# Partial Differential Equations - IITB

- Institution: IIT Bombay
- Department: Mathematics
- Teacher: Prof. Sivaji Ganesh
- Level: Graduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111101153)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111101153)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

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

1. [Lecture 1.1: Partial Differential Equations - Basic concepts and Nomenclature](https://invisiblenetwork.ai/learn/nptel/111101153/6d547bf940b5) (YouTube)
2. [Lecture 2.1: First Order Partial Differential Equations - How they arise? Cauchy Problems, IVPs, IBVPs](https://invisiblenetwork.ai/learn/nptel/111101153/992663136167) (YouTube)
3. [Lecture 2.2: First order Partial Differential Equations - Geometry of Quasilinear equations](https://invisiblenetwork.ai/learn/nptel/111101153/611e25ffc0b2) (YouTube)
4. [Lecture 2.3: First order Partial Differential Equations - General Solutions to Linear and Semilinear equations](https://invisiblenetwork.ai/learn/nptel/111101153/3112588c9064) (YouTube)
5. [Lecture 2.4: First order Partial Differential Equations - Lagrange's method for Quasilinear equations](https://invisiblenetwork.ai/learn/nptel/111101153/e569228e8307) (YouTube)
6. [Lecture 2.5: First order Partial Differential Equations - Relation between Characteristic curves and Integral surfaces for Quasilinear equations](https://invisiblenetwork.ai/learn/nptel/111101153/5a1ff831549f) (YouTube)
7. [Lecture 2.6: First order Partial Differential Equations - Method of characteristics for Quasilinear equations - 1](https://invisiblenetwork.ai/learn/nptel/111101153/118e1150227c) (YouTube)
8. [Lecture 2.7: First order Partial Differential Equations - Method of characteristics for Quasilinear equations - 2](https://invisiblenetwork.ai/learn/nptel/111101153/abd68fa470ed) (YouTube)
9. [Lecture 2.8: First order Partial Differential Equations - Failure of transversality condition](https://invisiblenetwork.ai/learn/nptel/111101153/a61cfc9bfe38) (YouTube)
10. [Lecture 2.9: First order Partial Differential Equations - Tutorial of Quasilinear equations](https://invisiblenetwork.ai/learn/nptel/111101153/97fa6428a9e4) (YouTube)
11. [Lecture 2.10: First order Partial Differential Equations - General nonlinear equations 1 - Search for a characteristic direction](https://invisiblenetwork.ai/learn/nptel/111101153/d25640a70590) (YouTube)
12. [Lecture 2.11: First order Partial Differential Equations - General nonlinear equations 2 - Characteristic direction and characteristic strip](https://invisiblenetwork.ai/learn/nptel/111101153/54ba3cc130c6) (YouTube)
13. [Lecture 2.12: First order Partial Differential Equations - General nonlinear equations 3 - Finding an initial strip](https://invisiblenetwork.ai/learn/nptel/111101153/1f7357be181c) (YouTube)
14. [Lecture 2.13: First order Partial Differential Equations - General nonlinear equations 4 - Local existence and uniqueness theorem](https://invisiblenetwork.ai/learn/nptel/111101153/7c9013202d6c) (YouTube)
15. [Lecture 2.14: First order Partial Differential Equations - Tutorial on General nonlinear equations](https://invisiblenetwork.ai/learn/nptel/111101153/8ebe4aed232f) (YouTube)
16. [Lecture 2.15: First order Partial Differential Equations - Initial value problems for Burgers equation](https://invisiblenetwork.ai/learn/nptel/111101153/f29efb202bb0) (YouTube)
17. [Lecture 2.16: First order Partial Differential Equations - Conservation laws with a view towards global solutions to Burgers equation](https://invisiblenetwork.ai/learn/nptel/111101153/51d435743515) (YouTube)
18. [Lecture 3.1: Second Order Partial Differential Equations - Special Curves associated to a PDE](https://invisiblenetwork.ai/learn/nptel/111101153/270f239a6126) (YouTube)
19. [Lecture 3.2: Second Order Partial Differential Equations - Curves of discontinuity](https://invisiblenetwork.ai/learn/nptel/111101153/7c3778e01502) (YouTube)
20. [Lecture 3.3: Second Order Partial Differential Equations - Classification](https://invisiblenetwork.ai/learn/nptel/111101153/4a2efa1b876c) (YouTube)
21. [Lecture 3.4: Second Order Partial Differential Equations - Canonical form for an equation of Hyperbolic type](https://invisiblenetwork.ai/learn/nptel/111101153/5e4c8889aa7d) (YouTube)
22. [Lecture 3.5: Second Order Partial Differential Equations -Canonical form for an equation of Parabolic type](https://invisiblenetwork.ai/learn/nptel/111101153/960fbd504b44) (YouTube)
23. [Lecture 3.6: Second Order Partial Differential Equations - Canonical form for an equation of Elliptic type](https://invisiblenetwork.ai/learn/nptel/111101153/33295381bf0a) (YouTube)
24. [Lecture 3.7: Second Order Partial Differential Equations - Characteristic Surfaces](https://invisiblenetwork.ai/learn/nptel/111101153/4ade36192eb4) (YouTube)
25. [Lecture 3.8: Second Order Partial Differential Equations - Canonical forms for constant coefficient PDEs](https://invisiblenetwork.ai/learn/nptel/111101153/b612445e79ec) (YouTube)
26. [Lecture 4.1: Wave Equation - A mathematical model for vibrating strings](https://invisiblenetwork.ai/learn/nptel/111101153/3e127e18dd94) (YouTube)
27. [Lecture 4.2: Wave Equation in one space dimension - d'Alembert formula](https://invisiblenetwork.ai/learn/nptel/111101153/9410906f7b04) (YouTube)
28. [Lecture 4.3: Tutorial on One dimensional wave equation](https://invisiblenetwork.ai/learn/nptel/111101153/c5d6d7a3e1c3) (YouTube)
29. [Lecture 4.4: Wave Equation in d space dimensions - Equivalent Cauchy problems via Spherical means](https://invisiblenetwork.ai/learn/nptel/111101153/59104e1f2260) (YouTube)
30. [Lecture 4.5: Cauchy problem for Wave Equation in 3 space dimensions - Poisson-Kirchhoff formulae](https://invisiblenetwork.ai/learn/nptel/111101153/862f4898c8c1) (YouTube)
31. [Lecture 4.6: Cauchy problem for Wave Equation in 2 space dimensions - Hadamard's method of descent](https://invisiblenetwork.ai/learn/nptel/111101153/7cfb2cf598de) (YouTube)
32. [Lecture 4.7: Nonhomogeneous Wave Equation - Duhamel principle](https://invisiblenetwork.ai/learn/nptel/111101153/aa6a1b5fc47c) (YouTube)
33. [Lecture 4.8: Wellposedness of Cauchy problem for Wave Equation](https://invisiblenetwork.ai/learn/nptel/111101153/00dab8cad724) (YouTube)
34. [Lecture 4.9: Wave Equation on an interval in ℝ - Solution to an IBVP from first principles](https://invisiblenetwork.ai/learn/nptel/111101153/5dc5c30e0a9e) (YouTube)
35. [Lecture 4.10: Tutorial on IBVPs for wave equation](https://invisiblenetwork.ai/learn/nptel/111101153/3f6ac07e2e6c) (YouTube)
36. [Lecture 4.11: IBVP for Wave Equation - Separation of Variables Method](https://invisiblenetwork.ai/learn/nptel/111101153/6b5d6dcc2ece) (YouTube)
37. [Lecture 4.12: Tutorial on Separation of variables method for wave equation](https://invisiblenetwork.ai/learn/nptel/111101153/e839bd6ac90e) (YouTube)
38. [Lecture 5.1: Qualitative analysis of Wave equation - Parallelogram identity](https://invisiblenetwork.ai/learn/nptel/111101153/19f6a98c636f) (YouTube)
39. [Lecture 5.2: Qualitative analysis of Wave equation - Domain of dependence, domain of influence](https://invisiblenetwork.ai/learn/nptel/111101153/5b3c2fa20f2d) (YouTube)
40. [Lecture 5.3: Qualitative analysis of Wave equation - Causality Principle, Finite speed of propagation](https://invisiblenetwork.ai/learn/nptel/111101153/b2bfe5d2e950) (YouTube)
41. [Lecture 5.4: Qualitative analysis of Wave equation - Uniqueness by Energy method](https://invisiblenetwork.ai/learn/nptel/111101153/60574c3e7964) (YouTube)
42. [Lecture 5.5: Qualitative analysis of Wave equation - Huygens Principle](https://invisiblenetwork.ai/learn/nptel/111101153/c2408ca5a299) (YouTube)
43. [Lecture 5.6: Qualitative analysis of Wave equation - Generalized solutions to Wave equation](https://invisiblenetwork.ai/learn/nptel/111101153/9bce3a3b1b02) (YouTube)
44. [Lecture 5.7: Qualitative analysis of Wave equation - Propagation of waves](https://invisiblenetwork.ai/learn/nptel/111101153/d7cdf19ba73b) (YouTube)
45. [Lecture 6:1: Laplace equation - Associated Boundary value problems](https://invisiblenetwork.ai/learn/nptel/111101153/980091985c9d) (YouTube)
46. [Lecture 6:2: Laplace equation - Fundamental solution](https://invisiblenetwork.ai/learn/nptel/111101153/f40603b3e462) (YouTube)
47. [Lecture 6:3: Dirichlet BVP for Laplace equation - Green's function and Poisson's formula](https://invisiblenetwork.ai/learn/nptel/111101153/139cf5b31295) (YouTube)
48. [Lecture 6:4: Laplace equation -Weak maximum principle and its applications](https://invisiblenetwork.ai/learn/nptel/111101153/e8a46f92d8ad) (YouTube)
49. [Lecture 6:5: Laplace equation - Dirichlet BVP on a disk in ℝ2 for Laplace equations](https://invisiblenetwork.ai/learn/nptel/111101153/2707ead493b1) (YouTube)
50. [Lecture 6:6: Tutorial 1 on Laplace equation](https://invisiblenetwork.ai/learn/nptel/111101153/04767003c801) (YouTube)
51. [Lecture 6:7: Laplace equation - Mean value property](https://invisiblenetwork.ai/learn/nptel/111101153/efc0e6c33d15) (YouTube)
52. [Lecture 6:8: Laplace equation - More qualitative properties](https://invisiblenetwork.ai/learn/nptel/111101153/1670e8b1a7ad) (YouTube)
53. [Lecture 6:9: Laplace equation - Strong Maximum Principle and Dirichlet Principle](https://invisiblenetwork.ai/learn/nptel/111101153/c8753b49ea33) (YouTube)
54. [Lecture 6:10: Tutorial 2 on Laplace equation](https://invisiblenetwork.ai/learn/nptel/111101153/f86b8301296b) (YouTube)
55. [Lecture 7.1: Cauchy Problem for Heat Equation - 1](https://invisiblenetwork.ai/learn/nptel/111101153/00a57edfb4ef) (YouTube)
56. [Lecture 7.2: Cauchy Problem for Heat Equation - 2](https://invisiblenetwork.ai/learn/nptel/111101153/c0e73dddb577) (YouTube)
57. [Lecture 7.3: IBVP for Heat equation Subtitle: Method of Separation of Variables](https://invisiblenetwork.ai/learn/nptel/111101153/9453c2ed323b) (YouTube)
58. [Lecture 7.4: Maximum principle for heat equation](https://invisiblenetwork.ai/learn/nptel/111101153/32b8ec2abfde) (YouTube)
59. [Lecture 7.5: Tutorial on heat equation](https://invisiblenetwork.ai/learn/nptel/111101153/15c308e069ca) (YouTube)
60. [Lecture 7.6: Heat equation Subheading : Infinite speed of propagation, Energy, Backward Problem](https://invisiblenetwork.ai/learn/nptel/111101153/b709fcf654d6) (YouTube)