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ANAND INSTITUTE OF HIGHER TECHNOLOGY

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

Unit-I Fourier Series

Part-A (Question and Answers)

1. State Dirichlet’s condition (May/June 2013,2009)

Sol: (i) f(x) is bounded, single valued

f(x) has atmost a finite no. of maxima and minima

f(x) has atmost a finite no. of discontinuities

2. Write the formula for finding Euler’s constant of a Fourier series in (0,2π) (A.U 2009)

Sol: [pic]

where [pic]

3. Sum the Fourier series for [pic] at x = 0 and x =1.(A.U N/D 2010)

Sol:

Here x =0 is discontinuous point

[pic]

Here x = 1 is discontinuous point

[pic]

4. What is the constant term ao and the coefficient of cosnx, an in the Fourier series expansion of f(x)=x-x3 in (-π, π).(A.U N/D 2010)

Sol:

[pic]

[pic][pic]

f(x) is odd function

ao = an = 0

5. State Parseval’s Identity for the full range expansion of f(x) as Fourier series in (0, 2 l ).(A.U 2008)

Sol: [pic]

6. Find a Fourier sine series for the function f(x) = 1, 0 < x < π (A.U 2007 nov 2013)

Sol:

[pic]

[pic]

[pic]

7. If the Fourier series for [pic] is [pic]

Prove that [pic](A.U 2009)

Sol:

Put [pic] in ( 1 )

Here [pic] is continuous point . [pic]

(1) [pic] [pic]

[pic]

8. Find bn in the expansion of x2 as a Fourier series in (-π, π).(A.U 2005,2008)

Sol:

Here [pic] is even function in (-π, π)

[pic]

9. If f(x) is an odd function defined in (-l , l ) what are the values of a0 and an.?(A.U A/M 2010)

Sol: Here f(x) is an odd function in (-l , l )

[pic]

10. What do you mean by Harmonic Analysis? (May/June 2013,2010)

Sol:

The process of finding the Fourier series of the function f(x) which is given in terms of numerical value.

11. Find an in expanding e -x as Fourier series in (-π, π)

Sol:

[pic][pic]

[pic]

[pic]

12. State Parseval’s Identity for the Half range cosine series expansion of f(x) in (0,1).A.U 2007)

Sol: [pic]

13. If for 0 ab = 2 => b = 2/a

Substitute b = 2/a in equation (2) we get

Z = ax + (2y)/a + c. This is the required solution.

13. Find the general solution in terms of arbitrary functions for the PDE 2p + 3q = 1.(A.U 2009)

Ans:

Subsidiary equation is[pic]

Consider[pic]integrating 3x – 2y = c1

Consider [pic]integrating y – 3z = c2

The Solution is f(3x – 2y, y – 3z) = 0.

14. Explain the method of solving Lagrange’s Linear equation (A.U 2010)

Ans.:

Lagrange’s linear P.D.E is of the form Pp + Qq = R where P, Q, R are functions of x,y,z,

To solve this, we first form the subsidiary simultaneous equations [pic]

If u = a and v = b Are two independent integrals of these simultaneous equations,

Then the solution of P.D.E. is Ф (u, v) = 0 where Ф is an arbitrary function.

It can also be put as u = f (v), f being arbitrary.

There are two methods of solving the simultaneous equations

(1) Method of grouping

(2) Multiplier Method.

15. Solve [pic] (A.U 2009)

Ans:

A.E is m2 + 2m = 0.

m (m+2) = 0 => m =0, m= – 2

The Solution is z = f1(y) +f2(y – 2x)

PART – B Questions

1. Solve [pic] (May 2007)

2. Solve [pic] (May 2007)

3. Solve [pic] (May 2006,2009)

4. Solve [pic] (May 2006,2010)

5. solve [pic] (May 2006)

6. solve [pic] (Dec 2006)

7. solve [pic] (may 2005,2012)

8. solve [pic] (Dec 2006)

9. solve [pic] (May 2005)

10. solve [pic] (Dec 2005)

11. solve [pic] (Nov 2005)

12.Solve [pic] (Nov2005)

13.Solve [pic] (May 2004, Apr 2000,2008,2013)

14.Solve [pic] (Dec 2004)

15.Solve[pic] (Dec 2003,2011)

16.Solve [pic] (Dec 2003)

17.Solve [pic] (Apr2000)

18.Solve [pic]

19.Solve[pic] (Apr2001)

20.Solve [pic] (Apr2004,2010)

21.Solve[pic] (May2007,2008)

22.Solve [pic] (May 2007)

23.Solve [pic] (May2007)

24.Solve [pic] (May2007,2008,2009,2012)

