8.      (6)

Or

             (b)     (i)      Using Taylor series method, solve  with ,  for x = 0.2 and x = 0.4.    (8)

                       (ii)    Also solve the problem using Runge–Kutta method to find (8)

  1. (a)     (i)     Solve     using                                              (12)

                       (ii)    Derive the Crank–Nicolson finite difference scheme for solving the parabolic equation   t > 0 and             (4)

Or

             (b)     (i)      Derive the explicit finite difference scheme for solving the one dimensional hyperbolic equation   subject to and                                                                                       (6)

                       (ii)    Solve  subject to   and  with  and  using Schmidt method for 2 time steps.                        (10)

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