Name of the Student: Unit I (Solution of Equations and Eigenvalue Problems)

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1 Engineering Mathematics 8 SUBJECT NAME : Numerical Methods SUBJECT CODE : MA6459 MATERIAL NAME : University Questions REGULATION : R3 UPDATED ON : November 7 (Upto N/D 7 Q.P) (Scan the above Q.R code for the direct download of this material) Name of the Student: Branch: Unit I (Solution of Equations and Eigenvalue Problems) Fied point iteration 3 ) Find the smallest positive root of 5 by the fied point iteration method, correct to three decimal places. (N/D 7) ) Solve y 3 e 3 by the method of fied point iteration. (M/J ) 3) Find a real root of the equation by iteration method. (M/J ) 3 4) Find a positive root of the equation cos 3 by using iteration method. (M/J 3) Newton s method (or) Newton Raphson method 4 ) Solve for a positive root of the equation using Newton Raphson method. (A/M ) ) Find by Newton-Raphson method a positive root of the equation 3 cos. (N/D 4) 3) Solve sin cos using Newton-Raphson method. (M/J ) 4) Find an iterative formula to find the reciprocal of a given number N and hence find the value of. (N/D ) 9 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page

2 Engineering Mathematics 8 5) Find thenewton s iterative formula to calculate the reciprocal N and hence find the value of. (N/D ),(A/M 5) 3 6) Find an iterative formula to find N, where N is a positive number and hence find 5. (N/D ) Solution of linear system by Gaussian elimination method ) Apply Gauss elimination method to find the solution of the following system: 3y z 5, 4 4y 3z 3, 3y z. (N/D ) Solution of linear system by Gaussian-Jordon method ) Apply Gauss-Jordan method to find the solution of the following system: y z ; y z 3; y 5z 7. (N/D ) ) Using Gauss-Jordan method to solve y 3z 8; y z 4; 3 y 4z. (N/D 4) 3) Solve the system of equation by Gauss-Jordan method: 5 9, 5 3 4, (M/J 4) Solution of linear system by Gaussian-Seidel method ) Using Gauss-Seidel method, solve the following system of linear equations 4 y z 4, 5y z, y 8z. (M/J 4) ) Apply Gauss-Seidal method to solve the equations y z 7; 3 y z 8; 3y z 5. (M/J ),(N/D 4) 3) Solve, by Gauss-Seidel method, the following system: 8 4y z 3; 3y z 4; 7 y 4z 35. (N/D ),(N/D 7) 4) Use Gauss Seidel iterative method to obtain the solution of the equations: 9 y z 9; y z 5; y 3z 7. (A/M ) 5) Solve the following system of equations using Gauss-Seidel method: y z 9; y z ; 3y z. (N/D ),(A/M 5) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page

3 Engineering Mathematics 8 6) Solve, by Gauss-Seidel method, the following system: 8 3y z ; 4 y z 33; 6 3y z 35. (N/D ) 7) Solve, by Gauss-Seidel method, the equations 7 6y z 85; 6 5 y z 7; y 54z. (M/J 3) Inverse of a matri by Gauss Jordon method ) Using Gauss-Jordan method, find the inverse of (M/J 4) ) Using Gauss Jordan method, find the inverse of the matri (M/J ) 4 4 3) Find the inverse of the matri by Gauss Jordan method: 3. (A/M ) 3 4) Using Gauss Jordan method, find the inverse of the matri (N/D ),(A/M 5) 5) Find, by Gauss-Jordan method, the inverse of the matri 4 A 3. (M/J 3),(N/D 7) Eigen value of matri by power method 5 ) Find the numerically largest eigenvalue of A 3 and its corresponding 4 eigenvector by power method, taking the initial eigenvector as T. (M/J 4),(N/D 4) ) Determine the largest eigenvalue and the corresponding eigenvector of the matri Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 3

4 Engineering Mathematics 8. (M/J ) 6 3) Find the largest eigenvalue of, by using Power method. 3 (A/M ),(N/D ),(M/J ),(N/D ),(N/D 7) Unit II (Interpolation and Approimation) Lagrange Polynomials and Divided differences ) Using Lagrange s interpolation, calculate the profit in the year from the following data: (M/J ) Year: Profit Lakhs Rs ) Using Lagrange s interpolation formula find the value of y when, if the values of and y are given below: (M/J ) : y : ) Find the polynomial f( ) by using Lagrange s formula and hence find f (3) for the following values of and y : (N/D 4) : 5 y : ) Find the interpolation polynomial f( ) by Lagrange s formula and hence find f (3) for,,,3,, and 5,47. (N/D 7) 5) Apply Lagrange s formula, to find y(7) to the data given below. (M/J 3),(A/M 5) : y : ) Use Lagrange s formula to find a polynomial which takes the values f (), f(), f(3) 6 and f (4). Hence find f (). (A/M ) 7) Using Lagrange s interpolation formula, find y () from the following data: y() ; y() ; y(3) 8; y(4) 56; y(5) 65. (M/J 4) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 4

5 Engineering Mathematics 8 8) Use Lagrange s method to find log 656, given that log , log , log and log (N/D ) 9) Determine f( ) as a polynomial in for the following data, using Newton s divided difference formulae. Also find f (). (N/D ) : 4 5 f( ) : ) Find f (3) by Newton s divided difference formula for the following data: (M/J 4) : 4 5 f( ) : ) Find the function f( ) from the following table using Newton s divided difference formula: (A/M ) : f( ): ) By using Newton s divided difference formula find f (8), given (N/D 4) : f( ): ) Using Newton s divided differences formula determine f (3) from the data: : 4 5 (M/J ) f( ): ) Use Newton s divided difference formula to find f( ) form the following data. : 7 8 (M/J 3),(A/M 5) y : ) Using Newton s divided difference formula, find f( ) from the following data and hence find f (4). (N/D ) 5 f( ) 3 47 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 5

6 Engineering Mathematics 8 Interpolating with a cubic spline ) Fit the cubic splines for the following data: (M/J 4) : y : ) The following values of and y are given: : 3 4 y : 5 Find the cubic splines and evaluate y(.5) and y (3). (M/J ) 3) From the following table: : 3 y : Compute y(.5) and y() using cubic sphere. (M/J ),(M/J 3),(A/M 5) 4) Obtain the cubic spline for the following data to find y (.5). (N/D ),(N/D 4) : - y : ) If f (), f(), f() 33 and f (3) 44. find a cubic spline approimation, assuming M() = M(3) =. Also, find f (.5). (A/M ) 6) Find the cubic spline approimation for the function give below. (N/D 7) : 3 f( ) : Assume that M() M(3). Hence find the value of f (.5). Newton s forward and backward difference formulae ) Find the cubic polynomial which takes the following values: (M/J ),(M/J 3),(N/D 4),(A/M 5) : 3 f( ): ) Find the interpolation polynomial f( ) by using Newton s forward difference interpolation formula and hence find the value of f (5) for (N/D ),(N/D 7) y Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 6

7 Engineering Mathematics 8 3) Find the value of y at and 8 from the data given below (N/D) 4) The population of a town is as follows: : y : Year: y Population in thousands: Estimate the population increase during the period 946 to 976. (N/D ) 5) Given the following table, find the number of students whose weight is between 6 and 7 lbs: (A/M ),(M/J ) Weight (in lbs) : No. of students: Unit III (Numerical Differentiation and Integration) Differentiation using interpolation formulae ) The following data gives the corresponding values for pressure ( p ) and specific volume ( v ) of a superheated steam. Find the rate of change of pressure with respect to volume when v. (N/D 7) v : p : ) A slider in a machine moves along a fied straight rod. Its distance cm along the rod is given below for various values of the time ' t ' seconds. Find the velocity of the slider when t.second. (M/J ) t : : ) For the given data, find the first two derivatives at.. (M/J 4) : y : ) Find f ( ) at.5 and 4. from the following data using Newton s formulae for differentiation. : y f ( ) : (N/D) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 7

8 Engineering Mathematics 8 5) Find the first three derivatives of f( ) at.5 by Newton s forward interpolation formula to the data given below. (M/J 3),(A/M 5) : y f ( ) : ) Given the following data, find y(6) and the maimum value of y (if it eists) 7) Find the first two derivatives of : (A/M ) y : and 56 /3 at 5, for the given table: (N/D ) : /3 y ) Find the first and second derivative of the function tabulated below at.6. (N/D ) : y : ) Find the first and second derivatives of y with respect to at from the following data: (N/D 7) : y : ) Find dy d and d y d at 5, from the following data: (M/J ) : y : Numerical integration using Trapezoidal, Simpson s /3, 3/8 rules & Romberg s method ) Evaluate d and correct to 3 decimal places using Romberg s method and hence find the value of loge. (N/D 4) d ) Use Romberg s method to compute correct to 4 decimal places. Also evaluate the same integral using tree-point Gaussian quadrature formula. Comment on the Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 8

