PESIT Bangalore South Campuspesitsouth.pes.edu/pdf/resources/CI-10MAT-31-MATHS.pdf5 x5-3.7 x 4 +7.4 x 3 – 10.8 x 2 + 10.8x – 6.8=0 3. Use Gauss-Seidel method to solve: 1) 20x+y-2z=17,

PESIT(BSC)– Course Information BE III-Sem CSE 10MAT31 - 1

PESIT Bangalore South Campus

10MAT31- ENGINEERING MATHEMATICS – III

Faculty Name: Mr.Nagesh.H./Mr.James Alex.

No of Units: 52

Class # Chapter Title Topics to be covered

% of portions

covered

Cumulative

1,2 UNIT-V

NUMERICAL

METHODS –I

T1:1009 t0

1015

1028 to

1032

1034 to

1036

Solution of algebraic and trasendental

method

Regula-falsi mehtod

12.5

3 Newton raphson method

4 Gauss- siedel method

5 Relaxation methods

6 Power method

7

UNIT-VI

NUMERICAL

METHODS-II

T1:1038-1045

1050-1054

1065-1080

Finite differences - forward, backward

25

8 Interpolation & extrapolation - Gregory

Newton formulae – problems

9 Lagrange's formula for unequal intervals –

inverse interpolation – problems

10 Newton's divided difference formula –

problems

11 Numerical differentiation – problems

12 Numerical integration - Simpson's 1/3 rule,

problems

Simpson's 3/8 rule – problems

13 Weddle's rule – problems

14

UNIT-VII

NUMERICAL

METHODS-III

T1:1127-1133

1135-1142

Numerical solution of pde-finite difference

approximation

37.5

15,16 Numerical solution of 2D Laplace equation

17,18 One dimensional heat equation

19,20 One dimensional wave equation


21 UNIT-IV

CURVE

FITTING AND

OPTIMIZATIO

N

T1:891 to 898

1144 to

1168

Curve fitting by the method of Least squares-

Fitting of the straight line y=ax+b

cbxaxy ++= 2

50 22 bbx

axyaey == ,

23,24,25 Optimization:LPP,mathematical formulation and

Graphical method

26,27 Simplex method

28

UNIT-I

Fourier series

T1:Page#:368-

393

T2:Page#:163-

282

T3:Page#:188-

274

Introduction, periodic, even and odd functions

62.5

29,30 Problems

31 Problems

32 Half range series

33 Problems

34 Practical Harmonic analysis – problems

35 UNIT-II

Fourier

Transforms

T1:Page#:711-

720

T2:Page#:283-

236

T3:Page#:275-

332

Fourier transforms- problems on infinite

transforms

75

36 Inverse Fourier transforms – problems

37 Fourier sine & cosine transforms - their

inverses

38,39 Problems

40 Properties

41 UNIT-VIII

Z-Transforms

T1:Page#:918-

920

929-974

T2:Page#:337-

356

T3:Page#:333-

360

Difference equation – basic definition

87.5

42 Z-Transforms- Definition, Standard forms

43 Linearity property, damping rule

44

Shifting rule, problems

Initial value theorem,final value theorem

45 Inverse Z transforms

46 Application of Z transforms

47,48

UNIT-III

Application of

PDE

T1:Page#: 557-

564 573-576

Various possible solution of 1D wave and heat equations

100

49 Solution of two dimentional Laplace equation- By

method of separation of variables.

50, 51 Solution of all these equations with specified

boundary conditions

52

D’Alembert’s solution of 1-D wave equation- problems

Literature:

Book type Code Title & Author Publication Information

Edition Publisher Year


Text Books: T1

“Higher Engineering

Mathematics”

Dr. B.S. Grewal

38th

Edition Khanna 2004

Text Books: T2 “Engineering Mathematics-III”

Dr. KSC 9rthEdition Sudha 2004

Text Books: T3 “Engineering Mathematics-III”

Dr. DSC 3rd Edition Prism 2004

Reference

Books: R1

“Advanced Engineering

Mathematics”

E Kreysizig

8th

Edition.

