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Partial Differential Equations - IITB

Prof. Sivaji Ganesh · IIT Bombay

Start with lecture 1 At NPTEL

About

Partial Differential Equations (PDEs) appear as mathematical models for many a physical phenomena. Closed-form solutions to most of these PDEs cannot be found. One of the possible ways to understand the models is by studying the qualitative properties exhibited by their solutions.

In this course, we study first order nonlinear PDEs, and the properties of the three important types of second order linear PDEs (Wave, Laplace, Heat) would be studied and compared.

Intended audience: Mathematics, Physics, Mechanical Engineering, Chemical Engineering

Prerequisites: Exposure to Multivariable calculus is needed. In addition, exposure to Linear algebra would be ideal.

Lectures

Week 1

  1. Lecture 1.1: Partial Differential Equations - Basic concepts and Nomenclature YouTube
  2. Lecture 2.1: First Order Partial Differential Equations - How they arise? Cauchy Problems, IVPs, IBVPs YouTube
  3. Lecture 2.2: First order Partial Differential Equations - Geometry of Quasilinear equations YouTube
  4. Lecture 2.3: First order Partial Differential Equations - General Solutions to Linear and Semilinear equations YouTube
  5. Lecture 2.4: First order Partial Differential Equations - Lagrange's method for Quasilinear equations YouTube

Week 2

  1. Lecture 2.5: First order Partial Differential Equations - Relation between Characteristic curves and Integral surfaces for Quasilinear equations YouTube
  2. Lecture 2.6: First order Partial Differential Equations - Method of characteristics for Quasilinear equations - 1 YouTube
  3. Lecture 2.7: First order Partial Differential Equations - Method of characteristics for Quasilinear equations - 2 YouTube
  4. Lecture 2.8: First order Partial Differential Equations - Failure of transversality condition YouTube
  5. Lecture 2.9: First order Partial Differential Equations - Tutorial of Quasilinear equations YouTube

Week 3

  1. Lecture 2.10: First order Partial Differential Equations - General nonlinear equations 1 - Search for a characteristic direction YouTube
  2. Lecture 2.11: First order Partial Differential Equations - General nonlinear equations 2 - Characteristic direction and characteristic strip YouTube
  3. Lecture 2.12: First order Partial Differential Equations - General nonlinear equations 3 - Finding an initial strip YouTube
  4. Lecture 2.13: First order Partial Differential Equations - General nonlinear equations 4 - Local existence and uniqueness theorem YouTube
  5. Lecture 2.14: First order Partial Differential Equations - Tutorial on General nonlinear equations YouTube

Week 4

  1. Lecture 2.15: First order Partial Differential Equations - Initial value problems for Burgers equation YouTube
  2. Lecture 2.16: First order Partial Differential Equations - Conservation laws with a view towards global solutions to Burgers equation YouTube
  3. Lecture 3.1: Second Order Partial Differential Equations - Special Curves associated to a PDE YouTube
  4. Lecture 3.2: Second Order Partial Differential Equations - Curves of discontinuity YouTube

Week 5

  1. Lecture 3.3: Second Order Partial Differential Equations - Classification YouTube
  2. Lecture 3.4: Second Order Partial Differential Equations - Canonical form for an equation of Hyperbolic type YouTube
  3. Lecture 3.5: Second Order Partial Differential Equations -Canonical form for an equation of Parabolic type YouTube
  4. Lecture 3.6: Second Order Partial Differential Equations - Canonical form for an equation of Elliptic type YouTube
  5. Lecture 3.7: Second Order Partial Differential Equations - Characteristic Surfaces YouTube
  6. Lecture 3.8: Second Order Partial Differential Equations - Canonical forms for constant coefficient PDEs YouTube

Week 6

  1. Lecture 4.1: Wave Equation - A mathematical model for vibrating strings YouTube
  2. Lecture 4.2: Wave Equation in one space dimension - d'Alembert formula YouTube
  3. Lecture 4.3: Tutorial on One dimensional wave equation YouTube
  4. Lecture 4.4: Wave Equation in d space dimensions - Equivalent Cauchy problems via Spherical means YouTube
  5. Lecture 4.5: Cauchy problem for Wave Equation in 3 space dimensions - Poisson-Kirchhoff formulae YouTube
  6. Lecture 4.6: Cauchy problem for Wave Equation in 2 space dimensions - Hadamard's method of descent YouTube

Week 7

  1. Lecture 4.7: Nonhomogeneous Wave Equation - Duhamel principle YouTube
  2. Lecture 4.8: Wellposedness of Cauchy problem for Wave Equation YouTube
  3. Lecture 4.9: Wave Equation on an interval in ℝ - Solution to an IBVP from first principles YouTube
  4. Lecture 4.10: Tutorial on IBVPs for wave equation YouTube

Week 8

  1. Lecture 4.11: IBVP for Wave Equation - Separation of Variables Method YouTube
  2. Lecture 4.12: Tutorial on Separation of variables method for wave equation YouTube
  3. Lecture 5.1: Qualitative analysis of Wave equation - Parallelogram identity YouTube
  4. Lecture 5.2: Qualitative analysis of Wave equation - Domain of dependence, domain of influence YouTube

Week 9

  1. Lecture 5.3: Qualitative analysis of Wave equation - Causality Principle, Finite speed of propagation YouTube
  2. Lecture 5.4: Qualitative analysis of Wave equation - Uniqueness by Energy method YouTube
  3. Lecture 5.5: Qualitative analysis of Wave equation - Huygens Principle YouTube
  4. Lecture 5.6: Qualitative analysis of Wave equation - Generalized solutions to Wave equation YouTube
  5. Lecture 5.7: Qualitative analysis of Wave equation - Propagation of waves YouTube

Week 10

  1. Lecture 6:1: Laplace equation - Associated Boundary value problems YouTube
  2. Lecture 6:2: Laplace equation - Fundamental solution YouTube
  3. Lecture 6:3: Dirichlet BVP for Laplace equation - Green's function and Poisson's formula YouTube
  4. Lecture 6:4: Laplace equation -Weak maximum principle and its applications YouTube
  5. Lecture 6:5: Laplace equation - Dirichlet BVP on a disk in ℝ2 for Laplace equations YouTube
  6. Lecture 6:6: Tutorial 1 on Laplace equation YouTube

Week 11

  1. Lecture 6:7: Laplace equation - Mean value property YouTube
  2. Lecture 6:8: Laplace equation - More qualitative properties YouTube
  3. Lecture 6:9: Laplace equation - Strong Maximum Principle and Dirichlet Principle YouTube
  4. Lecture 6:10: Tutorial 2 on Laplace equation YouTube

Week 12

  1. Lecture 7.1: Cauchy Problem for Heat Equation - 1 YouTube
  2. Lecture 7.2: Cauchy Problem for Heat Equation - 2 YouTube
  3. Lecture 7.3: IBVP for Heat equation Subtitle: Method of Separation of Variables YouTube
  4. Lecture 7.4: Maximum principle for heat equation YouTube
  5. Lecture 7.5: Tutorial on heat equation YouTube
  6. Lecture 7.6: Heat equation Subheading : Infinite speed of propagation, Energy, Backward Problem YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Bombay
Department
Mathematics
Teacher
Prof. Sivaji Ganesh
Level
Graduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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