NUMERICAL ANALYSIS WEEK 8 2004

INTEGRAL AND DERIVATIVE

 

PROBLEM 1

 

      Calculate integral of y=f(x)=1/(1 + x2) function between limits –1 ile 1 with 3 points Gauss-Legendre integration Formula

 

Note: Required roots and coefficients :

coefficients   

Roots

c0  0.555555555555553  

x0  0.774596669241483       

c1  0.888888888888889   

x1  0.000000000000000   

c2 0.555555555555553

x2  0.774596669241483


PROBLEM 2

 Calculate diferential  of y=f(x)=1/(1 + x2) function at point 1

 

PROBLEM 3

   solve with bole rule. Use n=2

 


Note
: Bole rule for n=1

I=(b-a)*[7*f(x0)+32*f(x1)+12*f(x2)+32*f(x3)+7*f(x4)]/90

Step size h=(b-a)/n

 

PROBLEM 4




Function is known as black body radiation shape function. The values of coefficients in the equation are:

c1 =3.743 x108 W.mm4/m2, c2 =1.4387x104 mm.K, s=5.67x10-8 W/(m2.K4) dür.

Use Gauss-Legendre integration and for a giving  lT calculate black body radiation shape factor. 

Create a table for the values of blackbody radiation shape factor from 5000 to infinity(?)

 

PROBLEM 5

Solve the integration

     


a) by analithical methods

b) by Trapezoidal rule with n=2(divide region into 2 parts) and calculate % error

c) Trapezoidal rule with n=2(divide region into 2 parts) and calculate % error

 

PROBLEM 6     

Depth(H) and velocity(U) profile of a channel is given in below table

 

x, m

0

2

4

6

8

10

12

14

16

18

20

H, m

0

1.8

2

4

4

6

4

3.6

3.4

2.8

0

U m/s

0

0.03

0.045

0.055

0.065

0.12

0.07

0.06

0.05

0.04

0

 

Cross sectiaonal area of the channel can be calculated as :

 


Flow
rate of the channel will be

  In this equations U is the velocity of the water measured at point x. Calculate cross sectional area and flow rate of the channel. 

 

PROBLEM 7

Temperature profle of a solid body can be given as

      



In
this equation T is the temperature, Ta is constant surface temperature.

A solid body with temperatures initially at  T=90 C and,  Ta =20 C put into a water container at temperature of  Ta = 20 C. The temperature profile as a function of time is given at the table.

 

Zaman , t, s

0

5

10

15

20

25

30

35

Sicaklik T C

90

55

40

28

24

22

21.5

20.6

 

Calculate k thermal conductivity coefficient.