MA1251 -NUMERICAL METHODS Questions Bank

Anna University, Chenna SRIIVASAN ENGINEERING COLLEGE MA1251 -NUMERICAL METHODS UNIT - I SOLUTIONS OF EQUATIONS AND EIGEN VALUE PROBLEMS PART – B 1. (a) Solve 3x – cos x – 1 = 0 by Newton's method (b) By using Gauss Seidel method. Solve the following system of equation. x + y +54z = 110, 27x + 6y – z = 85, 6x +15y + 2z = 72. 2. (a) Solve x3 - 4x +1 = 0 by using Regula falsi method. (b) Solve x + 3y + 3z = 16 ; x + 4y + 3z = 18 ; x + 3y + 4z = 19 by Gauss elimination method. 3. (a) Using the Gauss Jacobi method to solve the following equations. 10x + y + z = 12, 2x + 10y + z = 13, x +y + 5z = 7. (b) Find a positive root of 3x - 1+ sin x = 0 by fixed point method. 4. (a) Using the Gauss Jordan method to solve the following equations. 10x + y + z = 12, x + 10y - z = 10, x -2y + 10z = 9. (b) Solve x = cos x by Newton- Raphson method. é 1 1 1 ù 5. (a) Find the inverse of A = ê 0 1 -2ú by Gauss Jordan method. ê ú êë-1 1 1 úû (b) Solve x + y + 2z = 4 ; 3x + y - 3z = - 4 ; 2x – 3y – 5z = - 5 by Gauss elimination method. 6. (a) Find a positive root of 2x = 3 + cos x by fixed point method. (b) By using Gauss seidel method. Solve the following system of equation. 20x – y -2z = 17, 3x + 20y – z = -18, 2x – 3y + 20z = 25. 7. (a) Using the Gauss Jordan method to solve the following equations 2x – 6y + 8z = 24, 5x + 4y - 3z = 2, 3x + y + 2z = 16. (b) Solve xex = 2 by using Regula falsi method. 8. (a) Solve 4x + 2y + z = 14; x + 5y - z = 10; x + y + 8z = 20 by Gauss Jacobi method. é3 1 2 ù (b) Find the inverse of A = ê2 -3 -1ú by Gauss Jordan method. ê ú êë1 2 1 úû é1 -3 2 ù 9. Find the all Eigen values and eigen vectors of A = ê4 4 -1ú by Power method of iteration. 10. Find the all Eigen values and eigen vectors of A = ê ú êë6 3 5 úû é25 1 2 ù ê ú ê 1 3 0 ú by Power method of iteration. 2 0 -4 PART – B 1. The population of a town is shown below. By using corresponding interpolation Find the increase in the population b/w 1916 and 1948 Year 1911 1921 1931 1941 1951 1961 population 12 13 20 27 39 52 2. From the table find the pressure at t=142o and t=175o Tempo C 140 150 160 170 180 Pressure 3.685 4.854 6.302 8.076 10.225 3. From the data find the number of students whose weight is between 60 to 70 Weight 0-40 40-60 60-80 80-100 100-120 No. of students 250 120 100 70 50 4. (a) Estimate e-1.9 from the data. x 1.00 1.25 1.50 1.75 2.00 e- x 0.3679 0.2865 0.2231 0.1738 0.1353 (b) Using Lagrange's interpolation formula find y at x = 3 X 0 1 2 4 5 Y 0 16 48 88 0 5. (a) Using Lagrange's interpolation formula P.T y1 = y3 – 0.3(y5 – y-3) + 0.2(y-3 – y-5) (b) Construct a polynomial for the data and hence find y(5) given below. X 4 6 8 10 Y 1 3 8 16 6. (a) Construct a polynomial using Lagrange's method , hence find f(2.5), f(3.8) X 0 1 3 4 Y -12 0 6 12 (b) Express 3x2 + x + 1 ( x - 1)( x - 2)( x - 3) as a sum of partial fraction using Lagrange's Interpolation formula. 7. (a) Find a polynomial of degree two for the data X 0 1 2 3 4 5 6 7 Y 1 2 4 7 11 16 22 29 (b)Using Newton's divided difference formula find y at x = 5 X 0 1 3 6 Y 1 4 88 1309 ¢¢ ¢¢ 8. Fit a cubic spline for the giving data given y0 = y2 = 0 and hence find f(0.75), f(1.75). X 0 1 2 Y 1 2 3 ¢¢ ¢¢ 9. Fit a cubic spline for the giving data given y0 = y2 = 0 and hence find y(1.5), y'(1). X 1 2 3 Y -8 -1 18 10. Using Newton's divided difference formula, find the values of f(2), f(8) given below. X 4 5 7 10 11 13 Y 48 100 294 900 1210 2028 PART –B 1. (a) Find the first, second, third derivatives of the function tabulated below at x = 1.5 and x = 4. X 1.5 2 2.5 3 3.5 4 Y 3.375 7 13.625 24 38.875 59 (b) Find the value of cos (1.74 ) from the following table x 1.7 1.74 1.78 1.82 1.86 sinx 0.9916 09857 09781 09691 0.9584 2. (a) From the following data find f ¢(5), f ¢¢(5) & f ¢¢¢ (5) X 0 2 3 4 7 9 F(x) 4 26 58 112 466 922 3 dt (b) Using Gaussian two point and three point formula evaluate ò 2 1 + t 3. The table below gives velocity of a moving particle at time t seconds. Find the distance covered by a particle in 12 seconds and also acceleration at t = 2 seconds. t 0 2 4 6 8 10 12 v 4 6 16 34 60 94 136 4. (a) Find the