Transform and Boundary Value Problems / U20MABT03

Transform and Boundary Value Problems / U20MABT03

UNIT - 1

PART - A

1. Write down the Fourier series formula and define Fourier coefficients in the interval (0,2L)

2.Find a0 and an , if f(x)=x, - l < x < l.

3.Find bn, if f(x) = x2 in -l < x< l

4.Find a0 and an , if f(x) = x2 in - 𝝅 <x< 𝝅

5.find bn , if f(x) = x2 in - 𝝅<x< 𝝅

6.Define the Half range cosine series and sine series in the interval (0,l)

7.Define the Half range cosine series and sine series in the interval(0, 𝝅)

8.Write down the Parseval’s identity formula for sine and cosine series in the interval (𝟎, 𝝅).

9.Define Harmonic Analysis

10.Write down Root Mean Square Value formula.

UNIT - 1

PART - B

12. Find $b_{n}$ for $f (x) = x (2 L - x)$ in $(0, 2 L)$ :

14.Fourier series formula and $a_{0}$ for $f (x) = x - x^{2}$ , $(0, 2 π)$ :

16. Find $b_{n}$ for $f (x) = x$ in $(- L, L)$ :

17. Find the Half-Range Cosine Series for

$f (x) = x$ in $(0, L)$ :

UNIT - 1

PART - C

21 . a. Find the Fourier series for the function 𝑓(𝑥) = 𝑥2 + 𝑥 in (0,2𝑙).

21. b Find the Fourier series for the function in 𝑓(𝑥) = 𝑥(2𝑙 − 𝑥),i𝑛 (0,2𝑙). And hence deduce that the sum of the series 1/12 + 1/32 +1/52 +. ..

22. a. Find the Fourier series for the function in 𝑓(𝑥) = (𝜋 − 𝑥)2 𝑖𝑛 (0,2𝜋) and hence deduce that the sum of the series 1/12 + 1/32 + I Ap 1/52 +. ..

22.B. Find the fourier series for the function 𝑓(𝑥) = 𝑥(2𝜋 − 𝑥), 𝑖𝑛 (0,2𝜋), and hence deduce that the sum of the series 1/12 + 1/22 +1/32 +. ..

23.AFind the Fourier series for the function 𝑓(𝑥) = 𝑥2 + 𝑥 in (−𝑙, 𝑙).

24.A Find the Half range sine series for the function 𝑓(𝑥) = 𝑎, 𝑖𝑛 (0, 𝑙) and deduce that the sum of the series 1/12 + 1/32 +1/52 +. ..

24.b.Find the Half range cosine series for the function 𝑓(𝑥) = 𝑥2 𝑖𝑛 (0, 𝜋) and deduce that the sum of the series 1/14 + 1/24 + 34 +. ..= TT4/90

25.aFind the first two harmonic of the Fourier series of 𝑓(𝑥) given by the following table

UNIT - 2

PART - A

1.Form a partial differential equation by eliminating arbitrary constants from𝑍 = 𝑎𝑥 + 𝑏𝑦 + 𝑎2 + 𝑏2.

2.Form the Partial differential equation by eliminating the arbitrary constants ‘a’ and ‘b’ from 𝑍 = (𝑥2 + 𝑎2)(𝑦2 + 𝑏2).

3.Form the Partial differential equation by eliminating the arbitrary functions 𝑍 = 𝑓(𝑥/

𝑦).

4.Form the Partial differential equation by eliminating the arbitrary functions 𝑍 = 𝑓(𝑥2 + 𝑦2).

5.Find the general solution of 𝜕2𝑧 /𝜕𝑦2 = 𝑠𝑖𝑛 𝑥.

6.Find the general solution of 𝜕2𝑧 /𝜕𝑥2 = 0.

7.Solve (2𝐷2 + 7𝐷𝐷′ + 5𝐷′2 )𝑧 = 0.

8.Solve (𝐷2 − 7𝐷𝐷′ + 6𝐷′2 )𝑧 = 0.

9.Find the particular integral of (𝐷3 − 3𝐷2𝐷′ + 4𝐷′3)𝑧 = 𝑒𝑥+2𝑦

10.Find the particular integral of (𝐷2 + 𝐷𝐷′ − 6𝐷′2 )𝑧 = cos(3x+2y).

UNIT - 2

PART - B

11.Form the PDE by eliminating the functions from 𝑍 = 𝑓(2𝑥 + 𝑦) + 𝑔(3𝑥 − 𝑦).

12.Form the PDE by eliminating the arbitrary function ∅ from 𝜑(𝑥2 + 𝑦2 + 𝑧2 , 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧) = 0

13.Form the PDE by eliminating the functions from 𝑍 = 𝑓(2𝑥 + 3𝑦)𝑔(3𝑥 + 2𝑦)

14.Form the PDE by eliminating the arbitrary function ∅ from ∅(𝑥2 + 𝑦2 + 𝑧2 , 𝑥 + 𝑦 + 𝑧) = 0.

15.Solve p x + q y = z.

16.Find the Complete integral of the PDE 𝑍 = 𝑝𝑥 + 𝑞𝑦 + 𝑞2

17.Solve (𝐷2 + 𝐷𝐷′ − 6𝐷′2 )𝑧 = 𝑥 + 𝑦.

18.Obtain the general solution of p.d.e pzx + qzy = xy.

19.Solve (𝐷2 + 2𝐷𝐷′ + 𝐷′2 )𝑧 = 𝑒𝑥−𝑦

20.Solve :(𝐷2 − 3𝐷𝐷′ + 2𝐷′2 )𝑧 = sin(𝑥 − 2𝑦).

UNIT - 2

PART - C

21 .a.Solve (𝑚𝑧 − 𝑛𝑦)𝑝 + (𝑛𝑥 − 𝑙𝑧)𝑞 = 𝑙𝑦 − 𝑚𝑥.

22.a Solve (3𝑧 − 4𝑦)𝑝 + (4𝑥 − 2𝑧)𝑞 = 2𝑦 − 3𝑥

23.a.Solve (𝐷2− 𝐷𝐷′− 3𝐷𝐷′2 )𝑧 = 𝑒6𝑥+𝑦 + xy.

24.A.Solve (D2- 2DD’) z = x3y + 𝑒2𝑥

UNIT - 3

PART - A

1.Write the one-dimensional wave equation.

2.What are the various possible solutions of the one-dimensional wave equation?

3.Write the correct solution of the one-dimensional wave equation.

4.Write the one-dimensional heat equation.

5.What are the various possible solutions of the one-dimensional heat equation?

6.Write the correct solution of the one-dimensional heat equation.

7.State Fourier’s law of heat conduction.

8.Write the two-dimensional heat equation.

9.What are the various possible solutions of the two-dimensional heat equation?

10.Write the correct solution of the two-dimensional heat equation.

UNIT - 3

PART - B

11.A rod 40 cm long has its end A and B kept at and respectively, until steady state conditions prevail. Determine the temperature at steady state.

12.A rod 15 cm long has its end A and B kept at 30°C and 60°C respectively, until steady state conditions prevail. Determine the temperature at steady state

13.A rod 30 cm long has its end A and B kept at 40°C and 60°C respectively, until steady state conditions prevail. Determine the temperature at steady state.

14.A rod 30 cm long has its end A and B kept at 20°C and 80°C respectively, until steady state conditions prevail. Determine the temperature at steady state.

15.Derive the solution of one-dimensional heat flow equation in steady state condition.

16.Classify the partial differential equation 𝑈𝑥𝑥− 2𝑥𝑈𝑥𝑦 + 𝑥2𝑈𝑦𝑦− 2𝑈𝑦 = 0.

17.Classify the partial differential equation 2𝑈𝑥𝑥 + 𝑈𝑥𝑦− 2𝑈𝑦𝑦 = 0

18.Classify the partial differential equation 𝑦2𝑈𝑥𝑥 + 𝑥2𝑈𝑦𝑦 = 0.

19.Classify the partial differential equation 4𝑈𝑥𝑥− 12𝑈𝑥𝑦 + 9𝑈𝑦𝑦 = 0.

20.Write the three cases in classification of P.D.E

UNIT - 3

PART - C

21.a A string is stretched and fastened to two points 𝑥 = 0 and 𝑥 = 𝑙 apart. Motion is started by displacing the string into the form y=k(lx - x2)and then released it from this position at time t = 0. Find the displacement of the point of the string at a distance of x from one end at time t.

24.b.A square plate is bounded by the lines 𝑥 = 0, 𝑦 = 0 and 𝑥 = 𝑦 = 10, its faces are insulated.The temperature along upper horizontal line is given by u(𝑥, 10)= 𝑥(10 − 𝑥), when 0 < 𝑥 < 10. While other three edges are kept at 00C. Find steady state temperature in the plate.

25.A.A square plate is bounded by the lines 𝑥 = 0, 𝑦 = 0 and 𝑥 = 𝑦 = 20, its faces are insulated.The temperature along upper horizontal line is given by u(𝑥, 20)= 𝑥(20 − 𝑥), when 0 < 𝑥 < 20. While other three edges are kept at 00C. Find steady state temperature in the plate.

UNIT - 4

PART - A

1. State the Fourier Integral Theorem.