NPTEL PDE Course II: Elliptic & Evolution Equations | Prof. Nandakumaran
Course Details
| Exam Registration | 97 |
|---|---|
| Course Status | Ongoing |
| Course Type | Core |
| Language | English |
| Duration | 8 weeks |
| Categories | Mathematics |
| Credit Points | 2 |
| Level | Postgraduate |
| Start Date | 16 Feb 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 16 Feb 2026 |
| Exam Registration Ends | 27 Feb 2026 |
| Exam Date | 24 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Master Advanced PDEs: Enroll in the NPTEL Course on Partial Differential Equations - II
Building on the foundational principles established in its predecessor, the Advanced Course on Partial Differential Equations – II offers a rigorous 8-week journey into the heart of modern PDE theory. Designed for postgraduate students and professionals, this course delves into two critical areas: the in-depth analysis of elliptic equations and the powerful abstract framework of semi-group theory for evolution equations.
Course Overview and Instructor Excellence
This intensive program, spanning 20 hours of expert instruction, is led by distinguished academics Prof. A. K. Nandakumaran from the Indian Institute of Science (IISc) Bangalore and Prof. P.S. Datti, former faculty at TIFR-CAM, Bangalore. Their combined expertise ensures a learning experience grounded in both deep theoretical understanding and practical application.
Who Should Take This Course?
INTENDED AUDIENCE: This course is tailored for postgraduate students and researchers in Mathematics, Physics, and Engineering disciplines where PDEs are paramount—such as fluid dynamics, quantum mechanics, financial mathematics, and materials science.
PREREQUISITES: A solid foundation is required, including:
- Measure and Integration Theory
- Functional Analysis
- Basic Semi-group Theory
- Fundamentals of PDEs (as covered in Advanced Course on Partial Differential Equations – I)
Detailed 8-Week Course Layout
The curriculum is strategically divided into two core modules, each designed to build sophisticated problem-solving skills.
Module 1: Deep Dive into Elliptic Equations (Weeks 1-4)
This module transitions from classical to modern analysis of elliptic PDEs.
- Week 1: Introduction to weak (generalized) solutions for boundary value problems, focusing on existence and uniqueness theorems.
- Week 2: Exploration of eigenvalues and eigenfunctions, introducing the variational approach via the Rayleigh-Ritz quotient.
- Week 3: Analysis of the asymptotic behavior of eigenvalues and techniques for handling non-homogeneous boundary values.
- Week 4: Critical regularity results that establish the smoothness and differentiability of weak solutions under appropriate conditions.
Module 2: Evolution Equations via Semi-Group Theory (Weeks 5-8)
This module provides a unified abstract framework for time-dependent PDEs.
- Week 5: Abstract formulation of evolution equations. Introduction to semi-group theory and the pivotal Hille-Yosida theorem (key statements).
- Week 6: Application of the theory to classical parabolic (e.g., heat equation) and hyperbolic (e.g., wave equation) PDEs.
- Week 7: Focus on the Schrödinger equation, a cornerstone of quantum mechanics.
- Week 8: Advanced topics including perturbation results and analysis of the Schrödinger equation with a potential.
Essential Reference Materials
The course draws from a premier selection of textbooks to provide comprehensive coverage:
| Book Title | Author(s) | Focus Area |
|---|---|---|
| Topics in Functional Analysis and Applications | S. Kesavan | Functional Analysis |
| A Guide to Distribution Theory & Fourier Transforms | Robert S. Strichartz | Distributions, Fourier Analysis |
| Functional Analysis, Sobolev Spaces and PDEs | Haim Brezis | Sobolev Spaces, PDE Theory |
| Sobolev Spaces | Robert A. Adams & John J.F. Fournier | In-depth Sobolev Space Theory |
| Partial Differential Equations | L. C. Evans | Comprehensive PDE Reference |
| Partial Differential Equations – Methods & Applications | R. C. McOwen | PDE Methods |
| Partial Differential Equations: Classical Theory with a Modern Touch | A.K. Nandakumaran & P.S. Datti | Primary Course Text (Cambridge-IISc Series) |
| Ordinary Differential Equations – Principles and Applications | A.K. Nandakumaran, P.S. Datti & Raju K George | ODE Foundation (Cambridge-IISc Series) |
Why Enroll in This Advanced PDE Course?
This course is more than a series of lectures; it's a gateway to advanced research and application in mathematical sciences. You will gain:
- A rigorous understanding of weak solutions and regularity theory for elliptic PDEs.
- The ability to frame and solve evolution equations using the powerful, unifying language of operator semi-groups.
- Direct learning from IISc faculty through the acclaimed NPTEL platform.
- Preparation for cutting-edge research in pure and applied mathematics, theoretical physics, and advanced engineering.
Ready to advance your expertise in Partial Differential Equations? Explore the course links and enroll today to master the analytical tools that define modern mathematical analysis.
Enroll Now →