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

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

*ex*2

*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*8

*y*2

*z*7

3

*x*5

*y*2

*z*8

6

*x*2

*y*8

*z*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.

8

*x*3

*y*2

*z*20

4

*x*11

*y*

*z*33

6

*x*3

*y*12

*z*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 y*0 0 .

*dx*

Also determine the error by deriving it analytical solution.

**OR**

1

**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)**

*x*2

*yz*

*p*

*y*2

*xy*

*q*

*z*2

*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.

2

*u*

2

2

*u*

*c*

*t*

2

*x*

2

*u*( ,

0

*t*)

*u l*

( ,

*t*)

0 t 0

*u*

*x*0

,

*f*(

*x*f

) or 0

*x*

*l*

*u*

0

,

f or 0

*t*

*x*

*g*

*x*

*x l*

**OR**

**Q.4 (a)**

2

*pz*

*qz*

*z*

*x*

*y*2 .

**03**

**(b)**

3

*z*

3

*z*

3

*z*

**04**

*x*2

*y*

3

4

*e*

.

*x*3

*x*2

*y*

*y*3

**(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

*u*

*x*,0

*f*(

*x*f

) or 0

*x*

*l*

**Q.5 (a)**

*p*2

*q*2

*x*

*y*

**03**

**(b)**

*u*

*u*

**04**

Solve the given equation

2

*u*given that

*u*

*x*

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*

*u*

*x*,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*

*u*

*x*,0

*u*

*x*,

*u*

*a*,

*y*0

*u*(

*x*,0)

*f*

*x*

********************

2

This post was last modified on 04 March 2021