Welcome back. I hope that all of my readers had a restful weekend, and are having a smooth return to classes. In today’s blog post, I look forward to sharing some of the fundamental concepts of thermodynamics. If we were to define thermodynamics, we could say that it is the study of energy and the ways in which energy can be transferred. Therefore, we will begin our discussion … [Read more...] about A Crash Course in Thermodynamics (With Some Fluid Mechanics)

## Some Brief Words on Nonlinear Dynamical Systems and Chaos

Welcome back. I hope that all of my readers had a restful winter break and are having a fabulous start to their new semester. In this first blog post of 2024, I am excited to share some brief remarks on the rich and growing field of nonlinear dynamics and chaos. Fundamentally, this field is centered around studying nonlinear systems of ordinary differential equations (ODEs) … [Read more...] about Some Brief Words on Nonlinear Dynamical Systems and Chaos

## First-Order Nonlinear Partial Differential Equations and the Method of Characteristics

Partial differential equations govern the universe, and most of these equations are nonlinear. This poses a challenge to engineers because nonlinear equations are difficult to solve. In many cases, there is no explicit formula for a solution and it often takes several pages of mathematical arguments to prove whether or not a solution even exists. However, with some clever … [Read more...] about First-Order Nonlinear Partial Differential Equations and the Method of Characteristics

## Finding Reduced-Order Models for Nonlinearizable Dynamical Systems using Spectral Submanifolds

A fundamental theme throughout this blog series has been partial differential equations (PDEs). The reason for this is that PDEs are a language through which we can describe the world around us, from the smallest scales, to the very largest. Many PDEs, moreover, involve time derivatives. This makes sense, as phenomena such as fluid flows, beam vibrations, and gravitational … [Read more...] about Finding Reduced-Order Models for Nonlinearizable Dynamical Systems using Spectral Submanifolds

## The Fundamental Solution of Laplace’s Equation and its Importance in Physics

Welcome back. Today, we will explore a concept that is central to the study of linear partial differential equations: that of the fundamental solution. As its name suggests, a fundamental solution is a very special solution to a PDE that serves as a building block for more general solutions. In this blog, my goal is to derive the fundamental solution to Laplace’s … [Read more...] about The Fundamental Solution of Laplace’s Equation and its Importance in Physics