Central University of Rajasthan

End- term Exam -2010

Numerical Solution of Differential Equations

Paper-301

Max. Time allowed-Three hour Max. Mark-100

Attempt all questions

1. (a) What do you understand by the stability of a single step numerical method for the

solution to differential equations? (3)

(b) Discuss the convergence of Euler method for the solution to ODE’s. (2)

(c) Write the name of two multistep numerical methods for ODE’s. (2)

(d) Compare Euler and modified Euler method for the solution to ODE’s (3)

(e) Form PDE by eliminating the arbitrary function f from

U = y2 + 2 f (1/x + log y ) (3)

(f) Solve the PDE p tan x + q tan y = tan u (3)

(g) Write Charpit’s auxiliary equation for the solution of partial differential equation

f ( x, y, z, p, q ) = 0 (2)

(h) Write Schmidt method for solving the PDE Tt = α Txx (2)

(i) Write the name of a implicit method for solving the parabolic partial differential

equation, write the difference scheme for this method. (3)

(j) Write the name of any two methods for solving the BVP’s. (2)

2. (a) Solve the initial value problem y’ = -2 x y2 , y(0) = 1

with h = 0.2 on the interval[0 , 0.4] , use the fourth order classical R-K method.

(13)

(b) Find y (2.0 ) if y(x) is the solution of y’ = 0.5 ( x + y ) assuming y(0) = 2, y(0.5) = 2.636,

y(1.0) = 3.595 and y(1.5) = 4.68 using Milne’s P-C method. (12)

OR

(a) Solve the initial value problem y’ = -2 x y2, y(0) with h = 0.2 on the interval [0 , 0.4]

using P-C method . (13)

(b) Using modified Euler method to obtain the solution of the differential equation

y’ = x + √ y with the initial condition y(0) = 1 for the range 0 ≤ x ≤ 0.4 in steps of 0.2 (12)

3. (a) Find the complete integral of the equation P2 x + q2 y = z (12)

(b) Solve the PDE 2 s + ( r t – s2 ) = 1 (13)

OR

(a) Solve one dimensional diffusion equation TXX = (1/k) Tt using separation of variable

method. (12)

(b) Solve the PDE r + 5 s + 6 t = 0 (13)

4. (a) Discuss Crank Nicolson method for solving the parabolic partial differential

equation Tt = α Txx (12)

(b) Solve the Laplace equation uXX + uyy = 0 by employing five point formula , which

satisfy the following Dirichlet boundary conditions u (0, y) = 0 , u (x, 0) = 0, u (x, 1) = 100 x,

u (1, y) = 100 y consider ∆x = ∆y = h = 0.25 (13)

OR

(a) The function u satisfies the wave equation utt = uxx, 0 ≤ x ≤ 1. The boundary

conditions are u = 0 at x = 0 and x = 1 for t > 0 , and the initial conditions are u (x, 0) = sin πx/8,

ut (x, 0) = 0 compute the solution for x = 0 (0.1) 0.3 and t = 0 (0.1) 0.3 (13)

(b) Use Galerkin method for solving the following BVP described by the differential

equation D2u + u + x = 0, 0 < x < 1. Subject to the boundary conditions u (0) = 0, (du/dx)x=1 = 0

(12)

CENTRAL UNIVERSITY OF RAJASTHAN

END SEMESTER EXAMINATION

Third semester MTM 302 (Topology)

Date: 8-12-2010

Time: 3 hours Max marks: 100 Note : Unit 1 is compulsory. Attempt one question from each of the remaining units.

UNIT-1

1a) Let X = {a,b,c,d}. Construct a non-discrete topological space 𝑋, 𝜏 , having more

than 8 elements. (3)

b) Define interior Eo of a set E. Show that for any two subsets 𝐴 and 𝐵 of a topological

space X, (𝐴 ∩ 𝐵)° = 𝐴° ∩ 𝐵° . (3)

c) Let 𝑋 = {𝑎, 𝑏, 𝑐, 𝑑}. Construct a topological space 𝑋, 𝜏 for which

𝛽 = {∅, 𝑎 , 𝑏 , 𝑐 , {𝑑}} is a base. (3)

d) Define separated sets and the Hausdorff Lennes separation condition. Show that two

sets A and B are separated sets if and only if they satisfy the Hausdorff Lennes

separation condition. (3)

e) Give the statemants of Urysohn’s Lemma and Tietz Extension theorem. (3)

f) Let N be the set of all positive integers and 𝜏 be the family consisting of ∅, N

and all subsets of the form En = {1,2,…,n}. Check whether 𝜏 is a (i) To space

(ii) T1 space. (4)

g) Define connected and locally connected sets. Prove or disprove:

(i) Every locally connected space is connected.

