Integral Transforms

Paper Code: 
MAT324 A
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Understand the concept of popular and useful transformations techniques like; Laplace and inverse Laplace transform, Fourier transform, Hankel transform, Mellin transform with its properties and applications.
  2. Procure knowledge to solve ordinary and partial differential equations with different forms of initial and boundary conditions.

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

MAT 324A

 

 

 

 

Integral Transforms

 (Theory)

 

 

 

 

 

The students will be able to –

 

CO99: Gain the idea that by applying the theory of Integral transform the problem from its original domain can be mapped into a new domain where solving problems becomes easier.

CO100: Apply these techniques to solve research problems of signal processing, data analysis and processing, image processing, in scientific simulation algorithms etc.

CO101: Develop the ability of using the language of mathematics in analysing the real-world problems of sciences and engineering.

CO102: Think logically and mathematically and apply the knowledge of integral transform to solve complex problems.

CO103: Apply the transform methods to solve differential equations representing the models of specific problems.

CO104: Able to differentiate between various types of integral transforms

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
I
15.00
Laplace transform: Definition, Basic properties, Laplace transform of derivatives and integrals, Multiplication and division by power of independent variable, Evaluation of integrals by using Laplace transforms, Periodic functions, Initial-value and Final value theorem.
 
Unit II: 
II
15.00
Inverse Laplace transform: Definition, Basic properties, Inverse Laplace transform of derivatives and integrals, Multiplication and division by power of independent variable, Convolution theorem for Laplace transform, Evaluation of integrals by using inverse Laplace transform, Use of partial fractions, Heaviside expansion formula.
 
Unit III: 
III
15.00
Fourier transform: Definition and properties of Fourier complex sine, cosine, and complex transforms, Inversion theorem, Relationship between Fourier transform and Laplace transform, Modulation theorem, Convolution theorem for sine, cosine, and complex transforms, Parseval’s identity, Fourier transform of derivatives.
 
Unit IV: 
IV
15.00
Mellin transform: Definition and elementary properties, Mellin transforms of derivatives and integrals, Inversion theorem, Convolution theorem, Inverse Mellin transform of two functions, Infinite Hankel transform: Definition and elementary properties, Hankel transform of elementary function like exponential functions, Inversion formula, Hankel transform of derivatives, Basic operational property of Hankel transform, Parseval’s theorem.
 
Unit V: 
V
15.00
Solution of ordinary differential equations with constant and variable coefficients by Laplace transform, Application to the simple boundary value problem by Laplace, Fourier and infinite Hankel transforms.
 
Essential Readings: 
  • S.P. Goyal and A.K. Goyal, Integral Transforms and its Applications, Jaipur Publishing House, Jaipur, 2014.
  • D.C. Gokhroo and J.P.N. Ojha, Integral Transforms, Jaipur publishing House, Jaipur, 2000.
  • M. D. Raisinghania, Integral Transform, S. Chand & Co., New Delhi, 2013.
 
References: 
  • K. P. Gupta and J. K. Goyal, Integral Transforms, Pragati Prakashan, New Delhi, 2015.
  • Mohamed F. EL. Hewie, Laplace Transform, Createspace Independent Pub., 2013.
  • Joel L. Schiff, The Laplace Transform: Theory and Application, Springer Science & Business Media, 1999.
 
Academic Year: 
2022-2023
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