LINEAR NEUTRAL PARTIAL DIFFERENTIAL EQUATIONS: A SEMIGROUP ...

# Transforms and Partial Differential Equations

### Transcript of Transforms and Partial Differential Equations

Transforms and partial differential equationImportant questions

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VEL TECH Dr.RR & Dr.SR TECHNICAL UNIVERSITYDepartment of Mathematics

Transforms and Partial Differential EquationsSem:III Year :II

Unit :I FOURIER SERIES

Part A(2 marks)

1.Find the Fourier sine series for the function f(x)=1,2.What is the constant term and the coefficient of cosnx, in the Fourier expansion of f(x)=

3.What do you mean by harmonic analysis?4.State Parseval’s identity for Fourier range expansion of f(x) as Fourier series in (0,2l).5.Find in the expansion of as a Fourier series in ( ).

6.If f(x) is an odd function defined in (-l,l) what are the values of .

7.Find the Fourier constants for sinnx in ( ).8.State Parseval’s identity for the half range cosine expansion of f(x) in(0,1).9.Find in expanding as a Fourier series in ( ).10.If f(x) is discontinuous at x=a what does its Fourier series represent at that point.11.Find the root mean square value of the function f(x)=x in the interval (0,l).14.If f(x)=2x in the interval (0,4)then find the value of the in the Fourier series expansion

15.The Fourier series expansion of f(x) in(0,2 ) is f(x)= .Find the root mean square

value of f(x) in the interval (0,2 ).16.Determine the value of in the Fourier series expansion of f(x)= in .

17. Obtain the Fourier series for the function f(x)= .

28. If f(x)= , in a Fourier series if the period is 2 .

Part B(8 marks &16 marks)

1.Determine the Fourier series expansion of f(x)=xsinx in .2. Determine the Fourier series expansion of f(x)=xcosx in .

3.Obtain the Fourier series for the function f(x)=|x| in .Deduce that

.

5. Find the Fourier series of f(x)=xsinx in .6. Find the Fourier series of f(x)=|x| in .

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7. Find the Fourier series of f(x)= in the range . Hence deduce that

8.Obtain the Fourier series for f(x)= in ( ) . Deduce that .

9. Determine the Fourier series expansion of f(x)=x in the interval .10. Obtain the Fourier expansion for in the interval .

11. Find the Fourier series of f(x) with period 2l defined by f(x)= . Hence deduce

that .

12. Find the Fourier series of f(x)= with period 3 in the range (0,3).

13. Obtain the Fourier series for f(x)= .

14. Find the Fourier series of f(x)= . Deduce that .

15. Find the Fourier series expansion for f(x)= .

16.The half range cosine series for function f(x)= .

17.Obtain cosine series for f(x)=xsinx in 0<x<18. Find the half range cosine series for f(x)=x(2-x) in .

Deduce the sum of the series

19. Obtain the half range cosine series for function f(x)= and deduce that

.

20. The sine series for function f(x)=x in (0,l).21. Obtain the sine series for function

f(x)=

22.Compute the first three harmonic of the Fourier series for f(x) given by the following table

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x 0

f(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0

23.Find the Fourier series up to second harmonic representing the function given by the following table

x 0

f(x) 0.8 0.6 0.4 0.7 0.9 1.1 0.8

24. Find the Fourier series y=f(x) up to second harmonic from the following table

x 0 1 2 3 4 5

f(x) 9 18 24 28 26 20

Unit-II

PART – A

01. State Fourier integral Theorem

02. Define Fourier transform pair.

03. State convolution theorem for Fourier transforms.

04. If F(s) is the Fourier transform of f(x), then find the Fourier transform of f(x-a).

05. Express shifting theorem.

06. Prove =F(s+a).

07. Find the Fourier sine transform of 1/x.

08. Define self reciprocal.

09. Prove

10. Find FS [xe-ax]

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11. Prove that

12. Prove that

13. Solve the integral equation

14. State parseval’s identity theorem in Fourier transform

PART – B

1. Express the function as a Fourier integral, hence evaluate

and find value of .

2. Using Fourier Integral show that

3. Find Fourier Transform of f(x) if

Deduce that (i) ii)

4. (i) Find f (x), if its sine transform is Hence deduce that the inverse sine transform of .

ii) And show that is self-reciprocal with respect to Fourier Sine Transforms

5. Find the Fourier transform of

6. Find the Fourier transform of Hence prove that

7. Find the Fourier cosine transform of (i) e-ax cos ax and (ii) eiax sin ax.

8. State and prove convolution theorem for Fourier transforms and also parsevals identify.

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9. Show that the Fourier transforms of 2 2-x | |

( )0 , | |

a x af x

x a

is

Hence deduce that

Using parseval’s identify show that,

10. Evaluate using Fourier Transform

11. Find Fourier cosine transformer and hence find

12. Find the Fourier transform of if a > 0.

Deduce that if a > 0 [Anna May/ June 2003]

13. Find the fourier Transform of f(x) and hence deduce the value of

.

