Teaching & Academics

Partial Differential Equations (Poisson, Laplace, heat eq.)

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By using the Fourier Transform and the separation of variables, PDEs can be solved. Download the Partial Differential Equations (Poisson, Laplace, heat eq.) udemy course to know how.

What you'll learn

  • How to use the Fourier Trasforms to tackle the problem of solving PDE's
  • Fourier Transforms in one and multiple dimensions
  • Method of separation of variables to solve the Heat equation (with exercises)
  • Method of separation of variables to solve the Laplace equation in cartesian and polar coordinates (with exercises)
  • How to apply the Fourier Transform to solve 2nd order ODE's as well
  • concept of streamlines
  • Mathematical tricks

Requirements

  • Calculus (especially: derivatives, integrals)
  • Multivariable Calculus (especially: the Jacobian, the Laplacian, etc.)
  • Complex Calculus (basics of Fourier series and residues could help)

Description

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In the first part of the Partial Differential Equations (Poisson, Laplace, heat eq.) course, a Fourier Transform (FT) will be demonstrated as a useful tool to help solve partial differential equations (PDE).  For instance, the Fourier transform and its inverse (Inverse Fourier Transform, or simply IFT) derive from the Fourier series at the beginning of the course. Therefore, it may be useful for the student to already know the basics of this subject before beginning the course.

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Mathematics and multivariable calculus are prerequisites to the Partial Differential Equations (Poisson, Laplace, heat eq.) course, especially topics such as: calculation of derivatives and integrals, how to compute the gradient, spherical coordinates, the calculation of Jacobians, etc.

As well, you might find it useful to learn about residues used in Complex Calculus.

(February 2021): a second part has been added to the Partial Differential Equations (Poisson, Laplace, heat eq.) course, introducing the heat equation and the Laplace equation (in Cartesian and polar coordinates), as well as showing how to solve the exercises on PDEs step-by-step. All the steps leading to the solution are motivated in the exercises and are based on different boundary conditions. Second, we apply the Separation of Variables method, which allows us to transform a PDE into two ODE's (ordinary differential equations). This second portion of the Partial Differential Equations (Poisson, Laplace, heat eq.) course is self-contained and independent from the first. It may be useful to have some pre-requisite knowledge about ODEs.

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Additionally, exercises on the Wave equation and nonhomogeneous heat equations have been added.

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Who this course is for:

  • Students who want to understand how to solve Partial Differential equations (Poisson, Laplace, heat equation)
  • Students who would like to know more about Fourier Transforms
  • Students who want to understand how to use the Fourier Transform to solve 2nd order ODE's

Download Course
Source: https://www.udemy.com/course/partial-differential-equations-solved-by-fourier-transform/

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