25.Solve [pic] (May2006)

26.Solve [pic] (May2006)

27.Solve [pic]

28.Solve [pic] (A.U 2008)

29.Solve [pic] (Nov2006)

30.Solve [pic] (may 2006)

31. Solve [pic] (may 2005,2010,2012)

32. solve [pic] (Dec 2005)

33. Solve [pic] (Dec 2005,2011)

34. solve [pic] (Dec 2005)

35. Solve [pic] (Dec 2005,2011,2013, NOV 2013)

36. Solve [pic] (Dec 2004,2008,2013)

37. Solve [pic] (may 2004,2008,2011)

38. Solve [pic] (Dec 2004)

39. Solve [pic] (Dec 2002,2011)

40. Solve [pic] (Apr 2003)

41. Form the Partial Differential Equation from by eliminating the arbitrary constants ‘a’ and ‘b’ from

the expression [pic] (May 2007)

42.Find the Singular integral of [pic] (Nov2006)

43.Find the Singular integral of [pic] (Nov2003,2010)

44.Find the Singular solution of [pic] (Nov2002)

45. Form the PDE by eliminating f & [pic] from [pic] (Dec 2004)

46. Find the C.I of p +q = x + y

47. Form the PDE by eliminating the arbitrary function from the relation [pic] (April 2003,2009)

48. Form the PDE by eliminating the arbitrary function from the relation [pic] (Nov 2002,2011)

49. Form the PDE by eliminating the arbitrary function from the relation [pic] (April 2003,2007,NOV 2013)

50. Form the PDE by eliminating the arbitrary function from the relation [pic] (Nov 2003)

51. Form the PDE by eliminating the arbitrary function from the relation [pic] (April 2004)

52. Solve [pic] NOV 2013

53.Solve [pic] NOV 2013

*****

UNIT – IV APPLICATIONS OF PDE

1. Write all the solution of the wave equation (A.U 2010,2011)

Sol.: The wave equation is [pic].

The various possible solution of this equation is

y(x,t) = [pic]

=[pic]

= [pic].

The pde of a vibrating string is [pic] what is [pic]?

Solution: [pic]=T/m = Tension/mass per unit length of string.

2. State the assumptions made in the derivation of one dimensional wave equation.(A.U 2011)

Sol.:

The wave equation is [pic]

(i) The motion takes place entirely in one plane.ie xy plane.

(ii) We consider only transverse vibrations. The horizontal displacement of the particles of the string is negligible.

(iii) The tension T is constant at all times and at all points of the deflected string.

(iv) T is considered to be so large compared with the weight of the string and hence the force of gravity is negligible.

(v)The effect of friction is negligible.

(vi) The string is perfectly flexible.

3. Give reasons for choosing y = (Acospx+Bsinpx)(Ccospat+Dsinpat) as a suitable solution of the pde of the vibrating string.

Sol.: Since the vibrating of the string is a periodic motion with respect to time .we must get a solution for y(x,t) in

which trigonometric terms of t are present.

4. Write down the boundary conditions for the following boundary value problem.

“if a string of length ‘ℓ’ is initially at rest in its equilibrium position & each of its points is given the

velocity [pic] determine the displacement functions.

Sol.: wave equation [pic]boundary conditions are

y(0,t)=0

y(ℓ,t)=0}, t>0

y(x,0)=0

[pic]

5. Write all the solution of one dimensional heat equation. [pic](A.U M/J 2013)

Sol.:

u(x,t) = [pic]

u(x.t )= [pic]

u(x,t)= [pic].

6. The partial differential equation of one dimensional heat equation is [pic]what is [pic]?(A.U 2011)

Sol.:

[pic]is called the diffusivity of the material of the body through which heat flows.

If ρ be the density, c the specific heat & k thermal conductivity of the material.

We have the relation [pic]= k/ρc.

7. What is meant by steady state condition in heat flow? (A.U 2001)

Sol.:

steady state condition in heat flow means that the temperature at any point in the body does not vary with

time. It is independent of ‘t’ the time.

8. In steady state conditions derive the solution of one dimensional heat flow? (A.U 2011,nov 2013)

Sol.:

The pde of unsteady one dimensional heat flow is [pic][pic]

In steady state condition the temperature u depends only on x & not time t.

Hence [pic]. Eqn (1) reduces to [pic] solving 2.

The general solution of 2 is u = ax + b where a, b are arbitrary.

9. Find the steady state temperature of a rod of length l whose ends are kept at 30 and 40 .

Sol.:

The steady state equation of one dimensional heat flow is [pic]

Solving we get u = a x+ b [pic] .