John Wiley &

Sons 2004

PART-B

Unit-IV

NUMERICAL METHODS I:

Solution of System of algebraic and transcendental equations :

1. Use Regular - falsi method to find a real root of the given equations.

1) sinx-coshx+1=0

2) x2-logex –12 = 0

3) cosx-3x+1=0

4) ex-3x=0

5) x4+2x2-16x+5=0 in (0.1)

6) 2x+log10x=7 in (3.5,4) correct to 3 decimal places

2. Use Newton Raphson method to find the real root of given equations

1 12 to 4 decimal places

2 cosx = x

3 x3+5x+3=0 in (1,2)

4 x2+ 4 sinx = 0 to 4 decimal places

5 x5-3.7 x4 +7.4 x3 – 10.8 x2 + 10.8x – 6.8=0

3. Use Gauss-Seidel method to solve:

1) 20x+y-2z=17, 3x+20y-z=-18, 2x-3y+20z=25

2) 4x+2y+z=14, x+5y-z=10, x+y+8z=20. starting with initial set (1, 1, 1)

3) 8x+y+z-u = 18, -2x+12y-z=-17, 2x+16z+2u = 54, y+2z-20u= -14

starting with ( 1, 1, 1 )

4. Explain Power method to find the dominant eigen value and the corresponding

eigen vector for a given Matrix and use it for the following matrices:

21-0

1-21-

01-2

f)

24-20

612-10

3415-

e)

221

131

122

d)

31-2

132-

22-6

c)

100

121

112

b)

34-2

4-76-

26-8

a)

UNIT-V

NUMERICAL METHODS II:

FINITE DIFFERENCES AND INTERPOLATION FORMULAE.

1. Write the difference table for the following set of values.


X: 1 2 3 4 5

Y: 2 2.301 2.477 2.778 2.699

2. If f(x) is a cubic polynomial find the missing number.

X: 0 1 2 3 4

Y: 5 1 N 35 109

3. Evaluate )x(tan)i( 1−∆ )x2(cos)ii( 2∆ 22

xE

)i(

∆

4. Prove that

++≡∆

41

2

1 )(

22 δ

δδi

2 )( δ≡∆∇≡∇∆ii

2/12/1 )1( )1( )( −− ∇−∇≡∆+∆≡δiii

5. The values of ex for different x are tabulated below. Find the approximate values of e1.25 and

e1.65.

X: 1.1 1.2 1.3 1.4 1.5 1.6 1.7

ex: 3.004 3.320 3.669 4.055 4.482 4.953 5.474

6. The table below gives the distance in nautical miles of the visible horizon for the given heights, in feet, above the earth’s surface. Find d when x = 410 feet.

Height(x): 100 150 200 250 300 350 400

Distance(d): 10.63 13.03 15.04 16.81 18.42 19.90 21.27

7. Find the values of f(22) and f(42) from the following table.

x: 20 25 30 35 40 45

f(x): 354 332 291 260 231 204

8. Find f(x) when x = 0.1604 from the table given below :

X: 0.160 0.161 0.162

f(x): 0.15931821 0.16030535 0.16129134

9. Estimate the values of (i) sin 380 and (ii) sin 220 from the table given below.

x(degrees): 0 10 20 30 40

41 )iv(

22 δ

+≡µ


sin x: 0 0.17365 0.34202 0.50000 0.64279

10. Use Lagrange’s interpolation formula to find the value of y at x = 10 from the following table.

X: 5 6 9 11

Y: 12 13 14 16

11. Given log 654 = 2.8156, log 658 = 2.8182, log 659 = 2.8189, log 661 = 2.8202,

find log 656 using Lagrange’s interpolation formula.

12. Given the following table, find f(22) using Newton’s divided difference formula.

x: 1.0 2.0 2.5 3.5 4.0

f(x): 84.289 87.138 87.709 88.363 88.568

13. Apply Newton - Gregory forward difference formula to find f (5) given that

f (0) = 2, f (2) = 7, f (4)=10, f (6) = 14, f (8) = 19, f (10) = 24.

14. Find the polynomial approximating f (x) using Newton- Gregory forward difference formula, given

f (4) = 1,f (6) = 3, f (8) = 8, f(10) = 20.

15. Apply Newton- Gregory backward difference formula to find the polynomial

approximating f (x), given f (0) = 2, f(1) = 3, f (2) = 12, f (3) = 16.

16. Find f (7) using Newton - Gregory backward difference formula given that :

f (2) = 7, f (4) = 16, f (6) = 21, f (8) = 24, f (10) = 30, f (12) = 35.

17. Find f (32) using Gauss forward - central difference formula given that :

f (20) = 0.27, f (30) = 0.30, f (40) = 0.34, f (50) = 0.38.

18. Estimate f (½) using Gauss backward - central difference formula given that :

f (2) = 10, f (1) = 8, f (0) = 5, f (-1) = 10.

19. Apply Newton’s divided difference formula to find f (8) given that

f (4 ) = 48, f ( 5 ) = 100, f ( 7 ) = 294, f ( 10 ) = 900, f ( 11 ) = 1210.

20. Solve f (x) = 0 using Lagrange’s interpolation formula given that

f (30) = -30, f ( 34 ) = -13,f ( 38 ) = 3, f ( 42 ) = 18.

