Solving the Diffusion/Heat equation by Fourier Tranform

PDE solved by Fourier Transform (part 2)

This course aims to show how the Fourier Transform (FT) can be a powerful tool to solve Partial Differential Equations (PDE). The PDE that is treated in the course is the Diffusion/Heat equation. This equation is first derived from Physics principles described in the language of mathematics, then it is rigorously solved.

What you’ll learn

  • How to use the Fourier Trasforms to tackle the problem of solving Partial Differential Equations (PDE).
  • the physics and mathematics behind the diffusion process (related to particles contained in a fluid for example).
  • Mathematical details about the usage of the Fourier Transform.

Course Content

  • Introduction –> 1 lecture • 4min.
  • Derivation of the diffusion equation –> 3 lectures • 38min.
  • Solution to the Diffusion Equation –> 4 lectures • 1hr 3min.

Solving the Diffusion/Heat equation by Fourier Tranform

Requirements

  • This course comes after my previous course: “Partial Differential Equations solved by Fourier Transform”.
  • Multivariable Calculus (especially: the Jacobian, the Laplacian, etc.).
  • Complex Calculus (basics of Fourier series and residues could help).
  • Calculus (especially: derivatives, integrals).

This course aims to show how the Fourier Transform (FT) can be a powerful tool to solve Partial Differential Equations (PDE). The PDE that is treated in the course is the Diffusion/Heat equation. This equation is first derived from Physics principles described in the language of mathematics, then it is rigorously solved.

The course is the second installment of: “Partial Differential Equations solved by Fourier Transform”, which was previously published by the author. Therefore, it could be helpful to the student to already know the basics of those subjects treated in that course.

Calculus and Multivariable Calculus are other 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.