# Anna University Model Question Paper BE IV sem CIVIL NUMERICAL METHODS

**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)

- What is the order of convergence of Newton–Raphson method?
- Compare Gaussian elimination and Gauss–Jordon methods in solving the linear system .
- Given , use to show that .
- What is the order of interpolating polynomial could be constructed, if
*n*sets of are given? - Find , using Newton’s forward difference interpolation.
- What is the geometrical meaning of trapezoidal rule?
- What is single step method? Give examples.
- How do you apply Runge–Kutta method of order form to solve , and ?
- What is the order of convergence of Crank–Nicolson method for solving parabolic partial differential equation subject to , and ?
- 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)

- (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)

.

- (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 |

- (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)

- (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)

- (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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