NUMERICAL DIFFERENTIATION

Find the first and the second derivatives of the function at the given point , using the given formula: (a) Newton’s forword interpolation at x = 1.5 for

x : 1.5 2.0 2.5 3.0 3.5 4.0 y : 3.375 7.0 13.625 24 38.875 59

(b) Newton’s backward at x = 1961 for Year : 1931 1941 1951 1961 1971 Population : 40.6 60.8 79.9 103.6 132.7

NUMERICAL INTEGRATION

Use Simpson’s one-third rule

π/2 1 1


a) ∫ dx ; 6 intervals b) ∫ e-x2 dx ; 6intervals c) ∫ sin x dx ; 6 intervals

0 √(1 + x3) -1 0

Use Simpson’s three - eighth rule

1.5 5.2 1 a) ∫ x3 dx ; 6 intervals b) ∫ log x dx ; c) ∫ dx ; 6 intervals 1 ex - 1 4 0 1 + x2

Use Weddle’s rule to evaluate 6 1.4 3

a) ∫ dx b) ∫ log x dx c) ∫ dx 0 1 + x2 0.2 0 4x + 5

UNIT-VII

NUMERICAL METHODS III:

NUMERICAL SOLUTION OF PDE

1. Solve the Laplace’s equation 0=+ yyxx uu in the following square mesh with boundary values as shown

In the figure

2.Solve 0=+ yyxx uu in the following square region with the boundary conditions as indicated in the

figure.


3. Solve the Laplace’s equation 0=+ yyxx uu for 0<x<1, 0<y<1 given that u(x,0)=u(0,y)=0, u(x,1)=6x,

10 ≤< x and u(1,y)=3y, 0<y<1. Divide the region into 9 square meshes.

4. Solve the elliptic p.d.e for the following square mesh using the 5 point difference formula and by setting

up the linear equations at the unknown points P,Q,R,S.

5.Solve the wave equation 2

2

2

2

4x

u

t

u

∂

∂=

∂

∂ subject to u(0,t)=0, u(4,t)=0, 0)0,( =xu t and u(x,0)=x(4-x) by

taking h=1, k=0.5 upto four steps.

6. Solve the wave equation 2

2

2

2

4x

u

t

u

∂

∂=

∂

∂ given u(0,t)=u(5,t)=0, 0≥t , u(x,0)=x(5-x), 0)0,( =

∂

∂x

t

u,

0<x<5.Find u at t=2 given h=1, k=0.5.

7. Solve 2

2

2

2

x

u

t

u

∂

∂=

∂

∂ given that u(x,0)=0, u(0,t)=0, 0)0,( =xu t and ttu πsin100),,1( = in the range 10 ≤≤ t

by taking 4

1=h .

8. Consider the heat equation 2

2

2x

u

t

u

∂

∂=

∂∂

under the following conditions

1)u(0,t)=u(4,t)=0, 0≥t

2)u(x,0)=x(4-x), 0<x<4.


Employ the Bendre_Schmidt method with h=1 to find the solution of the equation, for 10 ≤< t .

9. Use the Bendre_Schmidt method to solve the equation 2

2

x

u

t

u

∂

∂=

∂∂

under the conditions

1)u(0,t)=u(1,t)=0, 0≥t

2)u(x,0)=sin ,xπ 0<x<1 by taking 4

1=h and

6

1=α . Carry out 3 steps in the time-level.

10. Evaluate the pivotal values of the equation xxtt uu 16= taking h=1 upto t=1.25. The boundary conditions

are u(0,t)=u(5,t)=0, 0)0,( =xu i and u(x.,0)= 2x (5-x).

PART-A

UNIT-IV CURVE FITTING

1. Fit the straight line of the form y= a + bx to the given data

x: 0 5 10 15 20 25

y: 12 15 17 22 24 30

2. Fit a parabola cbxaxy ++= 2 to the following data.

x: 20 40 60 80 100 120

y: 5.5 9.1 14.9 22.8 33.3 46.0

3. Fit a curve of the form y=axb for the data

x: 1 2 3 4 5 6

y: 2.98 4.26 5.21 6.1 6.8 7.5

4. The following table gives the marks obtained by a student in two subjects in ten tests. Find the

coefficient of correlation.

Sub A : 77 54 27 52 14 35 90 25 56 60

Sub B: 35 58 60 40 50 40 35 56 34 42

5. Show that there is a perfect correlation between x & y .

x: 10 12 14 16 18 20

y: 20 25 30 35 40 45

6. A computer while calculating the correlation coefficient bet x & y from 25 pairs of observations got

the following constants n = 25, Σ x = 125, Σ x2 = 650, Σ y = 100, Σy

2 = 460& Σ xy = 508. Later it

was discovered it had copied down the pairs (8, 12) & (6, 8) as (6, 14) & (8, 6) respectively. Obtain the

correct value of the correlation coefficient.