9 Engineering Mathematics 8 obtained values by comparing with the eact value of the integral which is equal to /4. (M/J ) d 3) Evaluate by using Romberg s method correct to 4 decimal places. Hence deduce an approimate value for. (M/J ) 4) Using Romberg s integration to evaluate. (A/M ) d 5) Evaluate d sin correct to three decimal places using Romberg s method. (M/J 4) d 6) Using Trapezoidal rule, evaluate by taking eight equal intervals. (M/J 3) 7) Compute / sin d using Simpson s 3/8 rule. (N/D ) 8) Evaluate I sin d by dividing the range into ten equal parts, using (i) Trapezoidal rule (N/D) (ii) Simpson s one-third rule, Verify your answer with actual integration. 9) Evaluate I 6 d by using (i) Direct integration (ii) Trapezoidal rule (iii) Simpson s one-third rule (iv) Simpson s three-eighth rule. (N/D ).3 ) Taking h.5, evaluate d using Trapezoidal rule and Simpson s three-eighth rule. (M/J 4) ) Using Simpson s one-third rule, evaluate.6 e d correct to three decimal places by step-size is.. (N/D 7) ) The velocity of a particle at a distance S from a point on its path is given by the table below: S (meter) (m / sec) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 9

10 Engineering Mathematics 8 Estimate the time taken to travel 6 meters by Simpson s /3 rd rule and Simpson s 3/8 th rule. (A/M ),(N/D 4) 3) The velocities of a car running on a straight road at intervals of minutes are given below: (A/M 5) Time (min): Velocity (km/hr): Using Simpson s /3 rd rule, find the distance covered by the car. Two and Three point Gaussian Quadrature formulae d ) Use Gaussian three-point formula and evaluate. (M/J ) 5 ) Evaluate 3) Evaluate 3 using 3 point Gaussian formula. (N/D 4) d d by Gaussian three point formula. (M/J 3) 4) Apply three points Gaussian quadrature formula to evaluate sin d. (A/M 5) Double integrals by Trapezoidal and Simpsons s rules ddy ) Using Trapezoidal rule, evaluate numerically with h. along - y direction and k.5 along y -direction. (M/J ) ) Evaluate ddy by trapezoidal rule. (N/D 4) y 3) Evaluate..4 ddy by trapezoidal formula by taking hk.. (A/M ) y 4) Evaluate y ddy by Trapezoidal rule taking hk.. (A/M 5) 4 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page

11 Engineering Mathematics 8 5) Evaluate 4 y ddy using Simpson s rule by taking h and 4 k.(n/d ) 6) Evaluate 7) Evaluate / /..4 sin( y) ddy using Simpson s rule with h k. y 4 (M/J ),(M/J 4) ddy using Simpon s one-third rule. (M/J 3) y 8) Evaluate f (, y) ddy by Trapezoidal rule for the following data, correct to three decimal places: (N/D 7) y / Unit IV (Initial Value Problems for Ordinary Differential Equations) Taylor series method ) Use Tailor series method to find y(.) and y(.) given that dy 3e y, d y (), correct to 4 decimal accuracy. (A/M ) dy ) Using Taylor series method solve y, y() at.,.,.3 d. Also compare the values with eact solution. (M/J ),(A/M 5),(N/D 7) 3) Using Taylor series method to find y(.) if y y, y(). (M/J 3) 4) Using Taylor s series method, find y at. by solving the equation dy y ; y(). Carryout the computations upto fourth order derivative. d (M/J 4) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page

12 Engineering Mathematics 8 5) Using Taylor s series method, find y at if Euler method for first order equation dy y d, y (). (N/D 4) y ) Solve y, y() at. by taking h. by using Euler s method. y ) Apply modified Euler s method to find y(.) and y(.4) given (M/J 3) y y, y() by taking h.. (N/D 4),(A/M 5) 3) Using Modified Euler s method, find y(4.) and y(4.) if dy d y (4). (N/D ) 5 y ; Fourth order Runge Kutta method for st order equation ) Use Runge Kutta method of fourth order to find y (.), given dy y d y y(), taking h.. (A/M ) ) Given y y y, y(), y() find the value of y (.) by Runge-Kutta s method of fourth order. (A/M 5) Milne s and Adam s predictor and corrector methods ) Given dy y y d, () y, y(.).69 and y(.).773, find (i) y (.3) by R-K method of fourth order and (ii) y (.4) by Milne s method. (N/D 7) ) Using Runge Kutta method of fourth order, find the value of y at dy 3.,.4,.6 given y, y(). Also find the value of y at.8 d using Milne s predictor and corrector method. (M/J 4) dy y ; y () ; y (.).6; y (.). d and y(.3)., evaluate y (.4) and y(.5) by Milne s predictor corrector method. 3) Given that Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page, (N/D )

