Anna University, Chennai

Department of Electrical and Electronics Engineering

Subject Code: MA8353

Subject Name: Transforms and Partial Differential Equations

Question Bank – All Units

PDF Format

Number of Pages: 61

Unit I: Fourier Series

Explain the convergence of Fourier series with suitable diagrams. (8 marks)

Find the Fourier series expansion of the function

�

(

�

)

=

�

f(x)=x defined in

[

−

�

,

�

]

[−π,π]. (10 marks)

Prove the following properties of Fourier series:

a. Linearity property

b. Symmetry property

c. Differentiation property (12 marks)

Discuss the Fourier series representation of even and odd functions. (8 marks)

Unit II: Fourier Transforms

Derive the expression for Fourier transform pair for a rectangular pulse and sketch its frequency spectrum. (10 marks)

State and prove the convolution theorem for Fourier transforms. (8 marks)

Find the Fourier transform of the function

�

(

�

)

=

�

−

�

�

�

(

�

)

f(t)=e

−at

u(t) using differentiation property. (10 marks)

Unit III: Laplace Transforms

Explain the properties of Laplace transforms with suitable examples. (10 marks)

Solve the following differential equation using Laplace transform method:

�

′

′

−

3

�

′

+

2

�

=

0

,

�

(

0

)

=

1

,

�

′

(

0

)

=

0

y

′′

−3y

′

+2y=0,y(0)=1,y

′

(0)=0 (12 marks)

Discuss the inverse Laplace transform using partial fraction method. (8 marks)

Unit IV: Z-Transforms

Define and explain the properties of Z-transforms with examples. (10 marks)

Find the Z-transform of the sequence

�

(

�

)

=

{

1

,

2

,

3

,

4

,

…

}

x(n)={1,2,3,4,…} using the definition. (8 marks)

Discuss the region of convergence (ROC) for Z-transform. (8 marks)

Unit V: Partial Differential Equations

Solve the wave equation

�

�

�

=

�

2

�

�

�

u

tt

=c

2

u

xx

subject to the boundary conditions

�

(

0

,

�

)

=

0

u(0,t)=0 and

�

(

1

,

�

)

=

0

u(1,t)=0, with initial conditions

�

(

�

,

0

)

=

sin

(

�

�

)

u(x,0)=sin(πx) and

�

�

(

�

,

0

)

=

0

u

t

(x,0)=0. (12 marks)

Derive the general solution of the heat equation

�

�

=

�

2

�

�

�

u

t

=α

2

u

xx

using separation of variables method. (10 marks)

Explain the concept of characteristics in solving first-order partial differential equations. (8 marks)

This comprehensive question bank covers all the essential topics of Transforms and Partial Differential Equations as per the syllabus of Anna University, Chennai. Each question is designed to assess the students' understanding and application of the theoretical concepts as well as their problem-solving skills. The detailed solutions to these questions will aid students in their preparation for examinations and enhance their proficiency in the subject matter.

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