Partial Differential Equations (Poisson, Laplace, heat equation)

Partial Differential Equations (Poisson, Laplace, heat equation)

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 Take a look at the home page of the course
If you prefer, you can enroll directly on Thinkific (it is just the same as doing it on this website, but checkouts are quicker there and require you insert less data)

 

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.

A second part to the course introduces the heat equation and the Laplace equation (in Cartesian and polar coordinates), and aims to show how to solve some exercises 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.

There is also another section related to the Diffusion/Heat equation. This equation is first derived from Physics principles described in the language of mathematics, then it is rigorously solved.