The first part of the course aims to show how the Fourier Transform (FT) can be a powerful tool to solve Partial Differential Equations (PDE). The FT and its inverse (Inverse Fourier Transform, or simply IFT), are derived from the concept of the Fourier series at the beginning of the course, therefore it could be helpful to the student to already know the basics of such subject.
Calculus and Multivariable Calculus are a necessary prerequisite to the course, especially the topics related to: calculation of derivatives and integrals, how to compute the gradient, the Laplacian of a function, spherical coordinates, the calculation of the Jacobian, etc.
Some knowledge of residues used in Complex Calculus might be useful as well.
Course update: a second part to the course has been added, which introduces the heat equation and the Laplace equation (in Cartesian and polar coordinates), and aims to show how to solve some exercises on PDE's step-by-step. The exercises contain different boundary conditions and all the steps leading to the solution are motivated. The method that is used in the second part is that of Separation of Variables, which allows the PDE to be transformed in two different ODE's (ordinary differential equations). This second part of the course is self-contained and independent of the first one. Some pre-requisite knowledge about ODE's could be very useful.
Exercises on nonhomogeneous heat equations have also been added, as well as exercises on the Wave equation.
Course update: I have added another section to the course, which treats the Diffusion/Heat equation. This equation is first derived from Physics principles described in the language of mathematics, then it is rigorously solved.
Intro to the course
FREE PREVIEWFourier series
FREE PREVIEWFourier Transforms
FREE PREVIEWDirac delta
FREE PREVIEWMultiple Fourier Transforms
FREE PREVIEWWhy the Dirac delta helps derive the Inverse Fourier Transform
Gradient and Laplacian: quick summary
FREE PREVIEWExample of pde
FREE PREVIEWSolution to the pde part 1
Solution to the pde part 2
Solution to the pde part 3
Physics behind the equation part 1
Physics behind the equation part 2
Setup of the diffusion problem
FREE PREVIEWIntegral equation satisfied by the function f(x,t)
Diffusion equation
Some possible boundary conditions of the diffusion equation
Solution of the diffusion equation part 1
Solution of the diffusion equation part 2
Solution of the diffusion equation part 3
Solution of the diffusion equation part 4
Solution to a PDE with the method of characteristics
Separation of variables to solve the heat equation (part 1)
FREE PREVIEWSeparation of variables to solve the heat equation (part 2)
Separation of variables to solve the heat equation (part 3)
Laplace Equation in Cartesian Coordinates (exercise)
FREE PREVIEWLaplace Equation in Polar coordinates (exercise 1)
Laplace Equation in Polar coordinates (exercise 2)
Laplace Equation in Polar coordinates (exercise 3)
Laplace Equation in Polar coordinates (exercise 4)
Concept of streamlines (with exercise)
Nonhomogeneous Heat Equation: Exercise 1
FREE PREVIEWNonhomogeneous Heat Equation: Exercise 2
Nonhomogeneous Heat Equation: Exercise 3
Nonhomogeneous Wave Equation (Exercise 1)
FREE PREVIEWNonhomogeneous Wave Equation: D'Alambert formula
Solving a wave equation using D'Alambert formula (exercise)
Energy conservation law for the wave equation
Bi-dimensional heat equation: exercise 1
Bi-dimensional heat equation: exercise 2
Bi-dimensional wave equation: exercise 1
Mathematical derivation of Navier Stokes equations part 1
Mathematical derivation of Navier Stokes equations part 2 (writing the stress tensor)
How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 1
FREE PREVIEWHow Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 2
How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 3
How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 4
How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 5
How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 6
derivation of Stokes law from Navier Stokes part 1
derivation of Stokes law from Navier Stokes part 2