14. Find Fourier sine and cosine transform of and hence find the Fourier sine transform of

and Fourier cosine transform of .

UNIT 3 Questions

S No 2 Marks(Min 10 Questions) Topic ID Degree Of Importance

1 Form the partial differential equation by eliminating the

arbitraryconstants a & b from z = ax + by.

Unit 3 A

2 Form the partial differential equation by eliminating the

Unit 3 A

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arbitraryconstants from z = ax + by + ab.

3 Form the partial differential equation by eliminating the arbitrary constants from z =

+

Unit 3 A

4 Find the partial differential equation of the family of sphereshaving their centres on the line

x=y=z.

Unit 3 A

5 Form the partial differential equation by eliminating the

arbitrary.

constants a & b from z=f(x² - y²)

Unit 3 A

6 Form the partial differential equation by eliminating the

arbitraryfunction from z=f(x²+y²)

Unit 3 A

7 Form the partial differential equation by eliminating the

functionFrom the relation z=f(x/y)

Unit 3 A

8 Form the partial differential equation by eliminating the

functionf from the relation z=y2+2f(1/x+logy)

Unit 3 A

9 Form the partial differential equation by eliminating the

arbitraryfunction φ from φ (x-

y,x+y+z)=0.

Unit 3 A

10 Find the particular integral of (D²-3DD'+2D'²)z = cos(x+2y).

Unit 3 A

UNIT 3 Questions

S No 8 Marks(Min 8 Questions) Topic ID Degree Of Importance

1 Solve:(D² -DD'+20 D'²)z = Unit 3 A

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+ sin(4x-y)

2 Solve: z = p²+q² Unit 3 A3 Solve:(D² -DD'-6D'²)z =x²y+ Unit 3 A

4 Solve:(y²+z²)p-xyq+xz = 0 Unit 3 A

5 Solve:(D² -6DD'+5D'²)z=

sinhy+xy

Unit 3 A

6 Solve:p(1-q²)=q(1-z) Unit 3 A

7 Solve: (D² -4DD'+4D'²)z = Unit 3 A

8 Find the singular integral of z = px+qy+p²+pq+q²

Unit 3 A

UNIT 3 Questions

S No 14 Marks(Min 5 Questions) Topic ID Degree Of Importance

1 Solve(D² +3DD'-4D'²)z =x+sin y Unit 3 A

2 Solve:z²(p²+q²)=x²+y² Unit 3 A

3 Solve:p(1+q)=qz Unit 3 A

4 Solve:x(y²+z)p+y(x²+z)q=z(x²-y²) Unit 3 A

5 Find a partial differential equation by eliminating a and b from theexpression (x-a) ²+(y-b) ²+z² =c²

Unit 3 A

UNIT 4 Questions

S No 2 Marks(Min 10 Questions) Topic ID Degree Of Importance

1 Explain the initial and boundary value problems?

Unit 4 A

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2 Explain the method of separation of variables.

Unit 4 A

3 By the method of separation of

variables solve q+ p=0.

Unit 4 A

4 State the assumption made in the derivation of one dimensional wave equation.

Unit 4 A

5 Write the one dimensional wave equation?

Unit 4 A

6 Write the three possible solutions of

.

Unit 4 A

7 In the PDE of a vibrating string

,what is .

Unit 4 A

8 Explain the various variables involved in one dimensional wave equation.

Unit 4 A

9 Write down the boundary conditions for the following boundary value problem “ if a string of length ‘l’ initially at rest in its equilibrium position and each of its point is

given the velocity =

, 0 determine the

displacement function y(x, t)

Unit 4 A

10 Define temperature gradient. Unit 4 A

11 Define steady state temperature distribution

Unit 4 A

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12 How many boundary conditions are required to solve completely for the PDE

Unit 4 A

13 State the laws assumed to derive the one dimensional heat equation

Unit 4 A

14 Say true or False :One dimensional heat conduction equation is

Unit 4 A

15 The steady state temperature of a rod of length ‘l’ whose ends are kept at

and is …….

Unit 4 A

16 State one dimensional heat equation with the initial and boundary conditions .

Unit 4 A

17 What is the basic difference between the solutions of one dimensional wave equation and one dimensional heat equation.

Unit 4 A

18 Give three possible solutions of the equation

Unit 4 A

19 State Fourier law of heat conduction Unit 4 A

20 When the ends of a rod length 20 cm are maintained at the temperature 10°C and 20°C respectively until

Unit 4 A

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steady state is prevailed. Determine the steady state temperature of the rod.

21If y(x,t)= is

the solution of wave motion satisfying certain conditions, then what will be the solution satisfying

y(x,0) =A sin .

