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What you will 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.

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

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Content

Method of Fourier Series

Introduction

Linear and Homogeneous PDEs

Separation of Variables for Linear Homogeneous PDEs

Eigenvalue Problems

Wave Equation

Laplace's Equation

Fourier Series

Fourier Sine & Cosine Series

Solving PDE Problems via Fourier Series

Nonhomogeneous PDE Problems

Method of Integral Transformation

Laplace Transform

Laplace Transform for the Heat Equation

Fourier Sine and Cosine Transforms

Fourier Transform

Method of Images

d'Alembert's Formula for the Wave Equation

Poisson's Formula for the Laplace Equation

Solving PDE Problems via Integral Transformation

PDEs in Higher Dimensions

Solving PDEs via Multiple Fourier Series

Laplace's Equation in a Disk

Bessel Functions

The Wave Equation in a Disk

Solving PDEs in Cylindrical Domains

Legendre Functions

Laplace Equation in a Ball

Heat Equation in a Ball

Solving PDEs via Special Functions

Final Exam

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