Advanced Partial Differential Equations

By Prof. Kaushik Bal   |   IIT Kanpur
Learners enrolled: 2097
The precise idea to study partial differential equations is to interpret physical phenomenon occurring in nature. Most often the systems encountered, fails to admit explicit solutions but fortunately qualitative methods were discovered which does provide ample information about the system without explicitly solving it. In this course we will explore the basic ideas of studying first order equations starting with the inner workings of method of characteristics followed by the three fundamental second order PDE’s namely Laplace equation, Heat equation and Wave equation.

Graduate students (MSc) and advanced undergraduate.
PREREQUISITES : A basic knowledge of several variable calculus is enough.
Course Status : Completed
Course Type : Core
Duration : 12 weeks
Start Date : 18 Jan 2021
End Date : 09 Apr 2021
Exam Date : 25 Apr 2021 IST
Enrollment Ends : 01 Feb 2021
Category :
  • Mathematics
Credit Points : 3
Level : Postgraduate

Page Visits

Course layout

Week 1-3: Module 1: First order Equations: Method of Characteristics, Transport Equation, Burgers Equation, Riemann problem, Lax-Oleinik Formulae, Entropy Solutions.
Week 4-6: Module 2: Laplace Equation: Rotational Invariance and Fundamental Solution, Green’s Function, Mean Value Theorem, Maximum Principle, Liouville theorem, Harnack inequality.
Week 7-9:  Module 3: Heat Equation: Self-Similarity and Fundamental solution, Maximum Principle, Duhamel’s Principle, Energy Method.
Week 10-12: Module 4: Wave equation: D’Alembert’s formulae, Kirchoff Techniques via reflections, Finite speed of Propogation.

Books and references

Evans, Lawrence C. Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. xviii+662 pp.
John,Fritz Partial differential equations. Fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag,New York, 1982. x+2

Instructor bio

Prof. Kaushik Bal

IIT Kanpur
I am Dr Kaushik Bal, assistant professor in the dept of Math and Stat, IIT Kanpur. I completed my PhD in 2011 under the supervision of Prof Jacques Giacomoni from UPPA, France with a specialization in elliptic and parabolic PDE’s. Currently my research interest revolves around nonlinear Schrodinger equation and nonlocal Hardy and Poincare inequalities in Sobolev space setting.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 25 April 2021 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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