Unit 4 A

22 Explain the term steady state. Unit 4 A

UNIT 4 QuestionsS No 8 Marks(Min 8 Questions) Topic ID Degree Of

Importance1 Derive the solutions of one dimensional

wave equations.Unit 4 A

2 Derive the solution of one dimensional heat equation by the method of variable separable method.

Unit 4 A

3 Derive the solution of two dimensional heat equation by the method of variable separable method.

Unit 4 A

4 A string stretched with fixed end points x=0 and x=l is initially in a position given by

y(x,0)= .If it is released from rest

from this position find the displacement ’y ‘at any distance ‘x’ from one end at any time t.

Unit 4 A

5 If a string of length ‘l’ is initially at rest in its equilibrium position and each of its

Unit 4 A

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points is given the velocity (

Determine

the displacement function y(x,t).

6 Find the solution to the equation

that satisfies the conditions (i)

u(0,t)=0 (ii) u(l,t) =0 for t (iii)

u(x,0) =

Unit 4 A

7 A rod of length l has its ends A and B kept at 0°c and 100°c until steady state condition prevails. If the temperature at B is reduced suddenly to 0°c and kept so while that of A is maintained, find the temperature u( x , t ) at a distance x from initial point and at time t.

Unit 4 A

8 A rod of length 20cm has its ends A and B kept at 50°c and 0°c until steady state condition prevails. If the temperature at A is reduced suddenly to 0°c and kept so while that of B is maintained, find the temperature at any point of the rod at time t.

Unit 4 A

UNIT 4 Questions

S No 14 Marks(Min 5 Questions) Topic ID Degree Of Importance

1 A homogeneous rod of conducting material of length ‘l’ units has ends kept at zero temperature and the

Unit 4 A

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temperature at the centre is T and falls uniformly to zero at the two ends. Find u ( x, t ).

2 A bar 10 cm long with insulated sides has its ends A and B kept at 20°c and 40°c respectively until steady state conditions prevailed. The temperature at A is then suddenly raised to 50°c and at the same instants that at B is lowered to 10°C. Find the subsequent temperature at any point of the bar at any time.

Unit 4 A

3 Two ends A and B of a rod of length 20 cms have the temperature at 30°c and 80°c respectively until the steady state condition prevails. Then the temperature at the ends A and B are changed to 40°c and 60°c respectively. Find u ( x , t ).

Unit 4 A

4 Find the solution of the equation

, 0

is bounded as t→

(ii) =0 for values of t

when x=0 and x=a

when

t = 0 and 0

Unit 4 A

5 A square plate is bounded by the lines x=0, y=0, x=20 `and y=20.Its faces hare insulated. The temperature along the upper horizontal edge is given by u

(x, 20) = x (20-x) when 0

while the other three edges are kept at 0 C. Find the steady state temperature

Unit 4 A

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in the plate.

6 Find the steady state temperature at any point of a square plate whose two adjacent edges are kept at 0° C and the other two edges are kept at the constant temperature 100° C.

Unit 4 A

7 An infinitely long rectangular plate with insulated surface is 10 cm wide. The two long edges and one short edge are kept at zero temperature,while the other short edge x=0 is kept at temperature given by u=

Find

the steady state temperature distribution in the plate.

Unit 4 A

UNIT: 5Part: A (2marks)

1. State and prove Initial value theorem in Z-Transform.2. Find Z-Transform of .

3. Show that Z = .

4. State and prove Final value theorem in Z-Transform.5. Show that

6. If .7. If Z[f(t)]=F(z),then Z[f(t +T)]=zF(z)-zf(0)8. If Z[f(n)]=F(z),then Z[f(n+1)]=z F(z)-z f(0)9. Determine the Z-transform of

10. Find the Z-transform of .

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11. Determine the Z-transform of x(n)= for n 0

12. Using initial and final value theorem ,find the value of .

13. Find the Z-Transform of

14. Find the Z-Transform of

15. Find the Z-Transform of .

PART:B

1. Find the Z-transforms of (i) (ii)n(n-1) (iii) (iv) (v) (vi)sin

(vii) .

2. Find the Z-transforms of (i) (ii) (iii) (iv) (v) (vi)

(vii) .

3. Find the inverse Z-transforms of X(z)= , : |z|>|a|.

4. Solve , .

5. Derive the differences equation from .

6. Solve using Z-transform given that .

7. Solve given that .

8. Using Z-transform solve , given that .

9. Using Z-transform solve , given that .

10. Find the inverse Z-transforms of .

11. Solve , given that .

12.Using convolution theorem find the inverse Z-transforms of .

13.Find the inverse Z-transforms of .

14. Find the Z-transforms of (i) (ii) .

15. Find the Z-transforms of (n+1)(n+2)16. Solve using Z-transform

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17. Find the inverse Z-transforms of using the method of residues.