## Description of the Workshop

Integral transforms are powerful mathematical tools with a range of applications across various fields. Integral transforms like the Laplace and Fourier transforms are used in engineering, physics, and applied mathematics to solve differential equations, analyze signals, and model systems. These transforms often turn complex differential equations into simpler algebraic equations, making it easier to find solutions. This ability to simplify and solve intricate problems is very practical. In this course, we will cover both Laplace and Fourier transform and we study their applications in ordinary and partial differential equations. We will start with the basic definitions and terminologies and later on we advance ourselves to solves different types of differential equations. I will try to make the course lucid and engaging and I hope the participants of this course will get to learn the basics of this course clearly.

## About the Instructor

Dr. Hari Shankar Mahato is currently working as an Assistant Professor at the Indian Institute of Technology Kharagpur. Before joining IIT Kharagpur, He worked as a postdoc at the University of Georgia, USA. He did his PhD from the University of Bremen, Germany and then I worked as a Postdoc at the University of ErlangenNuremberg and afterwards at the Technical University of Dortmund, both located in Germany. His research expertise are Partial Differential Equations, Applied Analysis, Variational Methods, Control Theory, Homogenization Theory and very recently have started working on Mathematical Biology. He can teach (both online and offline) any undergraduate courses from pre to advanced calculus, mechanics, ordinary differential equations, up to advanced graduate courses like linear and nonlinear PDEs, functional analysis, topology, mathematical modeling, fluid mechanics, homogenisation theory, control theory and data science.

## Module Details

#### Day 1 :

#### Module Name : Laplace Transform

##### Concepts Covered:

##### Laplace Transform :

- Definition of Laplace Transform, Linearity Property, Conditions for Existence of Laplace Transform.
- First and Second shifting properties, Laplace Transform of derivatives and Integrals, Unit Step functions, Dirac delta-function, Error Function.
- Differentiation and Integration of Transforms, Convolution Theorem, Inversion, Periodic functions.
- Evaluation of integrals by Laplace Transform.
- Solution of initial and boundary value problems.

#### Day 2:

#### Module Name : Fourier Series & Fourier Transform

##### Concepts Covered :

##### Fourier Series :

Periodic functions, Fourier series representation of a function, half range series, sine and cosine series, Fourier integral formula, Parseval’s identity.

##### Fourier Transform:

- Fourier Transform, Fourier sine and cosine transforms.
- Linearity, scaling, frequency shifting and time shifting properties.
- Self reciprocity of Fourier Transform, convolution theorem.
- Applications to boundary value problems.

## Eligibility Criteria & Intended Audience

At least 1 year in Undergraduate studies with exposure to Mathematics courses such as Differential and Integral Calculus, Differential Equations.

Intended Audience : 2nd year students of BTech and BSc.

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