(ii) Every connected space is locally connected. (6)

UNIT-2

2a) For any set 𝐸, define the derived set 𝑑(𝐸). If 𝐴, 𝐵, 𝐸 are subsets of a topological

space 𝑋, 𝜏 , then prove that the derived set has the following properties.

(i) 𝑑 𝜙 = 𝜙. (ii) If 𝐴 ⊂ 𝐵 then 𝑑 𝐴 ⊂ 𝑑(𝐵)

(iii) If 𝑥 ∈ 𝑑 𝐸 , then 𝑥 ∈ 𝑑(𝐸 ∖ 𝑥 ) (iv) 𝑑 𝐴 ∪ 𝐵 = 𝑑 𝐴 ∪ 𝑑(𝐵). (8)

b) Prove that a family B of sets is a base for a topology for the set X = ∪ { 𝐵: 𝐵 ∈ B } if

and only if for every B1, B2 ∈ B , and every 𝑥 ∈ B1 ∩ B2, there exists a B ∈ B such

that x ∈ B ⊂ B1 ∩ B2 . (8)

c) Define a closure operator on set X. Let c* be a closure operator on a set X. Let

F = { F ⊂ X | c*(F) = F }, and 𝜏= { G ⊂ X | G ′ ∈ F }. Show that 𝜏 is a topology

for X. Also show that if c is a closure operator defined by the topology 𝜏, then

c*(E) = c(E). (9)

OR

3a) Define the closure Ā of a set A. Prove that for any set A in a topological space ( X, 𝜏),

Ā = A ∪ d(A), where d(A) denotes the derived set of A. (8)

b) State the four Kuratowski closure axioms. Show that these axioms can be replaced by

A ∪ c(A) ∪ c(c(B)) = c( A ∪ B ) ∖ c ( ∅ ) for all A, B ⊂ X. (8)

c) Define an induced topological space X* of a topological space X. Show that X* is

indeed a topological space. Also show that if E is a subset of a subspace ( X*, 𝜏 ∗)

of a topological space ( X, 𝜏), then show that c*(E) = X* ∩ c(E). (9)

UNIT-3

4a) If C is a connected subset of a topological space ( X, 𝜏) which has a separation X = A | B,

then prove that either C ⊂ A or C ⊂ B. Deduce that if C is connected then c(C) is also

connected, where c(C) denotes closure of C. Also show that if two points of a set E are

contained in a connected subset of E, then E is connected. (8)

b) What is finite intersection property? Prove that a topological space ( X, 𝜏) is compact if

and only if any family of closed sets having finite intersection property has a non-empty

intersection. (8)

c) State and prove Alexandröff one point compactification. (9)

OR

5a) Define a subspace of a topological space. Let ( X*, 𝜏 ∗) be a subspace of a topological

space ( X, 𝜏). Let E ⊂X*. Show that E is 𝜏 ∗ compact if and only if E is 𝜏 compact. (8)

b) Let 𝑓: 𝑋 → 𝑋* be continuous map. Show that 𝑓 maps every arcwise connected subset

of 𝑋 onto an arcwise connected subset of 𝑋*. Also show that an open connected subset

of the plane is arcwise connected. (8)

c) Let 𝑓: 𝑋, 𝜏 → (𝑋*, 𝜏*) . Prove that the following conditions are equivalent.

