B.E. DEGREE EXAMINATION.
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 :
(ii) Use the Newton’s backward difference formula to approximate (0.65).
(iii) Use Stirlings formula to approximate (0.43).
(b) (i) Derive the Lagrange’s interpolation for unequal intervals. (6)
(ii) Find an approximate polynomial using Hermite’s interpolation. (10)
- (a) (i) Given the following table of values of and :
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)
(b) (i) For the following values of and , find the first derivative at
= 4. (6)
(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)
(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)
(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)