7. If θ is the angle between two regression lines show that

22

x

yx2

r-1 tan

yσσ

σσ

rθ

+= and explain the significance when r = 0.

8. Find the lines of regression for the following data:

x: 1 2 3 4 5 6 7 8 9 10

y; 10 12 16 28 25 36 41 49 40 50

9. If the mean of x is 65, mean of y is 67, σx = 7. 5, σx = 3.5 & r = 0.8 find the value of x corresponding to

y= 75 & y corresponding to x = 70.

10. The two regression lines are x = 4y + 5 & 16y = x + 64 find the mean values of x, y & r.


11. In a partially destroyed laboratory record of correlation data only the following results are legible.

variance of y is 16, regression equations are y = x + 5, 16x = 9y - 94, find the variance of x.

12. Fit a straight line to the data:

(a) x: 0 1 2 3 4

y: 1 1.8 3.3 4.5 6.3

(b) x: 1 2 3 4 5

y: 14 13 9 5 2

13. Fit a second degree parabola of the form y = ax2 + bx + c for the data:

x: 1 2 3 4 5

y: 1.8 5.1 8.9 14.1 19.8 . Estimate y for x = 2.5.

14. Fit an exponential curve of the form y = abx, for the following data:

x: 1 2 3 4 5 6 7

y: 87 97 113 129 202 195 193. Estimate y for x = 8.

LINEAR PROGRAMMING

Formulation of mathematical problem

1. A manufacturer produces two types of models M1 and M2.Each M1 model requires 4 hours of grinding

and 2hours of polishing; whereas each M2 model requires 2 hours of grinding and 5 hours of polishing. The

manufacturer has 2 grinders and 3 polishers. Each grinder works for 40 hours a week and each polisher

works for 60 hours a week Profit on an

M1 model is Rs.3 and on M2 is Rs.4. whatever is produced in a week is sold in the market. How should the

manufacturer allocate his production capacity to the two types of models so that he may the maximum profit

in a week.

2. A firm making castings uses electric furnace to melt iron with the following specifications:

Minimum maximum

Carbon 3.20% 3.40%

Silicon 2.25% 2.35%

Specifications and costs of various raw materials used for this purpose are given below:

Material Carbon% silicon% cost (Rs.)

Steel scrap 0.4 0.15 850/tonne

Cast iron scrap 3.80 2.40 900/tone

Remelt from foundry 3.50 2.30 500/tonne

If the total charge of iron metal required is 4 tonnes, find the weight in kg of each raw material that must be

used in the optimal mix at minimum cost.

3. An aeroplane can carry a maximum of 200 passengers. A profit of Rs.400 is made on each first class

ticket and a profit of Rs.300 is made on each economy class ticket. The airline reserves atleast 20 seats for

first class. However, atleast 4 times as many passengers prefer to travel by economy class than by the first

class. How many tickets of each class must be sold in order to maximize profit for the airline? Formulate the

problem as an L.P.P. problem.

4. A firm produces an alloy with the following specification:

(i) Specific gravity ≤ 0.97; (ii) chromium content ≥ 15% (iii) melting temperature ≥ 494 0 C.

The alloy requires three raw materials A, B, C, whose properties area as follows:

Properties of raw material

Property A B C

Sp.gravity 0.94 1.00 1.05

Chromium 10% 15% 17%

Melting pt. 470 0 C 500 0 C 520 0 C

Find the values of A, B, and C to be used to make 1 tonne of alloy of desired properties, keeping the raw

material cost at the maximum when they are Rs.105/tonne for A,Rs.245/tonne for B and Rs.165/tonne

.Formulate an L.P.P model for the problem.

Problems on Graphical Method.


1. Solve the L.P.P. Graphically.

A manufacturer produces two types of models M1 and M2.Each M1 model requires 4 hours of grinding

and 2hours of polishing; whereas each M2 model requires 2 hours of grinding and 5 hours of polishing. The

manufacturer has 2 grinders and 3 polishers. Each grinder works for 40 hours a week and each polisher

works for 60 hours a week Profit on an

M1 model is Rs.3 and on M2 is Rs.4. whatever is produced in a week is sold in the market. How should the

manufacturer allocate his production capacity to the two types of models so that he may the maximum profit

in a week.

2. Find the maximum value of z=2x+3y subject to the condition:

X+y ≤ 30, y ≥ 3,0 ≤ y ≤ 12,x-y ≥ 0 and 0 ≤ x ≤ 20.