(i) 𝑓 is continuous on 𝑋

(ii) the inverse of every open set in 𝑋* is open set in 𝑋

(iii) the inverse of every closed set in 𝑋* is closed set in 𝑋

(iv) 𝑓 𝑐 𝐸 ⊂ 𝑐* 𝑓 𝐸 for every 𝐸 ⊂ 𝑋. (9)

UNIT- 4

8a) Prove that a topological space is normal if and only if for any closed set F and an open

set G containing F there exists an open set G* such that F ⊂ G* and c(G*) ⊂ G. (8)

b) Define Fort’s space. Prove that Fort’s space is a compact, non-first axiom , Hausdorff

space. (8)

c) Prove that a T1 space is countably compact if and only if every open covering of X is

reducible to a finite subcover. (9)

OR

9a) Prove that a topological space X satisfying the first axiom of countability is a Hausdorff

space if and only if every convergent sequence has a unique limit. (8)

b) If an open set G has a non empty intersection with a connected set C in a T4 space X,

then show that either C consists of only one point or the set C ∩ G has cardinality

greater than or equal to the cardinality of the real number system. (8)

c) Define Appert’s space. Show that Appert’s space is a non first axiom Hausdorff

topological space. (9)

CENTRAL UNIVERSITY OF RAJASTHAN

End- term Exam -2010

Advance Numerical Methods

Paper MTM-304

Max. Time allowed-Three hour Max. Mark-100

Attempt all questions

1. (a) Define semi linear and quasilinear second order partial differential equations and

give one example of each. (4)

(b) Write the formula of a explicit and an implicit method for solving one dimensional

parabolic partial differential equation. (4)

(c) Define Discrete Fourier Transform. (2)

(d) Find the DFT of g = (1 , i , i2 , i3 ) = (1 , i ,-1 , -i ).Using frequency shift property.

(4)

(e) For f , g Є L1 (ZN) , then show that D(f*g)(n) is equal to D f (n) D g(n) for all n. (3)

(f) What do you understand about Buneman’s Algorithm? (4)

(g) What do understand by finite element? Explain line segment element. (4)

2. (a) Solve the differential equation

Tt = Txx , 0 ≤ x ≤ 0.5

Given that T = 0 when t = 0 , 0 ≤ x ≤ 0.5 , and the boundary conditions

TX = 0 at x = 0 and Tx = 1 at x = 0.5 for t >0

taking ∆x = 0.1 ∆t = 0.001 . Give solution for one time step only. (15)

(b) Explain Durfort – Frankel method for one dimensional diffusion equation. (10)

OR

(a) What do understand by weighted average implicit method for heat conduction equation,

explain it briefly. (10)

(b) Find the solution to the initial boundary value problem describe by the wave equation

utt = uxx , 0 ≤ x ≤ 1

subject to the boundary conditions

u (0 , t) = 0 = u(1 , t) , t > 0

and the initial conditions

u (x , 0) = sin (∏x) and ut(x , 0) = 0 , 0 ≤ x ≤ 1

using the explicit finite difference scheme. Assume ∆x = 1/8 , ∆t = 1/8 .

Compute u for two time levels. (15)

3. (a) Prove the Parseval’s Identities ,let f, g Є L2(ZN) . Then :

(i) ∑K=0N-1 f (k) g(k) = (1/N) ∑K=0

N-1D f (k) Dg(k)

(ii) ∑K=0N-1 │f (k) │2 = (1/N) ∑K=0N-1│D f (k) │2 (12)

(b) If f Є C8 calculate Df using Dfe and Dfo (13)

OR

(a) If f Є CN where N = 2k , prove that the number of multiplication required to compute Df

is 2N log2 N = 2k+1 .k (15)

(b) If f Є C4 calculate Df using Dfe and Dfo (10)

4. Solve by finite element method the boundary value problem

u’’ + (1+x2) u + 1 = 0, u (±1) = 0 (25)

OR

Use the finite element Garlerkin method to derive the difference schemes for the boundary

value problem

u’’ – K u’ = 0

u (0) = 1 , u (1) = 0

Where K > 0 is assumed constant.

Obtain the characteristic equation of the difference scheme .

The exact solution of the boundary value problem is

u = (eKX- eK)/ (1-eK ) (25)

CENTRAL UNIVERSITY OF RAJASTHANEnd Semester Examination (December 2010)

Third Semester M. Sc. Tech. Mathematics 2009-10Subject: MTM 304(Integral Transforms and Integral Equations)

Time Allowed: Three Hour Maximum Marks: 100

Note: Attempt all questions.