13 Engineering Mathematics 8 dy 4) Given that y ; y(.6).684, (.4).48, (.).7, d y y y(), find y(.) using Milne s method. (N/D ) 5) Use Milne s predictor corrector formula to find y (.4), given dy y, d y(), y(.).6, y (.).and y(.3).. (A/M ) 6) Given y, y(), y(.).933, y(.4).755, y(.6).493 y find y (.8) using Milne s method. (M/J ) 7) Given 5y y, y(4), y(4.).49, y(4.).97, y(4.3).43. Compute y (4.4) using Milne s method. (N/D 4),(A/M 5) 8) Using Adam s method, find y (.4) given y(.). and (.3).3 dy y d y (), (.). y. (N/D 7) dy 9) Given y y, y(), y(.).69, y(.).774, y(.3).54 d. Use Adam s method to estimate y (.4). (A/M ) ) Using Adams method find y (.4) given y ( y), y(), y(.).33, y(.).548 and y(.3).979. (M/J ) ) Given that y y ; y() ; y(.).8; y(.4).468 and y(.6).7379, evaluate y(.8) by Adam s predictor-corrector method. (N/D ) ) Using Adam s method to find y() if y ( y) /, y(), y(.5).636, y() 3.595, y(.5) (M/J 3) 3) Using Adam s Bashforth method, find y (4.4) given that 5y y, y(4), y(4.).49, y(4.).97 and y(4.3).43. (M/J 4) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 3

14 Engineering Mathematics 8 Unit V (Boundary Value Problems in ODE and PDE) Finite Difference Solution of Second Order ODE ) Solve the equation y y with the boundary conditions y() y() using finite differences by dividing the interval into four equal parts. (M/J ),(M/J 4),(A/M 5) ) Solve the boundary value problem y y with boundary conditions y() and y(), taking h by finite difference method. (N/D 7) 4 3) Solve y y with the boundary conditions y() and y(). (N/D ) 4) Solve, by finite difference method, the boundary value problem y( ) y( ), where y() and y(), taking h.5. (M/J ) 5) Using the finite difference method, compute y (.5), given y 64 y, (,), y() y(), subdividing the interval into (i) 4 equal parts (ii) equal parts. (N/D ),(N/D ) One Dimensional Heat equation by eplicit method ) Solve ut u in 4, t, given that u(, t), u(4, t), u(,) (4 ). Compute u upto t 4 with t. (N/D 7) u u ) Solve, subject to (, ) (, ) t u t u t and u (,) sin,, using Bender-Schmidt method. (M/J ) u u 3) Using Bender-Schmidt s method Solve t, given u(, t), u(, t) u(,) sin, and h.. Find the value of u upto t.. (M/J 4),(A/M 5) 4) Solve by Bender-Schmidt formula upto t 5 for the equation u ut, subject to and u(,) 5 u(, t), u(5, t),, taking h. (N/D ) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 4

15 Engineering Mathematics 8 u u 5) Solve with the condition u(, t) u(4, t), u(,) (4 ) taking t h employing Bender-Schmidt recurrence equation. Continue the solution through time steps. (M/J ) 6) Solve u 3 ut, h.5 for t,, u(, t), u(,), u(, t) t. (M/J 3) One Dimensional Heat equation by implicit method kc ) Obtain the Crank Nicholson finite difference method by taking. Hence, h u u find u(, t) in the rod for two times steps for the heat equation t, given u(,) sin( ), u(, t), u(, t). Take h =.. (A/M ) u u ) Solve by Crank-Nicolson s method for, t t given that u(, t), u(,t) and u(,) ( ). Compute u for one time step with h and k. (N/D 4) 4 64 One Dimensional Wave equation u u ) Solve the equation,, t satisfying the conditions t u u(,), (,), u(, t) and u(, t) sin t. Compute u(, t ) for 4 t time-steps by taking h. (N/D ) 4 u u ) Solve the wave equation,, t, t,.5 u u(, t) u(, t),t u (,) and (,), using,.5 t hk., find u for three time steps. (M/J 4),(A/M 5) u u 3) Find the pivotal values of the equation with given conditions 4 t u u(, t), u(4, t), u(,) (4 ) and (,) by taking h, for 4 t time steps. (M/J ) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 5

16 Engineering Mathematics 8 4) Solve 4 utt u, u(, t), u(4, t), u(,) (4 ), ut (,), h upto t 4. (M/J 3) Two Dimensional Laplace and Poisson equations ) Solve the elliptic equation u u for the following square mesh with boundary yy values as shown: (M/J ) ) By iteration method, solve the elliptic equation side 4, satisfying the boundary conditions. u u y over a square region of (i) u(, y), y 4 (ii) u(4, y) y, y 4 (ii) u(,) 3, 4 (iv) u (,4), 4 By dividing the square into 6 square meshes of side and always correcting the computed values to two places of decimals, obtain the values of u at 9 interior pivotal points. (N/D 4) 3) Solve the Poisson equation u y, y, 3 and y 3 with u over the square mesh with sides on the boundary and mesh length unit. (N/D ),(M/J 4),(A/M 5),(N/D 7) 4) Solve u 8 y for square mesh given u on the four boundaries dividing the square into 6 sub-squares of length unit. (N/D ) 5) Solve u 8 y over the square,, y, y with u on the boundary and mesh length =. (M/J 3) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 6

17 Engineering Mathematics All the Best---- Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: , ) Page 7