Oliver Khan

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

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

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

Welcome back. This week, I wanted to discuss the Laplace transform, which is an incredibly useful tool for solving linear ordinary and partial differential equations. The primary reason for its utility is that it converts differential equations in terms of one unknown function into algebraic equations in terms of another unknown function that can be solved relatively quickly. … [Read more...] about The Laplace Transform: A More Powerful Extension of the Fourier Transform

Welcome back. I hope my readers are having a great third week of the semester. In this installment of “Introduction to Flight,” I am very excited to discuss finite-time Lyapunov exponents (FTLEs), which are powerful tools in the fluids research community. When we think of turbulence, we might imagine an infinitely jumbled mess of fluid moving erratically throughout space. This … [Read more...] about Finite-Time Lyapunov Exponents: A Tool to Discover Coherent Structures in Turbulent Flows