PART-I

Q.1 Attempt all. Each question having 3 marks.(a) Show that

∫ xa g(t) dt2 =

∫ xa (x− t)g(t) dt.

(b) For what value of λ, the function g(x) = 1 + λx is solution of the integral equation.

−x =∫ x

0ex−tg(t) dt

(c) Define complex Hilbert space.(d) Show that the homogeneous integral equation

φ(x) = λ

∫ 1

0(3x− 2) t φ(t) dt = 0

has no eigenvalues and eigenfunctions.(e) Show that

M[sin x; p] = γ(p) sin(πp

2

), 0 < <(p) < 1.

(f) Show that

H1

{e−ax

x; p

}=

1p− a

p√

a2 + p2.

(g) EvaluateH0

{cos ax

x; p

}.

(h) Evaluate

H1

{df

dx; 1

}where f(x) =

e−ax

x.

PART-II

Q.2 (a) Reduce the following Boundary value problem into an integral equation

d2u

dx2+ λu = 0 with u(0) = 0, u(l) = 0. (8)

(b) Prove that the eigenvalues of symmetric kernel are real. (9)(c) Solve the non-homogeneous Fredholm integral equations of second kind, by the method of

successive approximation to the third order

g(x) = 2x + λ

∫ 1

0(x + t) g(t) dt, g0(x) = 1 (9)

ORQ.3 (a) Prove that the eigenfunctions of a symmetric kernel, corresponding to different eigenvalues,

1

2

are orthogonal. (8)(b) Solve the following Fredholm integral equation of second kind

g(s) = s + λ

∫ 1

0(st2 + s2t) g(t) dt (9)

(c) Show that the integral equation

g(x) = f(x) +1π

∫ π

0sin(x + t) g(t) dt (9)

possesses no solution for f(x) = x, but having infinitely many solution for f(x) = 1.

PART-III

Q.4 (a) LetY (x) be generated from a continuous function g(x) by the operator

λ

∫ b

aK(x, t) g(t) dt,

where K(x, t) is continuous real and symmetric, so that

Y (x) = λ

∫ b

aK(x, t) g(t) dt,

then prove that Y (x) can be represented over interval (a, b) by a linear combination of thenormalized eigen function of homogeneous integral equation

g(x) = λ

∫ b

aK(x, t) g(t) dt. (15)

(b) Solve the Volterra’s integral equation

g(x) = 29 + 6x +∫ x

0[5− 6(x− t)] g(t) dt. (10)

ORQ.5 (a) Solve the following symmetrical integral equation with the help of Hilbert-Schmidt theorem

g(x) = 1 + λ

∫ π

0cos(x + t) g(t) dt. (15)

(b) Find the Neumann series for the solution of the following integral equation

g(x) = 1 + x + λ

∫ x

0(x + t) g(t) dt. (10)

PART-IV

Q. 6 (a) State and prove the Parseval Theorem of Hankel Transform and use to evaluate

Hν

{Jν(ax)

x2; p

}. (13)

3

(b) If Hn{f(r)} = fn(k). Then prove that

Hn

{1r

d

dr

(r

df

dr

)− n2

r2f(r)

}= −k2 fn(k) (12)

provided both rf ′(r) and rf(r) vanish as r → 0 and r →∞.

OR

Q. 7 (a) If Hn{f(r)} = fn(k), then prove that

Hn{f ′(r)} =k

2n

[(n− 1) fn+1(k)− (n + 1) fn−1(k)

], n ≥ 1,

H1{f ′(r)} = −kf0(k), (12)

provided [rf(r)] vanishes r → 0 and r →∞.

(b) Find the potential φ(r, θ) that satisfies the Laplace equation

r2φrr + rφr + φθθ = 0

in an infinite wedge 0 < r <∞, −α < θ < α, with the boundary conditions

φ(r, α) = f(r), φ(r,−α) = g(r), 0 ≤ r <∞,

φ(r, θ)→ 0 as r →∞ for all θ in − α < θ < α. (13)

1

CENTRAL UNIVERSITY OF RAJASTHAN, KISHANGARH Department of Computer Science & Engineering

End Semester Examination-2010

Programme: M.Sc. Tech Mathematics III Semester

Subject: MTM 306: Computer Networks

MAX MARKS: 100 DATE: 22/12/2010 TIME: 3hrs

Section A: Note-: Attempt all questions.

