Partial Differential Equations: Comprehensive Course
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.
Solution of a PDE equation
Some more physics behind the pde
Solving the Diffusion/Heat equation by Fourier Tranform
2nd order ODE solved via Fourier Transform
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Setup of the diffusion problem -
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Integral equation satisfied by the function f(x,t)
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Diffusion equation
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Some possible boundary conditions of the diffusion equation
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Solution of the diffusion equation part 1
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Solution of the diffusion equation part 2
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Solution of the diffusion equation part 3
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Solution of the diffusion equation part 4
PDE solved with the method of characteristics
Heat equation solution via Separation of Variables
Laplace Equation solved via the method of Separation of Variables
Nonhomogeneous Heat Equation
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Laplace Equation in Cartesian Coordinates (exercise)
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Laplace Equation in Polar coordinates (exercise 1)
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Laplace Equation in Polar coordinates (exercise 2)
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Laplace Equation in Polar coordinates (exercise 3)
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Laplace Equation in Polar coordinates (exercise 4)
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Concept of streamlines (with exercise)
Wave Equation (Exercises)
Bi-dimensional problems (heat and wave equation)
Derivation of the Navier-Stokes equations and their solution in a 2D case
How Einstein mastered Navier-Stokes equations in his PhD dissertation
Stokes law obtained from Navier-Stokes equations
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How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 1
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How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 2
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How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 3
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How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 4
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How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 5
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How Einstein mastered Navier-Stokes equations in hid PhD dissertation Part 6
Appendix on PDE's
Bonus section: Introduction to the Heisenberg Uncertainty Principle
Bonus Section: Uncertainty Principle derivation
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Mathematical summary of how to prove the uncertainty principle
Note: the following lectures will prove the Uncertainty principle more slowly. In this lecture we do it quickly so you can get the idea of how it can be done.
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Introduction to the short course on the Heisenberg Uncertainty Principle
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Probability that a particle exists at a certain time
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Probability that a particle has a certain_energy
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Uncertainty in the localization in time and in the energy of the particle
Bonus Section: Consequences of the Uncertainty principle
