# Anna University Coimbatore 2009 B.E Transforms and Partial Differential Equation - Model - Question Paper

**MODEL EXAMINATION**

**TRANSFORMS AND
PARTIAL DIFFERENTIAL EQUATION**

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**Year/Semester & Branch: II / III Common to all
Branches **

** Max. Marks: 100 Time: 180 min **

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** PART-A
Answer ALL Questions (20X2=40)**

1. Determine the value of in the Fourier series expansion of . (A/M 08)

2. Define Root mean square of over the range . (Tri-N/D 08)

3. If in the interval (0,4), then find the value of in the Fourier series expansion. (Cbe-N/D 08)

- State Dirichlets condition for Fourier series. (Tnl-N/D 08) (Nov 05)
- Let be the Fourier cosine transform of . Prove that . (Cbe-N/D 08)
- State Fourier integral theorem. (A/M 08) (Cbe-N/D 08)
- If then prove that (Tnl-N/D 08)
- what is the sine transform of if is the Fourier sine transform of .(Tri-N/D 08)
- Form the partial differential equation by eliminating arbitrary constants from (Cbe-N/D 08)
- Find the complete integral of (N/D 08).
- Eliminate the function f from (Tnl-N/D 08).
- Write the complete integral of (Tnl-N/D 08)
- A rod 50cm long with insulated sides has its ends A and B kept at and respectively. Find the steady state temperature distribution of the rod. (Cbe-N/D 08)
- Classify the partial differential equation . (A/M 08)
- Write the initial conditions of the wave equation if the string has an initial displacement but no initial velocity. (Tnl-N/D 08)
- List all the possible solutions of the one dimensional wave equation and state the proper solution. (Tri-N/D 08)
- Form the difference equation from . (A/M 08)
- Find (A/M 08) (Tnl-N/D 08)
- If What is ? (Tri-N/D 08)
- Find the Z- transform of . (Tri-N/D 08)

** PART-B
(Answer ANY 5 questions) (5 X 12 = 60)**

- a) Obtain the Fourier series of of period 2
*l*and defined as follows .Hence deduce (Tnl-N/D 08)

b) Find the half range sine series of in (Tnl-N/D 08)

- Find the Fourier cosine transform of . Hence prove that (Tnl-N/D 08)
- a) Solve .(Tri-N/D 08)

b) Solve .(Tri-N/D 08)

- 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 . Determine the displacement function at any time*t*.(Cbe N/D 08) - a) Find the Z-transform of and . Hence find .

b) Solve using Z-transform given .

- a) Obtain the Fourier series upto second harmonic from the data

x : 0

: 0.8 0.6 0.4 0.7 0.9 1.1 0.8 (Cbe N/D 08)

b) Find the Fourier cosine transform of . Hence deduce the value of

.(Tri-N/D 08)

- The ends A and B of a rod
*l*cm long have their temperatures kept at and , until steady state conditions prevail. The temperature at the end B is suddenly reduced to and that of A is increased to .Find temperature distribution in the rod after time*t*.(Tnl-N/D 08)

- a) Find using Convolution theorem. (Tnl-N/D 08)

b) Find the singular integral of . (Cbe N/D 08)

Earning: Approval pending. |