Reg. No.: B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2010 Fifth Semester Chemical Engineering MA 2264 — NUMERICAL METHODS (Common to Fifth Semester Polymer Technology and Plastic Technology) (Also common to Fourth Semester Aeronautical Engineering, Electrical and Electronics Engineering and Civil Engineering) (Regulation 2008) Time : Three hours Scientific calculator is permitted Answer ALL questions PART A — (10 × 2 = 20 Marks) 1. 2. 3. 4. What is Newton’s algorithm to solve the equations x 2 = 12? To what kind of a matrix, can the Jacobi’s method be applied to obtain the eigenvalues of a matrix? Find the first and second divided differences with arguments a, b, c of the 1 function f(x) = . x 5. Write down the formulae for finding the first derivative using Newton’s forward difference at x = x 0 and Newton’s backward difference at x = x n . 2 6. Evaluate ∫e 0 −x 2 dx by two point Gaussian quadrature formula. 7. Find y(0.1) by using Euler’s method given that 9. 10. 21 11. (a) (i) 9 Find, by 8. What are multi-step methods? How are they better than single step methods? What is the error for solving Laplace and Poisson’s equations by finite difference method? Write down the Crank-Nicolson formula to solve parabolic equation. PART B — (5 × 16 = 80 Marks) power method, the largest eigenvalue and the 1 3 − 1 corresponding eigenvector of a matrix A = 3 2 4 with initial − 1 4 10 vector (1 1 1)T . 21 When to use Newton’s forward interpolation and when to use Newton’s backward interpolation? 9 dy = x + y , y( 0) = 1 . dx CENTRAL LIBRARY-SVCE, SRIPERUMBUDUR 21 9 Maximum : 100 Marks (8) Question Paper Code : 53190 15037 1. f (0) = 3 . (8) (ii) Find the value of tan 45° 15′ by using Newton’s forward difference interpolation formula for x° : 45 46 (8) 49 50 47 48 (b) Find the natural cubic spline approximation for the function f(x) defined by the following data : x: 0 1 2 3 Hence find the value of f(2.359 4. f ( 2) = 1 and f (3) = 2 .(ii) Solve. (a) (i) 21 Or f(x) : 1 2 33 244 15 17 19 21 1 tan x° : 1. x: (8) 25 5 21 9 y : 3. (a) (i) Use Lagrange’s interpolation formula to fit a polynomial to the given data f ( −1) = −8 . the system of equations (8) 3x + 20 y − z = −18 2x − 3 y + 20z = 25.00000 1.873 4. (8) 12.125.583 4.07237 1. 0.25 and 0.5) .796 (ii) Using Romberg’s rule evaluate ∫ 1 + x dx 0 1 places by taking h = 0. by Gauss-Seidal method. Or CENTRAL LIBRARY-SVCE.123 4.11061 1.5) and f ′( 2.03553 1. Hence find the value of f (1) .5.19175 9 23 From the following table of values of x and y. SRIPERUMBUDUR 2 21 9 (16) 20x + y − 2z = 17 correct to three decimal (8) 53190 . Or (b) (i) Find the inverse of method. obtain y′( x ) and y′′( x ) for x = 16. 1 2 4 A = 2 3 − 1 by using Gauss-Jordan 1 − 2 2 (8) (ii) Find the smallest positive root of 3x = 1 + sin x correct to three decimal places by iterative method. 13. (b) (i) (ii) Evaluate ∫∫ 4 1 dx dy by Trapezoidal rule in x-direction with 1 x + y 1 5 h = 1 and Simpson’s one-third rule in y-direction with k = 1. t ) = 0 . (8) Or (b) Given dy = xy + y 2 . y(0) = 1 .2 given Taylor’s series method up to four terms. u(5. (8) Solve ut = u xx in 0 < x < 5 . taking h = 1/3.1) by Runge-Kutta method of fourth order given y′′ + xy′ + y = 0 . using Newton’s divided difference formula.2774 . (8) 21 9 (8) dy = x 2 y − 1. (8) (b) (i) Solve ∇ 2u = 8x 2 y 2 in the square region − 2 ≤ x .1169 and y(0. y ≤ 2 with u = 0 on the boundaries after dividing the region into 16 sub intervals of length 1 unit. t ) = 0 and u(1. (ii) Find the value of y(0. 9 Or 3 (ii) Solve ∂u ∂ 2 u ∂ 2u ( x . f(1) = 15. by finite difference method. y(1) = 1 . y(0) = 1 .1) = 1.2) = 1.4) by Milne’s method.1. Compute u up to t = 2 with ∆ x = 1 . SRIPERUMBUDUR 53190 21 ————————— Solve the boundary value problem y′′ = xy subject to the conditions y(0) + y′(0) = 1 . by using Bender-Schmidt formula. y(0. (8) (16) . 0) = x 2 (25 − x 2 ) . by dx (8) Find the first derivative of f(x) at x = 2 for the data f(–1) = –21. Compute u( x . (8) 21 9 (ii) CENTRAL LIBRARY-SVCE. 14. t > 0 given that u(0. t ) = 100 sin π t . t > 0 given u( x . find dx y(0. 0 < x < 1 .25. = ∂t ∂t 2 ∂x 2 u(0. f(2) = 12 and f(3) = 3. t ) for four times steps with h = 0. 0. u( x . t ) = 0 . . 0) = 0 . 0 ) = 0 .3) by Runge-Kutta method of forth order and (i) (ii) 15. y( 0) = 1 and y′(0) = 0 . (a) (i) y(0. (a) (i) Find the value of y at x = 0.