first and second derivative of the function tabulated below at x = 0.6. x 0.4 0.5 0.6 0.7 0.8 y 1.5836 1.7974 2.0442 2.3275 2.6511 (b) Using Gaussian two point formula evaluate 1 2 p 2 ò log(1 + x)dx 0 5. (a) Find the value of log 2 1/3 from x dx ò 3 by using Simpson's 1/3 rule, h = 0.25 (b) Evaluate 1.4 2.4 ò ò 1 2 1 dxdy xy 0 1 + x by using Simpson's rule. 6 dx 6. Evaluate ò 2 0 1 + x by i) Trapezoidal rule ii) Simpson's 1/3 rule iii) Simpson's3/8 rule iv) Actual integration 2 2 7. Evaluate ò dx ( n= 8), by i) Trapezoidal rule ii) Simpson's 1/3 rule 0 1 + x + x iii) Simpson's3/8 rule iv) Actual integration 1 dx 8. (a) Evaluate ò 2 by using Romerg's method correct to 4 decimal places. 0 1 + x Hence deduce an approximate value of Π. 1 3 2 4 1 (b) Evaluate òò (x + y)2 dxdy by using Trapezoidal, Simpsons rules with h=k=0.5. 9. (a)Evaluate 2 dx ò by i) Trapezoidal rule ii) Simpson's 1/3 rule Take h=0.25. 1 x 1 1 1 (b) Evaluate òò dxdy by using Trapezoidal, Simpsons rules. 0 0 1 + x + y 4 ex dx 10. (a) Using Simpson's rule find ò , with h=1 0 12 dx x (b) Using Gaussian two point and three point formula evaluate ò 5 PART – B 1. (a) Using Taylor series method , find , correct to four decimal places , the value of y(0.1) and y(0.2), given dy = x2 + y 2 and y(0) = 1 dx (b) Using Modified Euler and Euler method find y(0.2) given dy = y - x2 + 1 y(0) = 0.5 dx 2.(a) Using Taylor series method , find the value of y(0.2) and y(0.4), given dy =1 - 2 xy and y(0) = 0 dx (b) Using Modified Euler and Euler method find y(0.2), y(0.1) given dy = y 2 + x2 y(0) = 1 dx dy y 2 - x2 3. Using R.K Method of fourth order, Solve given y(0) = 1 find y at x = 0.2 x = 0.4. = dx y 2 + x2 4. Using R.K Method of fourth order Solve given y(0) = 2 find y at x = 0.2 x = 0.4 dy = x3 + y dx 5.Using Milne's method find y(2) given y¢ = æ 1 ö ( x + y) given y(0) = 2 2 ç ÷ è ø y(0.5) = 2.636 y(1) = 3.595 y(1.5) = 4.968. 6. Using Milne's method find y(4.4) given 5xy¢ + y2 - 2 = 0 given y(4) = 1 y(4.1) = 1.0049 y(4.2) = 1.0097 y(4.3) = 1.0143. 7. Given y¢= xy , y(0) = 1, y(0.1) = 1.01, y(0.2) = 1.022, y(0.3) = 1.023, find y(0.4) using 2 Adam's method. 8. Given y' = 1 + xy , y(0) = 2, find y(0.4) by using Adam's method. 9. Given y¢ = x2 + y , y(0) = 1, find y(0.1) by Taylors method, y(0.2) by modified Euler's method, y(0.3) by Runge-kutta method and y(0.4) by Milne's method. 10. Given y¢ = x + y 2 , y(0) = 1, find y(0.1) by Taylors method, y(0.2) by modified Euler's method, y(0.3) by Runge-kutta method and y(0.4) by Adam's method. PART – B 1. Solve y'' – y = x, x Î ( 0, 1) , given y(0) = y(1) = 0 using finite differences dividing the interval into four equal parts. 2. Solve uxx - 2ut =0, given u(0.t)=0, u(4,t)=0 and u(x, 0) = x(4 - x) , taking h=1 find the values of u up to t=5. 2 3. Given ¶ f = ¶f , f (0, t ) = f (5, t) = 0, f ( x, 0) = x2 (25 - x2 ) find f in the range taking h=1 ¶x2 ¶t and up to 5 seconds. 4. Using Crank-Nicholson's scheme, solve uxx = 16ut , 0<x<1, t>0 given u(x,0)=0, u(0,t)=0,u(1,t)=100t compute u for one step in t direction taking h=1/4 5. Using Crank-Nicholson's scheme, solve uxx = ut , 0<x<1, t>0 given u(x,0)=0, u(0,t)=0, u(1,t)=t compute u for one step in t direction taking h=1/4 6. Solve Ñ2u = -10( x2 + y2 + 10) over the square mesh with sides x = 0 , y = 0 , x = 3 , y = 3 with u = 0 on the boundary and mesh length 1 unit. 7. Solve Uxx + Uyy = 0 over the square mesh of side 4 units satisfying the following boundary conditions i) u( 0 , y ) = 0 for 0 £ y £ 4 ii) u( 4 , y ) = 12 + y for 0 £ y £ 4 for 0 £ x £ 4 iv) u( x , 4 ) = x2 for 0 £ x £ 4 iii) u( x , 0 ) = 3x 8. Solve numerically, 4uxx = utt with the boundary conditions u(0,t)=0, u(4,t)=0 and the initial conditions ut ( x, 0) = 0 and u(x, 0) = x(4 - x) , taking h=1 (for 4 time steps). 9. Solve 25uxx = utt for u at the pivotal points given ì2x , 0 £ x £ 2.5 u(0, t) = u(5, t) = 0, ut ( x, 0) = 0 for one half period of vibration. and u( x, 0) = í î10 - 2x , 2.5 £ x £ 5 10. Solve Ñ2u = 0 at the nodal points for the following square region given the boundary conditions. 0 10 20 30 20 40 40 50 60 60 60 60

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