Transform Calculus & Differential Equations | IIT Kharagpur Course Guide
Course Details
| Exam Registration | 209 |
|---|---|
| Course Status | Ongoing |
| Course Type | Core |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 25 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Transform Calculus and its Applications in Differential Equations: A Comprehensive Guide
For postgraduate students and professionals in mathematics, engineering, and the physical sciences, mastering Transform Calculus is a critical step towards solving complex real-world problems. This powerful branch of mathematics provides elegant and efficient techniques for tackling differential equations, which are the cornerstone of modeling dynamic systems in physics, engineering, economics, and beyond.
This article delves into a detailed 12-week postgraduate course structure, designed and taught by Prof. Adrijit Goswami of IIT Kharagpur, that systematically unravels the theory and application of integral transforms.
About the Course Instructor: Prof. Adrijit Goswami
Prof. Adrijit Goswami brings a wealth of academic and research experience to this course. Joining IIT Kharagpur in 1992, he holds an M.Sc. and Ph.D. from Jadavpur University. His diverse research portfolio spans Operations Research, Theoretical Computer Science, Data Mining, Cryptography, and Network Security. With over 90 international publications and guidance provided to 14 Ph.D. scholars, Prof. Goswami's interdisciplinary expertise ensures the course connects fundamental mathematical theory with advanced applications, preparing students for cutting-edge research and development.
Who is This Course For?
This course is primarily intended for postgraduate students in Mathematics. However, it is also highly beneficial for students and professionals from disciplines that rely heavily on mathematical modeling, including:
- Physics and Engineering (Electrical, Mechanical, Civil)
- Computer Science (particularly algorithms and scientific computing)
- Economics and Finance (for modeling dynamic systems)
- Any curriculum with an advanced course in Mathematics.
Course Overview and Objectives
Transform Calculus has become an integral part of the undergraduate and postgraduate mathematics curriculum. This course is meticulously designed to:
- Train students in fundamental Integral Transform techniques, primarily Laplace and Fourier Transforms.
- Demonstrate the direct application of these transforms in solving Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs).
- Introduce higher-level concepts to build a strong foundation for future research.
- Combine theoretical overviews with solved numerical examples for concrete understanding.
- Assess progress through weekly graded assignments.
Detailed 12-Week Course Layout
| Week | Topic | Key Focus Areas |
|---|---|---|
| 1 | Introduction to Laplace Transform | Definition, existence conditions, basic properties (linearity, shifting). |
| 2 | Laplace Transform of Derivatives and Integrals | Key theorems for transforming ODEs, solving initial value problems. |
| 3 | Laplace Transform of Special Functions | Unit step, Dirac delta, periodic functions, and convolution theorem. |
| 4 | Inverse Laplace Transform | Methods using partial fractions, convolution, and complex inversion formula. |
| 5 | Application to ODEs & Integral Equations | Solving linear ODEs with constant coefficients, Volterra integral equations. |
| 6 | Fourier Series | Representation of periodic functions, convergence theorems. |
| 7 | Introduction to Fourier Transforms | Definition (complex form), properties, relation to Fourier series. |
| 8 | Fourier Sine and Cosine Transforms | Definitions, properties, transforms of elementary functions. |
| 9 | Parseval’s Identity | Energy conservation principle for Fourier Sine and Cosine Transforms. |
| 10 | Application of Fourier Transform to ODEs | Solving ODEs on infinite domains, integral equations. |
| 11 | Application of Fourier Transform to PDEs | Solving classic PDEs (Heat, Wave, Laplace equations) on infinite domains. |
| 12 | Finite Fourier Transform | Introduction and application to Boundary Value Problems on finite intervals. |
Why Learn Transform Calculus?
Transform techniques simplify complex problems. They convert differential equations into algebraic equations, which are far easier to manipulate and solve. Once solved in the transformed domain, the inverse transform provides the solution to the original problem. This method is indispensable for:
- Engineering: Circuit analysis, control systems, signal processing.
- Physics: Quantum mechanics, heat transfer, wave propagation.
- Applied Mathematics: Solving boundary value problems, integral equations.
- Research & Development: Formulating and solving new models in emerging scientific fields.
Preparing for the Course
To succeed in this advanced course, a solid foundation in the following areas is recommended:
- Calculus (single and multivariable)
- Ordinary Differential Equations
- Basic Complex Analysis (familiarity with complex numbers and integration)
- Linear Algebra
This 12-week journey through Transform Calculus, under the guidance of an experienced educator like Prof. Goswami, offers more than just mathematical techniques. It provides a powerful toolkit for analytical thinking and problem-solving, equipping postgraduate students with the skills necessary to contribute to academic research and technological innovation. By bridging the gap between abstract theory and practical application, this course stands as a vital component in the education of any aspiring mathematician, engineer, or scientist.
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