1. What is the difference between bit-rate and baud rate?An analog signal carries 4 bits in each signal unit. If 1000 signal units are sent per second, find the baud rate and the bit rate. [3]

2. Identify the five components of data communication system.[2] 3. How performance of network is measured on the basis of Packet delay and

throughput as function of load. [4] 4. Draw a hybrid topology with a star backbone and four ring networks. [3] 5. List and brief various transmission impairments. [3] 6. Differentiate between circuit switching and packet switching. [2] 7. Give some examples of generic and new generic domains. What is the port

number used by DNS? [4] 8. Explain the concept of Netid and Hosted with an example. [2] 9. What is the purpose of a firewall in a network? [2]

Section B:

UNIT-I (Attempt any one group)

1. For n devices in a network, what is the number of cable links required for a mesh,

ring, bus and star topology? [4]

2. Draw polar line encoding schemes for 011001110. [5]

3. What are internetwork connecting devices? Discuss the properties of each. [1+3]

4. Classify multiplexing and explain TDM in brief. A multiplexer combines four 100-

Kbps channels using a time slot of 2 bits. Show the output with four arbitrary

inputs. What is the frame rate? What is the frame duration? What is the bit rate?

What is the bit duration? [3+4]

5. How are OSI and ISO related to each other? What are the responsibilities of the

transport layer in the Internet model? [2+3]

OR

1. Discuss are the advantages and disadvantages of optical fibres over metallic

cables. [4]

2. Classify Transmission Media. Write down the characteristics of Microwaves and

Radiowaves. [2+3]

2

3. How do the layers of internet model correlate to the layers of the OSI model?

What are the responsibilities of the Presentation layer in the Internet model?

[2+3]

4. Classify multiplexing and explain WDM in brief. Five channels, each with a 100-

KHz bandwidth, are to be multiplexed together. What is the minimum bandwidth

of the link if there is a need for a guard band of 10 KHz between the channels to

prevent interference? [3+2]

5. Explain how analog data are converted to digital signals. What sampling rate is

needed for a signal with a bandwidth of 20,000 Hz (1000 to 21,000 Hz)? [4+2]

UNIT-II (Attempt any one group)

1. Given three IP addresses are 32.46.7.3, 200.132.110.35 and 140.75.8.92. Find

their classes, network addresses and their subnet marks. [6]

2. Discuss CSMA/CD and Token Ring protocol as applied to LAN networks with relevant diagrams. [6]

3. Write short note on the following protocols

a. ICMP

b. DHCP [3X2=6]

4. What are the responsibilities of the Network layer in the Internet model? Explain

different routing techniques used in networking. [3+4]

OR

1. Compare and contrast the Go-Back-N ARQ Protocol with Selective-Repeat ARQ.

[6]

2. Write short note on the following protocols

a. DNS

b. IPv4 [3X2=6]

3. What are the responsibilities of the Network layer in the Internet model? Classify

and explain unicast and multicast routing protocols. [4+5]

4. Find the error, if any, in the following IP addresses:

a. 111.56.045.78 b. 221.34.7.8.20

c. 75.45.301.14 d. 11100010.23.14.67 [4]

3

UNIT-III (Internal Choice in Question No 3 and 4)

1. How IP Address, Port Number and Socket Address are related? Explain with an

example. [3]

2. Differentiate between TCP and UDP.Imagine a TCP connection is transferring a file

of 6000 bytes. The first byte is numbered 10010. What are the sequence numbers

for each segment if data are sent in five segments with the first four segments

carrying 1000 bytes and the last segment carrying 2000 bytes? [4+3]

3. How data delivery in transport layer is distinguished from datalink and network

layer? [3]

4. Explain the process of connection establishment and connection termination.

OR

Compare the services offered by POP3, IMPA4 and SMTP e-mail servers.

[6]

5. Explain TCP segment format and brief the various Control field availiable.

OR

Compare the strategies followed by Open-loop and Closed-loop congestion

control techniques.

[6]