Partial Differential Equations: Comprehensive Course
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
- How to derive the Black Scholes equation in Finance
- How to derive (and in some cases solve) the Navier-Stokes equations
- concept of streamlines
- Mathematical tricks
- How to derive Heisenberg Uncertainty Principle using concepts of Probability Theory
Requirements
- Calculus (especially: derivatives, integrals)
- Multivariable Calculus (especially: the Jacobian, the Laplacian, etc.)
- Complex Calculus (basics of Fourier series and residues could help)
- Some notions of probability theory (distributions, mean, variance)
- Complex numbers
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.
Who this course is for:
- Students who are interested in Physics and in mathematical derivations of concepts
- engineers
- mathematicians
- physicists
- data scientists
- computer programmers
Instructor
- 4.5 Instructor Rating
- 686 Reviews
- 33,172 Students
- 23 Courses
I obtained my PhD in "Mechanics and Advanced Engineering Sciences" in 2021.
I attained a Bachelor of Science and Master of Science in Mechanical engineering in 2015 and 2017 respectively, from the University of Bologna.
I was the teaching tutor for the course of Mechanics of Machines from the academic year 2018 until the end of 2021 at the University of Bologna (branch of Forlì).
My passion for mathematics, physics and teaching has motivated me to lecture high school and university students.
My approach as a teacher is to prove to students that memory is less important for an engineer, mathematician, or physicist, than learning how to tackle a problem through logical reasoning. I believe that a teacher of scientific subjects should try to develop his students’ curiosity about the subject, rather than just concentrating on acquisition of knowledge, however important that may also be. Students should be encouraged to dig deeper and build on their knowledge by continually questioning it, rather than accepting everything at face value without a thorough understanding.
For enquiries (e.g. about tutoring, or advice related to the subjects spanned by my courses), you can either contact me on LinkedIn, or you can post questions in my courses' message boards, or you can also contact me via email or on my website.
You can also find the updated versions of my courses on my website.