Download GTU B.Tech 2020 Summer 3rd Sem 3130107 Partial Differential Equations And Numerical Methods Question Paper

Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Summer 3rd Sem 3130107 Partial Differential Equations And Numerical Methods Previous Question Paper

Seat No.: ________
Enrolment No.___________


GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER? III EXAMINATION ? SUMMER 2020
Subject Code: 3130107 Date:27/10/2020
Subject Name: Partial Differential Equations and Numerical Methods
Time: 02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.
Q.1 (a) Use Iteration method to find the real root of the equation 3
x x 1 0 correct
03
to six decimal places starting with x 1.
0

(b) Use Bisection method to find the real root of the equation x cos x 0 correct
04
upto four decimal places.

(c) Explain the Newton-Raphson method briefly. Also find an iterative formula
07
for N and hence find 7 correct to three decimal places.


Q.2 (a)
1
03
Evaluate ex2 dx
by Simpson's-1/3 rule with n=10 and estimate the error.
0

(b) Solve the following linear system of equations by Gauss elimination method.
04
0 x 8y 2z 7
3x 5y 2z 8
6x 2y 8z 26

(c) Compute cosh 0.56using Newton's forward difference formula and also
07
estimate the error for the following table.
x
0.5
0.6
0.7
0.8
f x
1.127626
1.185465
1.255169
1.337435


OR


(c) The speed, v meters per second, of a car, t seconds after it starts, is show in the
07
following table.
t 0 12
24
36
48
60
72
84
96 108 120
v 0 3.60 10.08 18.90 21.60 18.54 10.26 4.50 4.5 5.4 9.0
Using Simpson's 13 rule, find the distance travelled by the car in 2 minutes.
Q.3 (a)
1
03
Use Trapezoidal rule to evaluate 3
x dx using five subintervals.
0

(b) Check whether the following system is diagonally dominant or not. If not,
04
rearrange the system and solve it using Gauss-Seidel method.
8x 3y 2z 20
4x 11y z 33
6x 3y 12z 35

(c) Explain Euler's method briefly and apply it to the following initial value
07
dy
problem by choosing h 2
.
0 and hence obtain y 0
.
1 .
x y y0 0 .
dx
Also determine the error by deriving it analytical solution.


OR

1

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Q.3 (a) Use Runge?Kutta second order method to find the approximate value of
03
dy
y(0.2) given that
2
x y & y(0) 1 & h 0
.1

dx

(b) Find the Lagrange interpolating polynomial from the following data
04
x
0
1
4
5
f x
1
3
24
39

(c) Derive Secant iterative method from the Newton-Raphson method and use it
07
to find the root of the equation cos x
x
xe 0 correct to four decimal places.
Q.4 (a) x2 yzp y2 xyq z2 xy .
03

(b)
2
z
z
04
Solve
z 0 given that when
y
x ,
0 z e and
1.
2
x
x

(c) Obtain the solution of following one-dimensional Wave equation together with
07
following initial and boundary conditions by the method of separation of
variables.
2u
2
2
u
c
t
2
x
2
u( ,
0 t) u l
( ,t)
0 t 0
ux 0
, f (x f
) or 0
x l
u
0
,
f or 0
t x
gx
x l

OR

Q.4 (a)
2
pz qz z x y2 .
03

(b)
3
z
3
z
3
z
04
x2 y
3
4
e
.
x3
x2
y
y3

(c) Obtain the solution of following one-dimensional heat equation with insulated
07
sides by the method of separation of variables.
u
2
2
u
c
t
x
2
u( ,
0 t) u l
( , t)
0 t 0
ux,0 f (x f
) or 0
x l
Q.5 (a) p2 q2 x y
03

(b)
u
u
04
Solve the given equation
2
u given that ux
3
x
0
, e
6
.
x
t

(c) Obtain the solution of following one-dimensional heat equation with insulated
07
ends by the method of separation of variables.
u
2
2
u
c
t
x
2
u ( ,
0 t) u l
( ,t)
0 t 0
x
x
ux,0 f (x f
) or 0
x l


OR
Q.5 (a)
2
z 2
2
p q
2
1 a ,
03
(b) Using method least squares, find the best fit straight line for the following data.
04
x
1
2
3
4
5
y
1
3
5
6
5
(c) Obtain the solution following two-dimensional Laplace equation.
07
u u 0
xx
yy
ux,0 ux, ua, y 0
u(x,0) f x
********************
2

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