Integral Transforms

Paper Code: 
24MAT324(A)
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

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: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24MAT

324(A)

 

 

 

 

Integral Transforms

 (Theory)

 

 

 

 

 

CO101: Compute Laplace transforms of elementary functions and piecewise continuous functions.

CO102: Demonstrate inverse Laplace transforms using various techniques such as partial fraction decomposition, convolution and contour integration.

CO103: Explore the concept of Fourier transform and its properties.

CO104: Compute Mellin and Hankel transforms of special functions such as Bessel functions, gamma functions and hypergeometric functions.

CO105: Apply the  different transforms to solve ordinary and partial differential equations

CO106: Contribute effectively in course- specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Laplace transform:
15.00

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: 
Inverse Laplace transform:
15.00

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: 
Fourier transform:
15.00

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: 
Mellin transform:
15.00

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: 
Solution of ordinary differential equations by transforms:
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.

 

References: 
  • 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.
  • S.P. Goyal and A.K. Goyal, Integral Transforms and its Applications, Jaipur Publishing House, Jaipur, 2014.

 

SUGGESTED READING

  • 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.

e- RESOURCES

 

JOURNALS             

 

Academic Year: 
2024-2025
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