INTEGRAL TRANSFORM (Optional Paper)

Paper Code: 
MAT 324A
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. To expose the students with 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. To solve ordinary and partial differential equations with different forms of initial and boundary conditions.

Learning Outcomes

Learning and teaching strategies

Assessment

After the completion of the course the students will be able to:

CLO76- 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.

CLO77-  Apply these techniques to solve research problems of signal processing, data analysis and processing, image processing, in scientific simulation algorithms etc.

CLO78- Develop the ability of using the language of mathematics in analysing the real-world problems of sciences and engineering.

CLO79-  Think logically and mathematically and apply the knowledge of integral transform to solve complex problems.

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching

Learning activities for the students:

Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical

 

 

 

 

Presentations by Individual Students.

Class Tests at the end of each unit.

Written assignment(s)

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 ofHankel 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: 
  1. S.P. Goyal, A.K. Goyal, Integral Transforms and its Applications, Jaipur Publishing House, Jaipur, 2014.
  2. D.C. Gokhroo,J.P.N. Ojha, Integral Transforms, Jaipur publishing House, Jaipur, 2000.
  3. M.D.Raisinghania, Integral Transform, S.Chand & Co., New Delhi,2013.
 
       
References: 
  1. K P Gupta, J K Goyal, Integral Transforms, Pragati Prakashan, New Delhi,2015
  2. Mohamed F. EL. Hewie, Laplace Transform,Createspace Independent Pub., 2013.
  3. Joel L.Schiff, The Laplace Transform: Theory and Application, Springer Science & Business Media, 1999.
  4. John Miles, Integral Transform in Applied Mathematics, Cambridge University Press, 1971.
 
Academic Year: 
2020-2021
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