3. A company manufactures two types of cloth, using three different of wools. One yard length of type a

cloth requires 4 oz of red wool, 5 oz of green wool and 3oz of yellow wool. One yard of type B cloth

requires 5 oz of red wool, 2 oz of green wool and 8 oz of yellow wool .The wool available for manufacture

is 1000 0z of red wool, 1000 oz of green wool and 12oo oz of yellow wool. The manufacture can make a

profit of Rs.5 on the one yard of type A cloth and Rs3. on yard of type B cloth. Find the best combination of

the quantities of type and type B cloth which gives him maximum profit by solving the L.P.P. graphically.

4. A company making cold drinks has two bottling plants located at town T1 and T2.Each plant produces

three drinks A, B and C and their production capacity per day is shown below.

Cold drinks plant at

T1 T2

A 6000 2000

B 1000 2500

C 3000 3000

The marketing department of the company forecasts a demand of 80000 bottles of A, 22000 bottles of B and

40000 bottles of C during the month June .The operating costs per day of plants at T1 and T2 are Rs.6000

and Rs.4ooo respectively .Find the number of days for which each plant must be run in June so as to

minimize the operating costs while meeting the market demand.

5. A firm uses milling machines, grinding machines and lathes to produce two motor parts. The machining

times required for each part , the machining available on different machines and the profit on each motor

part are given below.

Type of machine Machining time reqd. for the Max time available

motor part (mts) per week(minutes)

I II

Milling machine 10 4 2000

Grinding machine 3 2 900

Lathes 6 12 3000

Profit/unit (Rs.) 100 40

6 .Using graphical method, solve the following L.P.P.

Maximize: Z=2x1+3x2

Subject to: x1-x2 ≤ 2

x1+x2 ≥ 4

x1,x2 ≥ 0

7. Solve the L.P.P. problem graphically.

Maximize: Z=4x1+3x2

Subject to: x1-x2 ≤ -1

-x1+x2 ≤ 0

x1,x2 ≥ 0

Problems on Simple Method 1. Convert the following L.P.P. to the standard form

(i) Maximize: Z=3x1+5x2+7x3


Subject to: 6 x1-4x2 ≤ 5

3x1+2x2+5x ≥ 11

4x1+3x3 ≤ 2

x1, x2 ≥ 0

(ii) Minimize: Z=3x1+4x2

Subject to: 2 x1-x2-3x3=-4

3x1+5x2+5x4=10

x1-4x2=12

x1, x3, x4 ≥ 0

2. Find the basic solutions of the following system of equations identifying in each case the basic and non

basic of the system.

2x1+x2+4x2=11, 3x1+x2+5x3=14.

Investigate whether the basic solutions are degenerate basic solutions or not. Hence find the basic feasible

Solutions of the system.

3. Find an optimal solution to the following L.P.P. by computing all basic solutions and then finding one that

maximizes the objective function:

2x1+3x2-x3+4x4=8, x1-2x2+6x3-7x4=-3 ,x1,x2,x3,x4 ≥ 0.

4. Obtain all the basic solutions of the following system.

x1+2x2+x3=4, 2x1+x2+5x3=5.

5. Using Simplex method

(i) Maximize: Z=5x1+3x2

Subject to: x1+x2 ≤ 2

5x1+2x2 ≤ 10

3x1+8x2 ≤ 12

x1, x2 ≥ 0

(ii) Minimize: Z=x1-3x2+3x3

Subject to: 3x1-x2+2x3 ≤ 7

2x1+4x2 ≥ -12

-4x1+3x2+8x3 ≤ 12

x1, x2,x3 ≥ 0

(iii) Maximize: Z=107x1+x2+2x3

Subject to: 14x1+x2-6x3+3x4 =7

16x1+(1/2)x2-6x3 ≤ 5

3x1-x2-x3 ≤ 0

x1, x2,x3,x4 ≥ 0

6. A firm produces three products which are processed on three machines. The relevant data is given

below.

Machine Time per unit (mins) Machine capacity

Product A Product B Product C (mins/day)

M1 2 3 2 440

M2 4 -- 3 470

M3 2 5 --- 430

The profit per units for products A, B, C is Rs.4 Rs.3 and Rs.6 respectively. Determine the daily number

of units to be manufactured for each product. Assume that all the units produced are consumed in the

market.

7. A company makes two types of products .Each product of the first type requires twice as much labour

time as the second type. If all products are of type only, the company can produce a total of 500 units a day,

The market limits daily sales of the first and the second type to 150 to 250 units respectively. Assuming that

the profits per units are Rs.8 for type I and Rs.5 for type II,determine the number of units of each type to be

produced to maximize profit.

UNIT-I FOURIER SERIES:

INFINITE SERIES


I Obtain the Fourier series expansion for the following functions in the given intervals:

1. e-x in ( -π , π ).

2. x2 in ( -π , π ) and hence deduce that π2 = 1 - 1 + 1- - - - - - - -

12 12 22 32 3. x - 1 in ( -π , π ).

