Introduction to Partial Differential Equations

Name: Introduction to Partial Differential Equations
Rating: 4.8 (24 reviews)

Introduction to PDEs

Created byUgur Abdulla

Last updated 6/2021

English

What you'll learn

The course exposes basic ideas critical to comprehend the concept of partial differential equation (PDE) and master the methods for solving classical PDEs of mathematical physics - wave, potential and heat equations.

Course content

3 sections • 27 lectures • 20h 16m total length

Introduction44:53
Linear and Homogeneous PDEs45:11
Separation of Variables for Linear Homogeneous PDEs40:12
Apply separation of variables to linear homogeneous PDEs to construct all product solutions by converting to ODEs, illustrated with the heat equation and polar coordinates.
Eigenvalue Problems46:24
Wave Equation45:36
Laplace's Equation47:58
Fourier Series48:22
Fourier Sine & Cosine Series46:10
Study Fourier sine and cosine series, even and odd extensions, and coefficient formulas, with wave and heat equation applications and x^2 on 0 to pi as example.
Solving PDE Problems via Fourier Series45:07
Nonhomogeneous PDE Problems48:33
The lecture shows how the method of Fourier can extend to nonhomogeneous PDEs, solving wave, heat, and potential equations with boundary and initial conditions via eigenfunction expansions.

Laplace Transform44:46
Learn how the Laplace transform converts PDEs in unbounded domains into simpler ODEs, using linearity, differentiation properties, and convolution to solve transport problems and recover solutions.
Laplace Transform for the Heat Equation43:41
Fourier Sine and Cosine Transforms47:15
Explore the Fourier sine and cosine transforms on the semi-infinite axis, derive forward and inverse transforms, and apply them to solve the heat equation with boundary conditions, illustrating Green's representation.
Fourier Transform47:08
Explains Fourier transform and inverse Fourier transform and how they turn the heat equation into an ODE, then solves via convolution with the initial temperature and the fundamental solution.
Method of Images45:04
The method of images for the heat equation on a semi-infinite rod yields a green function as a difference of fundamental solutions.
d'Alembert's Formula for the Wave Equation44:09
Explore d'Alembert's formula for the wave equation on an infinite string, decomposing initial displacement and velocity into left and right moving traveling waves at speed c.
Poisson's Formula for the Laplace Equation33:26
Derives Poisson's formula for the Laplace equation in the half-plane via Fourier transforms, yielding the Poisson kernel and a convolution representation.
Solving PDE Problems via Integral Transformation36:58
Apply Laplace transforms to turn PDEs into solvable ODEs, solving a non-homogeneous wave equation with initial and boundary conditions, and a convection–diffusion heat equation via the fundamental solution and convolution.

Solving PDEs via Multiple Fourier Series50:07
Laplace's Equation in a Disk50:36
Bessel Functions48:15
Explore the Bessel equation and its Bessel functions of the first kind, including half-integer cases, and see how the gamma function helps reveal their properties in circular domains.
The Wave Equation in a Disk45:37
Explore solving the two-dimensional wave equation on a unit disk using separation of variables in polar coordinates, yielding eigenfunctions with Bessel functions and a center regularity condition.
Solving PDEs in Cylindrical Domains46:04
Legendre Functions47:41
Laplace Equation in a Ball47:32
Solve the Laplace equation inside a ball using spherical coordinates and boundary data, then obtain solutions through separation of variables and spherical harmonics, guided by eigenvalue problems.
Heat Equation in a Ball42:26
Solve the heat equation in a unit ball by separation of variables, using spherical harmonics and spherical Bessel functions, to form a complete initial-value solution with zero boundary conditions.
Solving PDEs via Special Functions37:42
Final Exam

Requirements

Calculus and Differential Equations courses

Description

Partial Differential Equations (PDEs) are pivotal in both pure and applied mathematics. They emerge as mathematical models of processes in nature in which some quantities change continuously in space and time. The temperature or the magnetic field on earth, the velocity of a fluid or gas, electrostatic potential of the conductor, the density of a cancerous tumor in the body, neuronal activity of the brain, stock price fluctuations, and the population of biological species are just a few of the complex systems modeled by this ubiquitous equations. Supported with the power of modern software and the emerging fields of Artificial Intelligence and Deep Learning, PDEs are expanding to all areas of modern science and technology. This course presents a foundation for PDEs, starting from their physical origin and motivation. In particular, it introduces the classical equations of mathematical physics, namely the heat, wave, and Laplace equations. In this course, you will learn key ideas critical to the study of PDEs - separation of variables, integral transforms, special functions of mathematical physics, Fourier series, and related topics.Though the topics focus on the foundational mathematics, connections are constantly made to the underlying physics from which they emerge. Substantial practical part of the course is dedicated to solving explicitly various physical problems by using these methods

Who this course is for:

undergraduate and graduate math, science and engineering students, high school seniors, STEM professionals who possess a prerequisite knowledge on calculus and differential equations

Introduction to Partial Differential Equations

What you'll learn

Explore related topics

Course content

Method of Fourier Series10 lectures • 7hr 38min

Method of Integral Transformation8 lectures • 5hr 42min

PDEs in Higher Dimensions9 lectures • 6hr 56min

Requirements

Description

Who this course is for: