Partial Differential Equations: Comprehensive Course

Course Curriculum

Chapter 1: Fourier Transform and its inverse

Lecture 1: Fourier series

Lecture 2: Fourier Transforms

Lecture 3: How to interpret improper integrals of sinusoids

Lecture 4: Dirac delta

Lecture 5: Multiple Fourier Transforms

Lecture 6: Why the Dirac delta helps derive the Inverse Fourier Transform

Chapter 2: Solution of a PDE equation

Lecture 1: Gradient and Laplacian: quick summary

Lecture 2: Example of pde

Lecture 3: Solution to the pde part 1

Lecture 4: Solution to the pde part 2

Lecture 5: Solution to the pde part 3

Chapter 3: Some more physics behind the pde

Lecture 1: Physics behind the equation part 1

Lecture 2: Physics behind the equation part 2

Chapter 4: Solving the Diffusion/Heat equation by Fourier Tranform

Lecture 1: Setup of the diffusion problem

Lecture 2: Integral equation satisfied by the function f(x,t)

Lecture 3: Diffusion equation

Lecture 4: Some possible boundary conditions of the diffusion equation

Lecture 5: Solution of the diffusion equation part 1

Lecture 6: Solution of the diffusion equation part 2

Lecture 7: Solution of the diffusion equation part 3

Lecture 8: Solution of the diffusion equation part 4

Chapter 5: 2nd order ODE solved via Fourier Transform

Lecture 1: 2nd order non-homogeneous ODE solved via Fourier Transform

Chapter 6: PDE solved with the method of characteristics

Lecture 1: Non linear first order PDE solved with the method of characteristics

Chapter 7: Heat equation solution via Separation of Variables

Lecture 1: Separation of variables to solve the heat equation (part 1)

Lecture 2: Separation of variables to solve the heat equation (part 2)

Lecture 3: Separation of variables to solve the heat equation (part 3)

Chapter 8: Laplace Equation solved via the method of Separation of Variables

Lecture 1: Laplace Equation in Cartesian Coordinates (exercise)

Lecture 2: Laplace Equation in Polar coordinates (exercise 1)

Lecture 3: Laplace Equation in Polar coordinates (exercise 2)

Lecture 4: Laplace Equation in Polar coordinates (exercise 3)

Lecture 5: Laplace Equation in Polar coordinates (exercise 4)

Lecture 6: Concept of streamlines (with exercise)

Chapter 9: Nonhomogeneous Heat Equation

Lecture 1: Nonhomogeneous Heat Equation: Exercise 1

Lecture 2: Nonhomogeneous Heat Equation: Exercise 2

Lecture 3: Nonhomogeneous Heat Equation: Exercise 3

Chapter 10: Wave Equation (Exercises)

Lecture 1: Nonhomogeneous Wave Equation (Exercise 1)

Lecture 2: Nonhomogeneous Wave Equation: D'Alambert formula

Lecture 3: Solving a wave equation using D'Alambert formula (exercise)

Lecture 4: Energy conservation law for the wave equation

Chapter 11: Bi-dimensional problems (heat and wave equation)

Lecture 1: Bi-dimensional heat equation: exercise 1

Lecture 2: Bi-dimensional heat equation: exercise 2

Lecture 3: Bi-dimensional wave equation: exercise 1

Chapter 12: Black Scholes equation

Lecture 1: Mathematical Derivation of the Black-Scholes equation

Chapter 13: Derivation of the Navier-Stokes equations and their solution in a 2D case

Lecture 1: Mathematical derivation of Navier Stokes equations part 1

Lecture 2: Mathematical derivation of Navier Stokes equations part 2

Chapter 14: How Einstein mastered Navier-Stokes equations in his PhD dissertation

Lecture 1: How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 1

Lecture 2: How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 2

Lecture 3: How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 3

Lecture 4: How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 4

Lecture 5: How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 5

Lecture 6: How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 6

Chapter 15: Stokes law obtained from Navier-Stokes equations

Lecture 1: derivation of Stokes law from Navier Stokes part 1

Lecture 2: derivation of Stokes law from Navier Stokes part 2

Chapter 16: Appendix on PDE's

Lecture 1: Derivation of the incompressible fluid equation

Chapter 17: Bonus section: Introduction to the Heisenberg Uncertainty Principle

Lecture 1: Mathematical summary of how to prove the uncertainty principle

Lecture 2: Introduction to the short course on the Heisenberg Uncertainty Principle

Lecture 3: Probability that a particle exists at a certain time

Lecture 4: Probability that a particle has a certain_energy

Lecture 5: Uncertainty in the localization in time and in the energy of the particle

Chapter 18: Bonus Section: Uncertainty Principle derivation

Lecture 1: Derivation of the uncertainty principle part 1

Lecture 2: Derivation of the uncertainty principle part 2

Lecture 3: Derivation of the uncertainty principle part 3

Chapter 19: Bonus Section: Consequences of the Uncertainty principle

Lecture 1: Probability that particles come into existence with high energy

Lecture 2: Distribution for which we have the minimum uncertainty

Chapter 20: Appendix

Lecture 1: Derivation of some formulas used in previous lectures