4. x in ( -1, +1 ).

5. x - x2 in ( -1, +1 ).

6. π 2 - x2 in ( -π , π ).

7. x( 2π - x) in ( 0, 2π ).

8. 2x - x2 in ( 0, 3 ).

9. x in ( -π , π ). and hence deduce the value of π2/8

10. x sinx in ( 0, 2π ) and hence deduce that π = 1 +2 [ 1 - 1 + 1_ - + - - - - - - ]

2 3 3.5 5.7

11. x2 in ( 0, 2π ) and hence deduce that

π2 = 1 + 1 + 1 + . . . . . . . . . and π2 = 1 - 1 + 1 - + . . . . . . . .. . ..

6 12 22 32 12 12 22 32

12. x3 in ( -π, π ).

13. 1 - x + x2 in (-π, π ).

14 ( x - π )2 in ( 0, 2π ).

15. e-ax in ( -π, π ). and hence obtain the series for __π__

sinhx 1 + x , -π ≤ x ≤ 0

16. f (x) = 1 - x , 0 ≤ x ≤ π

π, -π ≤ x ≤ 0

17. f (x ) = x , 0 ≤ x ≤ π

-k , - π ≤ x ≤ 0

18. f(x) = k, 0 ≤ x ≤ π

x , 0 ≤ x ≤ 1

19. f(x) = x - 21, 1 ≤ x ≤ 21

x2 , 0 ≤ x ≤ π 20. f(x) =

-( 2π - x)2 π ≤ x ≤ 2π πx, 0 ≤ x ≤ 1

21. f(x) = π ( 2 - x ) 1 ≤ x ≤ 2

x, 0 ≤ x ≤ π 22. f(x) = 2π - x, π ≤ x ≤ 2π

Hence deduce that π2 = 1 + 1 + 1 + - - - - - -

8 12 32 52


2, -2 ≤ x ≤ 0

23. f(x) = x, 0 ≤ x ≤ 2

24. . Find the half - range cosine and half - range sine series for the following. ( a ). f (x) = x ( π - x ) in ( 0, π ). ( b ) f (x ) = x2 in ( 0, π ).

2k x, ( 0, 1/2 ) (c ) f (x ) =

2k (1 - x ), ( 1/2, 1 )

25.. Find the half range cosine series for the following: (a) f (x ) = x in ( 0, 2 )

(b) f (x ) = (x - 1 )2 in ( 0, 1 )

x, 0 ≤ x ≤ 1

(c) f (x) =

2 - x, 1 ≤ x ≤ 2

cosx , 0 ≤ x ≤ ( π/2 )

(d) f (x) =

0, ( π/2 ) ≤ x ≤ π 26. . Find the half - range sine series for the following: (a) f (x) = x3 in ( 0, 2 ).

(b) f (x) = 1+x2 in ( 0, π ).

(1/4) - x, in ( 0, 1/2 )

(c) f (x) =

x - (3/4), in ( 1/2, 1 )

cosx, 0 ≤ x ≤ ( π/4 )

(d) f (x) = sinx, (π/4) ≤ x ≤ ( π/2 ) 27. Obtain the complex Fourier series expansion of e-x in (-1, 1).

28. Find the complex Fourier series for cos ax in (- π, π).

29. Expand y in terms of Fourier series using the table below.

X: 0 π /6 π/3 π/2 2π/3 5π /6

Y: 0 9.2 14.4 17.8 17.3 11.7

30. Determine the Fourier expansion , upto third harmonics, for the function f(θ) defined by the

following table.

θ0 30 60 90 120 150 180 210 240 270 300 330 360

F(θ) 0.6 0.83 1.0 0.8 0.42 0 -0.34

-0.5 -0.2 0.67 0.7 0.5

31. The turning moment T on the crank-shaft of a steam engine for crank angle degrees is given

below:


θ: 0 15 30 45 60 75 90 105 120 135 150 165

T: 0 2.7 5.2 7.0 8.1 8.3 7.9 6.8 5.5 4.1 2.6 1.2

Express T in a series of sine up to second harmonics.

UNIT-II

FOURIER TRANSFORMS :

I Define the complex Fourier transform of a function f(x) and give the inversion

formula. Use ‘α ’ as the parameter .

