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
êë1 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

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  040  4060  6080  80100  100120 
No. of students  250  120  100  70  50 
4. (a) Estimate
e1.9
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 – y3) + 0.2(y3 – y5)
(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
(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
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

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.

(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
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

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


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 Rungekutta 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 Rungekutta 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 CrankNicholson’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 CrankNicholson’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

ReplyDeletesolve the following equation by gauss seidel method up to 3 iterations and find the value of x. 14x3yz=40,x6y+2z=28, x2y+12z=86