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PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 1
10MAT31: ENGINEERING MATHEMATICS – III
FacultyName:Mrs.Jagadeeswari/Mr.Girish.V.R./Mr.Nagesh.H.M/Mr.Ravikumar/Mrs.Sivashankari/Mrs.James Alex No of Units: 52
Class # Chapter Title Topics to be covered
% of portions
covered
Cumulative
1 UNIT-I
FOURIER
SERIES
Introduction, periodic, even and odd
functions
12.5
2,3,4 Problems
5 Half range series
6 Problems
7 Practical Harmonic analysis – problems
8,9 UNIT-V
NUMERICAL
METHODS –I
Solution of algebraic and trasendental
method
Regula-falsimehtod
25
10 Newtonraphson method
11 Gauss- siedel method
12 Relaxation methods
13 Power method
14
UNIT-IV
CURVE
FITTING AND
OPTIMIZATIO
N
Curve fitting by the method of Least squares-
Fitting of the straight line y=ax+b
cbxaxy ++= 2
37.5
15,16 bbxaxyaey == ,
17,18 Optimization:LPP,mathematical formulation and
Graphical method
19,20 Simplex method
21
UNIT-VI
NUMERICAL
METHODS II
Finite differences - forward, backward
50 22,23 Interpolation & extrapolation - Gregory
Newton formulae – problems
24 Lagrange's formula for unequal intervals –
inverse interpolation – problems
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 2
25 Newton's divided difference formula – problems
26 Numerical Integration –Simpsons 1/3rd rule
27 Simpson's 3/8 rule – problems,Weddle's rule –
problems
28,29 UNIT-II
Fourier
Transforms
Fourier transforms- problems on infinite
transforms
62.5
30 Inverse Fourier transforms – problems
31 Fourier sine & cosine transforms - their
inverses
32 Problems
33 Properties
34,35 UNIT-VII
NUMERICAL
METHODS III
Numerical solution of pde-finite difference
approximation
75
36,37 Numerical solution of 2D Laplace equation
38 One dimensional heat equation
39,40 One dimensional wave equation
41 UNIT-VIII
Z
TRANSFORMS
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
Various possible solution of 1D wave and heat
equations
100
49 Solution of two dimentionalLaplace 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 Title & Author Publication Information
Edition Publisher Year
Text Books:
“Higher Engineering
Mathematics”
Dr. B.S. Grewal
42th
Edition Khanna 2012
Reference
Books:
“Advanced Engineering
Mathematics”
E Kreysizig
9th
Edition.
John Wiley &
Sons 2014
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 3
PART-B
Unit-IV
NUMERICAL METHODSI:
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 METHODSII:
FINITE DIFFERENCES AND INTERPOLATION FORMULAE.
1. Write the difference table for the following set of values.
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 4
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
41 )iv(
22 δ
+≡µ
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 5
9. Estimate the values of (i) sin 380 and (ii) sin 220 from the table given below.
x(degrees): 0 10 20 30 40
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
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 6
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 40 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 METHODSIII:
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.
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 7
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,( =
∂∂
xt
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 .
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 8
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 ++= 2to 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&Σxy = 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.
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 9
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
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 10
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
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 11
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
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 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
1. Find the nature of the series .......................9.7.5
3
7.5.3
2
5.3.1
1+++
2. Find the nature of the series ...................321
321
21
21
1
1222222
+++
+++
+
++
3. Test for convergence ∑∞
= ++1 1
1
n nn
4. Test for convergence ( )∑∞
=
−+1
2 1n
nn
5. Test for convergence the series ...................2
3
2
2
2
13
2
2
2
+++
6. Test for convergence the series ...................!3
3
!2
21
22
+++
7. Test for convergence the series ...................10.7.4
9.6.3
7.4
6.3
4
3+++
8. Discuss the nature of the series ...................4.33.22.1
32
+++xxx
9. Find the nature of the series ( )∑∞
=1
2
n
nnxa , a<1
10. Discuss the convergence of
2
1
31
n
n n∑
∞
=
−
PESIT SOUTHCAMPUS
P E S INSTITUE OF TECHNOLOGY (BSC)– Education for the Real World – Course Information – BE III-Sem 10MAT31 - 13
11. Find the nature of the series ...................5
4
4
3
3
21 3
3
2
2
+
+
++ xxx
12. Find the nature ...................6
5
5
4
4
3 3
3
2
2
+
+
+ xxx
13. Examine the nature of the series ...................32194
+++eee
14. Apply Cauchy’s integral test to discuss the nature of the harmonic series ∑∞
=
1
1
np
n
15. Find the nature of the series ∑∞
=
2 )(log
1
npnn
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. π2 - 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
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x , 0 ≤ x ≤ 1
19. f(x) = x - 21, 1 ≤ x ≤ 21
x2 , 0 ≤ x ≤π
20. f(x) = -( 2π - x)2π≤ 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 )
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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 tdt = 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
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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/√2πeiαa
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
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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) IfZ(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)!]
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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) thenlt (un) lt (Z-1)(Z-2). Z→∞ Z→1
(iii) Ifu(Z)= 4
2
)1(
1452
−
++
Z
ZZEvaluate 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(
322
−++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).
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(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 0 at 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
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Question Paper
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