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.
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
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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
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
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
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
0 1 + x
by using Simpson’s rule.
6 dx
0 1 + x
by i) Trapezoidal rule ii) Simpson’s 1/3 rule
iii) Simpson’s3/8 rule iv) Actual integration
2
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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
by using Romerg’s method correct to 4 decimal places.
0 1 + x
Hence deduce an approximate value of Π.
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(b) Evaluate
òò (x + y)2
dxdy
by using Trapezoidal, Simpsons rules with h=k=0.5.
9. (a)Evaluate
2 dx
1 x
1 1 1
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
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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
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
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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.
20
40
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ReplyDeletesolve the following equation by gauss seidel method up to 3 iterations and find the value of x. 14x-3y-z=40,x-6y+2z=-28, x-2y+12z=-86