Partial Differential Equations: Comprehensive Course

PDE solved by Fourier Transform, Fourier Series, method of separation of variables + section on uncertainty principle

4.35 (58 reviews)
Udemy
platform
English
language
Science
category
5,991
students
17 hours
content
Oct 2023
last update
$59.99
regular price

What you will 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

How to derive Heisenberg Uncertainty Principle using concepts of Probability Theory

Description

Solving Partial Differential Equations using the Fourier Transform: A Step-by-Step Guide

Course Description:

This course is designed to provide a comprehensive understanding of how the Fourier Transform can be used as a powerful tool to solve Partial Differential Equations (PDE). The course is divided into three parts, each building on the previous one, and includes bonus sections on the mathematical derivation of the Heisenberg Uncertainty Principle.

Part 1: In this part, we will start with the basics of the Fourier series and derive the Fourier Transform and its inverse. We will then apply these concepts to solve PDE's using the Fourier Transform. Prerequisites for this section are Calculus and Multivariable Calculus, with a focus on topics related to derivatives, integrals, gradient, Laplacian, and spherical coordinates.

Part 2: This section introduces the heat equation and the Laplace equation in Cartesian and polar coordinates. We will solve exercises with different boundary conditions using the Separation of Variables method. This section is self-contained and independent of the first one, but prior knowledge of ODEs is recommended.

Part 3: This section is dedicated to the Diffusion/Heat equation, where we will derive the equation from physics principles and solve it rigorously. Bonus sections are included on the mathematical derivation of the Heisenberg Uncertainty Principle.

Course Benefits:

  • Gain a thorough understanding of the Fourier Transform and its application to solving PDE's.

  • Learn how to apply Separation of Variables method to solve exercises with different boundary conditions.

  • Gain insight into the Diffusion/Heat equation and how it can be solved.

  • Bonus sections on the Heisenberg Uncertainty Principle provide a deeper understanding of the mathematical principles behind quantum mechanics.

Prerequisites:

  • Calculus and Multivariable Calculus with a focus on derivatives, integrals, gradient, Laplacian, and spherical coordinates.

  • Prior knowledge of ODEs is recommended.

  • Some knowledge of Complex Calculus and residues may be useful.

Who is this course for?

  • Students and professionals with a background in Mathematics or Physics looking to gain a deeper understanding of solving PDE's using the Fourier Transform.

  • Those interested in the mathematical principles behind quantum mechanics and the Heisenberg Uncertainty Principle.

Content

Introduction

Probability that a particle exists at a certain time
Probability that a particle has a certain_energy
Uncertainty in the localization in time and in the energy of the particle

Uncertainty Principle derivation

Derivation of the uncertainty principle part 1
Derivation of the uncertainty principle part 2
Derivation of the uncertainty principle part 3

Consequences of the Uncertainty principle

Probability that particles come into existence with high energy
Distribution for which we have the minimum uncertainty

Appendix

Derivation of some formulae used in lecture 8

Screenshots

Partial Differential Equations: Comprehensive Course - Screenshot_01Partial Differential Equations: Comprehensive Course - Screenshot_02Partial Differential Equations: Comprehensive Course - Screenshot_03Partial Differential Equations: Comprehensive Course - Screenshot_04

Reviews

Yuqi
September 1, 2023
This course is heavy-loaded in physics-related concepts, not ideal for some one that is not a physics major but just want to learn about PDE, especially the later 6 sections.

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Related Topics

3660626
udemy ID
11/25/2020
course created date
12/9/2020
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