II Find the complex F.T. of the functions defined as follows: eikx, a<x<b 1. f(x) =

0, otherwise

1 - x2, x < 1

2. f(x) =

0, otherwise 1 - x , x < a

3. f(x) = Hence show that sin2 t dt = π 0 , x >a>0 t2 2

0, x<p 4. Show that the F.T. of f(x) = 1, p<x<q is (1/√2π) eiqα - eipα

0, x>q iα

√2π , x < a

5. f(x) = 2a

0, otherwise 6. f(x) = x e- a x , ‘a’ is a positive constant

(a2 - x2)-1/2 , x < a

7. Show that F.T. of f(x) = is √π/2 J0(aα)

0, otherwise

8. Show that the F.T. of the Dirac delta function F{δ (x - a) } is 1/√2π eiαa

9. (i) f(x) = cosax2 and (ii) f(x) = sinax2 . Use the results

∫∫∞

∞−

∞

∞−

== 2

πdusinu ducosu 22

Show that the function e-x /2 is self reciprocal with respect to the complex F.T.

by finding the F.T. of e-a x , a>0 10. Find the inverse complex F.T. of (i) sin(aα) (ii) e-a α , a>0

α

11. Find the F.T. of (i) x sin 4x2 (ii)x2e-4x (iii) The Heaviside Unit Step function H(x - a) using δ(x - a) = H(x - a)

12. State and prove the Convolution Theorem for the complex F.T. of two functions f(x) and g(x).

13. Verify the Convolution Theorem for the functions 1, x < 1

(i) f(x) = e-x= g(x) (ii) f(x) = g(x) =

0, x >1

14. Find the inverse F.T. of 1 using the Convolution theorem (1 + α2)2

15. State and prove Parseval’s identity for Complex Fourier Transforms.


16. Using Parseval’s identity show that

(i) ( ) ( )∫

∞

>

<−==

−

0

4

2

1 |x| 0,

1 |x| |,|1 xffor

6

cos1 xdx

x

x π

(ii) ( ) ( )∫

∞

>

<−==

−

0

2

6

2

1 |x| 0,

1 |x| ,1 xf for

15

sincos xdx

x

xxx π

17. Define Fc(α) and Fs(α), the Fourier cosine and sine transforms, respectively,

of a function f(x).

18. Find the Fourier Cosine and Fourier sine transforms of e-ax, a>0. Hence deduce the

inversion formulae.

19. Find the Fourier Cosine transforms of :

(i) e-x (ii) 1 (iii) x2e-ax (iv) e-ax (v) cosx, 0<x<a

1 + x2 x f(x) = 0, x ≥ a 20. Find the Fourier Sine transforms of :

(i) e-x (ii) x (iii) xe-ax (iv) e-ax (v) sinx, 0<x<a

1 + x2 x f(x) = 0, x ≥ a

1, 0<x<a

21. Find Fc(α) and Fs(α) of f(x) = 0, x ≥a

22. Find the F.S.T. of e- x and hence deduce that ∫∞

−=+

0

m

2e

2

πdxsinmx

x1

x

28. Find f(x) if its sine transform is e-aα . Hence find the inverse sine transform of 1

α α

29. Find the Fourier Sine and Cosine transforms of xm-1, 0<m<1. Hence find the same for (i) √x & (ii) 1/√x. 30. State and prove Parseval’s identity for Fourier Sine and Cosine transforms

PART-B

UINIT-VIII

Z TRANSFORMS

I. Define a linear Difference Equations.

II. Solve the following difference Equation. (i) un+2-2un+1+un=0.

(ii) yn+1-2yncosα+yn-1=0. (iii) un+2-6un+1+9un=0. (iv) uk+3-3uk+2+4uk=0.

(v) un+1-2un+2un-1=0. (vi) 4yn-yn+2=0.Given that y0=0,y1=2.

III. Define Z Transform of a function un and give the inversion.

IV. (i) Find the Z transform of Z(an).

(ii) Show that Z(np)= -Zdz

d Z(np-1),p being a +ve integer.

(iii) Show that Z(aun+bvn-cwn)=aZ(un)+bZ(vn)-cZ(wn).

(iv) Find the Z Transform of Z(nan).

(v) Show that Z(n2an)=3

22

)( aZ

ZaaZ

−

+.


(vi) Show that Z(cosnθ)=1cos2

)cos(2 +−

−

θθ

ZZ

ZZ

(vii) Show that Z(sinnθ)=1cos2

sin2 +− θ

θZZ

Z.

(viii) Find the Z transform of Z(ancosnθ). (ix) Find the Z Transform of Z(ansinnθ). (x) Find the Z Transform of (n+1)2.

V. Find the Z Transform of the following. (i) ean (ii) nean (iii) n2ean (iv) ancoshnθ

(v) etsin2t (vi) cos

+42

ππn

(vi) nCp(0≤p≤n). (vii) n+pCp.

VI. (i) If Z(un)=U(Z) then Z(un-k)= Z-kU(Z). (ii) If Z(un)=U(Z) then Z(un+k) =Zk[U(Z)-u0-u1Z

-1-u2Z-2-------uk-1Z-(k-1)]

(iii) If Z(un)=U(Z) then Z(nun) = -ZdZ

ZdU )(.