The boundary conditions

u = 30 when x = 0, u = 40 when x = ℓ

30 = a(0) + b, => b = 30

40 = a (ℓ) + 30, => a ℓ = 10, => a = 10/ℓ

Sub in 2 we get u = 10x/ ℓ + 30.

10. Explain the term “ Thermally insulated ends ‘

Sol.:

If an end of a heat conducting body is thermally insulated. It means that no heat passes through that

section Mathematically the temperature gradient is zero at that point [pic].

11. Express the boundary conditions in respect of insulated ends of a bar of length ‘a’ and also the temperature distribution f(x). (A.U 2011)

Sol.:

The boundary conditions are

[pic] For all values of t

u(x,0) = f(x) for 0 < x < a.

12. What are the assumptions made while deriving one dimensional heat equation?

Sol.:

1. Heat flows from a higher to lower temperature.

2. The quantity of heat required to produce a given temperature change in a body is proportional to

the mass of the body and the temperature change .The constant of proportionality is known as the

specific heat(c) of the material.

3. The rate at which heat flows across any area is jointly proportional to the area and to the temperature

gradient normal to the area. Ie The rate of change of temperature w.r.t the distance normal to the area .

The constant of proportionality is known as the thermal conductivity (k).This is known as Fourier law of

heat conduction.

13. What is steady state heat equation in two dimensions in Cartesian form?

Sol.:

The required equation is [pic]

14. Write the different solutions of Laplace equation in Cartesian coordinates? (A.U 2008)

Sol.:

[pic]

15. For [pic]write a solution which is periodic in y?

Sol.: [pic]

16. Write the general solution y(x,t) of vibrating motion of a string of length ‘ℓ’ with fixed end point and zero initial velocity.

Sol.:

y(x,t) = [pic]

17. A rectangular plate is bounded by the lines x = 0, y = 0, x = a and y = b. Its surfaces are insulated and the temperature along the adjacent sides x = a; y = b are kept at 100 & the temperature along the two sides x = 0 & y = 0 are kept at 0.write the boundary conditions.

Sol.:

The boundary conditions are

u(0,y) = 0

u(a,y) = 100}, 0[pic]

u(x,0) = 0

u(x,b) = 100}, 0[pic]

18. classify the pde (i)[pic] (ii)[pic](

Sol.:

(i) Here A = 1; B = 0; C = x

[pic]– 4AC = 0 – 4x = – 4x [pic]

If x is negative (x < 0) then [pic] ie (i) is Hyperbolic

If x > 0 then [pic] is elliptic

If x = 0 then [pic] is parabolic

(ii) Here A =[pic]; B = – 2xy; C = [pic]

Now [pic] (ii) is parabolic.

19. Classify the following

(i) [pic] (ii) [pic]

Sol.:

(i). A = x; B = 0; C = y

[pic] = – 4xy < 0

[pic] [pic] is elliptic

(ii). A = 1; B = – 2; C = 0

[pic] [pic] is Hyperbolic.

20. Classify the following pde?

[pic]

Sol.:

Here A =[pic]; B = 0; c = 1

[pic]– 4AC = 0 – 4[pic] = – 4[pic]< 0

[pic]

[pic] is elliptic

21. An insulated rod of length I cm has its ends A and B maintained at [pic] and 80⁰crespectively. Find the steady state solution of the rod.(nov 2013)

Soln.

U=ax+b

X=0 => U(0)=b, b=0

X=1 => u(l)=al+b

80=al

a=80/l

Part-B Questions

1. A tightly stretched flexible string has its ends fixed at x = 0 and x = l. At the time t = 0, the

String is given a shape defined by [pic]where k is a constant and then

Released from rest. Find the displacement at any point ‘x’ of the string at any time [pic][pic].

(May2005 NOV 2013)

2. A uniform string is stretched and fastened to two points ‘l’ apart. Motion is started by displacing

the string into the form of the curve [pic] and then releasing it from this position at

time t = 0. Find the displacement of the point of the string at a distance x from one end at time t.

(Nov 2003, 2007, May 2007, 2008,2009,2013)

3. An elastic string of length 2l fixed at both ends is disturbed from its position at equilibrium

Position by imparting to each point an initial velocity of magnitude[pic]. Find the

displacement function[pic]. (May 2006, Nov 2007)

4. A tightly stretched string with fixed end points [pic] is initially in a position

equation by [pic].It is released from rest from this position. Find the displacement

at any time‘t’ (Nov 2004)[pic]

[pic]

5. A string is tightly stretched and its ends are fastened at two points at [pic] .The midpoint

of the string is displaced transversely through a small distance ‘b’ and the string is released from

rest in that position. Find an expression for the transverse displacement of the string at

any time during the subsequent motion? (Nov 2002, 2005, May 2005, April 2001,2010)

6. A tightly stretched string of length 2l is fixed at both ends the midpoint of the string is displaced

by a distance ’b’ transversely and the string is released form rest in this position. Find the

displacement of any point of the string at any subsequent time.

(Nov 2006, 2005, 2002, May 2005,2012)

7. A taut string of length l has its ends[pic]fixed. The point where[pic]is drawn aside a

small distances h, the displacement y(x, t) satisfies[pic]. Determine [pic] at any time‘t’? (May 2006)

8. A stretched string with fixed end points [pic] is initially at rest is in equilibrium position.

if it is set vibrating giving each point a velocity [pic]then show that

[pic](Nov 2003, April 2001)

9. If a string of length l is initially at rest in its equilibrium position and each of its points is given

a velocity V such that V [pic]S.T the displacement any time t is given by

[pic] (May 2007)

10. A string is stretched between two fixed points at a distance 2 l apart and the points of the

string are given initial velocities V and V = [pic] x being the distance from an

end point. Find the displacement of the string at any time? (May 2004, 2008, Dec 2008)

11. A tightly stretched string with fixed end points [pic]is initially at rest in its equilibrium

Position. If it is set to vibrate by giving each point a velocity [pic], find the displacement

Of the string at any subsequent time. (Dec 2008,2012)

12. A rod of length l has its end A and B kept at [pic]c and [pic]c respectively. Until steady state

conditions prevail. If the temperature at B is reduced suddenly to [pic]c and at the same time the

temperature at A raised to [pic]c . find the temperature [pic]at a distance x from A and at time t.

(May 2003).

13. A metal bar 10cm long with insulated sides, has its ends A and B kept at [pic]c and [pic]c respectively.

until steady state conditions prevail. The temperature at A is suddenly raised to [pic]c and that the same

same time that at B is lowered to [pic]c. find the subsequent temperature at any point of the bar at any time.

Hence prove the temperature at the mid point of the rod remains for all time , regardless of the material

of the rod. (May 2001, 2003, Nov 2005)

14. The ends A and B of a rod l c.m. long have their temperatures kept at [pic]c and [pic]c., until steady state

conditions prevail. The temperature of the end B is suddenly reduced to [pic]c and that of a is

increased to [pic]c. find the temperature distribution on the rod after time t. (May 2007,2012)

15. A rod of length 20cm has its ends A and B kept at temperature [pic]c and [pic]c respectively

until steady state conditions prevail. It the temperature at each end is then suddenly reduced

to [pic]c and maintained [pic]c. find the temperature distribution at a distance from A at time ‘t’.(Nov 2005)

16. A rod of length 30 cm has its ends A and B kept at [pic]respectively until steady state

conditions prevail. If the temperature of A is suddenly raised to [pic]while that the other end B is

reduced to [pic], find the temperature distribution at any point in the rod. (Dec 2008,2012)

17. An infinitely long rectangular plate with insulated surface is 10cm wide. The two long edges and

one short edge are kept at zero temperature while the other short edge [pic] is kept at temperature

given by

[pic] , Find the steady state temperature distribution in the plate?

(May 2006, Nov 2004, 2005,2013)

18. A rectangular plate with insulated surface is 8cm wide and 10cm long compared to its width that

it may be considered infinite in the length without introducing an appreciable error. It the

temperature along one short-edge[pic] is given by [pic]. While the

two long edges[pic]and [pic]as well as the other short edges are kept at [pic]c. find the

steady state temp. for [pic]. (Nov 2003)

19. A rectangular plate with insulated surfaces is ‘a’ an wide and so long compared to its

width that it may be considered infinite in length, without introducing an appreciable error. It the

two long edges [pic]and the short edge at infinity are kept at temperature [pic]c, while

the other short edge [pic]is kept at temperature [pic], find the steady state temperature

at any point [pic]of the plate. (May 2007)

20. An infinitely long plate in the form of an area is enclosed between the line[pic][pic] of.[pic]

the temperature is zero along the edges[pic][pic] and the edge at infinity. If the edge[pic] is

kept at temperature [pic] find the steady state temperature distribution in the plate. (May 2006)

21. An infinitely long uniform plate is bounded by two parallel edges and an end at right angle to them.

The breadth of this edge[pic]is [pic], this end is maintained at temperature as [pic]at

all points while the other edges are at zero temperature. Find the temperature [pic]at any point

of the plate in the steady state. (Nov 2006).