(iV) Show that Z(1/n!)=e1/z Hence Evaluate (a) Z{1/(n+1)!] (b) Z[1/(n+2)!]

VII Find the Z Transform of the following.

(i) nsinnθ (ii) n2enθ.

VIII (i) If Z(un)=U(Z) then u0 = lt U(Z). Z→∞

(ii) If Z(un)=U(Z) then lt (un) lt (Z-1)(Z-2). Z→∞ Z→1

(iii) If u(Z)= 4

2

)1(

1452

−

++

Z

ZZ Evaluate u2 and u3.

(iv) ) If u(Z)= 4

2

)1(

1232

−

++

Z

ZZEvaluate u2 and u3.

IX. Find the inverse Transform of (i) By Power Series Method

(a) log)1( +Z

Z (b)

2)1( +Z

Z

(ii) By the Partial functions.

(a) )4)(2(

32 2

−++ZZ

ZZ

(b) Evaluate Z-1

−+ )3)(2(

1

ZZ ,(i)│Z│<2 (ii) 2<│Z│<3 (iii) │Z│>3

(iii) By inverse integral method

(a) )2)(1(

10

−− ZZ

Z (b)

)2)(1(

3

−++

zZ

Z

X. Using Z Transforms Solve;


(i) un+2+4un+1+3un=3n. with u0=0,u1=1. (ii) yn+2+6yn+1+9yn=2n with y0=y1=0.

(iii) un+2-2un+1+un=3n+5. (iv) yn+2-4un=0. y0=0,y1=2 (v) yn+1+1/4yn-1=un+1/3un-1 where un is a unit step sequence.

PART-A

UNIT-III

PARTIAL DIFFERENTIAL EQUATIONS

VIII (a) Solve completely the equation ∂2y/∂t2 = c2 ∂2y/∂x2, representing the vibrations of

a string of length l, fixed at both the ends, given that y(0,t) =0; y (x,0) = f(x) and ∂y (x,0) / ∂t =0, 0<x<l.

(b) A tightly stretched string with fixed end points x = 0 and x = l is initially in a

position given by y = yo Sin3 (πx / l). If it is released from rest from this

position, find the displacement y(x,t). (c) A homogeneous rod of conducting material of length 100 cm has its ends kept at zero temperature and the temperature initially is u(x,0) = x , 0 ≤ x ≤ 50

100-x , 50 ≤ x ≤ 100.

Find the temperature u (x,t) at any time.

(d) Solve ut = 4uxx subject to the conditions, u(0,t) = 0 for all t; u(1,t) = 0 ;

u(x,0) = x-x2 (0 ≤ x ≤ 1)

(e) A string is stretched and fastened to two points at a distance ‘l’ apart. Motion is

started by displacing the string into the form, y = k( lx - x2 ) from which it is released at time t = 0 Find the displacement of any point on the string at a distance x from one end at time t.

dudu

dv)k(u1

U2

U1

2

22

∫

++

IX. a) Find the D’Alemberts solution of the wave equation

b) Find the deflection of the vibration string of the unt length having fixed ends with initial velocity zero and initial deflection f(x) =k(sinx-sin2x) c) A tightly streached string with fixed end points x=0 and x=l is initially in a position given

by y=

l

xy

π3

0 sin

.If it is released from rest from this position,find the displacement y(x,t). d) Using D’Alemberts method find the deflection of a vibrating string of unit length having

fixed ends with initial velocity zero and initial deflection

1)f(x)=ax(1-x2) 2)f(x)=asin xπ2

e) Find the solution of Laplace equation. f) An infinitely long plane uniform plate is bounded by two parallel edges and an end at right angles to them. The breadth

is π ; this end is maintained at an temperature u 0at all points and other edges are at zero

temperature. Determine the

temperature at any point of the plate in the steady state. g) A long rectangular plate of width a cms with insulated surface as its temperature v=0 on both the long sides and one of

the short sides so that v(0,y)= 0, v(a,y)=0, v(x, ∞ )=0, v(x,0)=kx. Show that the steady

state temperature within the plate is

V(x,y)=a

xne

n

aka

yn

n

n ππ

π

sin)1(2

1

1 −∞

=

+

∑−


h) Solve 02

2

2

2

=∂

∂+

∂

∂

y

u

x

u subject to the condition u(0,y)=u(l,y)=u(x,0)=0 and u(x,a)=sin(

l

xnπ)

i) Solve 02

2

2

2

=∂

∂+

∂

∂

y

u

x

u for 0<x<π , 0<y<π subject to the condition u(0,y)=u(π ,y)=u(x, π )=0

and u(x,0)=sin2x

Question Paper


]





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