22. A rectangular plate with insulate surface is 10cm wide and so long compared to it’s with that it may

be considered infinite in length without introducing appreciable error. The temperature at short edge

y = 0 is given by

[pic]and all the other three edges are kept at [pic]c. Find the steady state

temperature at any point in the plate. (May 2005, 2008)

23. A rectangular plate with insulated surface is 10cm wide so long compared to its width that

it may be considered infinite in the length without introducing an appreciable error. It the

temperature along one short-edge[pic] is given by [pic]. While the

two long edges[pic]and [pic]as well as the other short edges are kept at [pic]c. find the

steady state temp. for [pic]. (Nov 2003,2012)

24. A rectangular plate with insulate surface is 10cm wide and so long compared to it’s width that it may

be considered infinite in length without introducing appreciable error. The temperature at short edge

y = 0 is given by

[pic]as well as the other short edges are kept at [pic]c. Find the temperature

[pic]at any point [pic]of the plate in the steady state. (May 2009)

25. Find the solution to the equation [pic]. That satisfies the condition u(0,t)=0, u(l,t)=0, for t>0 and

u(x,0) [pic](nov 2013)

*****

UNIT-V Z – Transforms

Part – A

1. Define Z- transform. (A.U 2013)

Sol.:

If {x(n)} is a causal sequence if x(n) = 0 for n< 0, then Z transform is called one sided or unilateral

Z transform of {x(n)} and is defined as Z{x(n)} = [pic]

2. Prove that Z-transform is linear. (Or) Prove that [pic]

Sol.:

[pic]

3. Find Z{an}. (A.U 2008)

Sol.:

[pic]

4. Find Z{eat} (A.U 2010)

Sol.:

[pic]

5. Find [pic]

Sol.:

[pic]

6. Find Z-transform of ‘n’

Sol.:

[pic]

7. Find Z{n2} (A.U 2011)

Sol.:

[pic]

8. Find Z transform of 1/n.

Sol.:

[pic]

9. Find Z transforms of [pic] (A.U 2009)

Sol.:

[pic]

10. Prove that [pic](A.U 2011)

Sol.:

[pic]

[pic]

11. State and prove initial value theorem. (A.U 2010)

Sol.:

[pic]

12. If [pic]

Sol.:

[pic]

13. Find Z{nan}

Sol.:

[pic]

14. Find Z{n(n – 1)} (A.U 2010)

Sol.:

[pic]

15. Find Z-transform of ‘t’

Sol.:

[pic]

16. Prove That [pic](A.U 2008)

Sol.:

[pic]

17. Find [pic](A.U 2007)

Sol.:

[pic]

[pic]

18. Find [pic]( A.U 2011)

Sol.:

[pic]

19. Find [pic] (A.U 2010)

Sol.:

[pic]

20. Find [pic]

Sol.:

[pic][pic]

21Find Z transforms of [pic] (A.U nov 2013)

Sol.:

[pic]

PART – B Questions

1. Find [pic] using the partial fractions. (A.U 2010)

2. Solve the difference equations [pic]

where y(0) =1, Y(1)=0 (A.U 2008)

3. Prove that [pic]( A.U 2007)

4. State and prove second shifting theorem in Z- transform (A.U 2013)

5. Using convolution theorem evaluate inverse Z- transform of [pic](A.U 2009,nov 2013)

6. Solve the difference equation [pic] given that Y(0)=3 and Y(1)=-2.

7. Solve the equation [pic]given that Yo=Y1=0 (A.U 2012)

8. Find [pic] ( A.U 2013)

9. Solve [pic] given Yo=3( A.U 2007)

10. Derive the Difference equation from [pic](A.U 2009)

11. Using Z- transform solve [pic] given that Yo=3 and Y1=-5.(A.U 2010)

12. Prove that [pic]

13. State and prove final value theorem.(A.U 2008)

14. State and prove Convolution theorem. on Z transform (A.U 2008)

15. Find [pic] using Convolution theorem.(A.U 2010)

16. Find Z [pic] and Z [pic] (A.U 2010 nov 2013)

17. Derive the difference equation from [pic] (A.U 2011)

18. From [pic], derive a difference equation by eliminating the constants.

19. Derive the difference equation from [pic].(A.U 2010)

20. Solve the difference equation [pic] where y(0) = 1 and y(1) = 0.(A.U 2011)

21. Using Z- Transform, solve [pic].(A.U 2011)

22. Using Z- Transform, solve [pic]given uo=1 and u1=2 (A.U 2009,2010)

23.Using Z- Transform, solve [pic]given yo=0 =y1 (A.U nov 2013)

24. Form the difference equation fromy(n)=(A+Bn)[pic](Nov 2013)

*****

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