Channel Title: Jason Bramburger Channel Description: I am an assistant professor in applied mathematics at Concordia University with a passion for teaching. I am also actively researching at the interface of dynamical systems, machine learning, and computation. I have broad interests in the application of dynamical systems theory, including pattern formation, chaos theory, and data analysis. I received my PhD in applied mathematics from the University of Ottawa in 2017 under the supervision of Benoit Dionne and Victor LeBlanc. I have also held postdoctoral fellowships at Brown University, the University of Victoria, and the University of Washington. I was named a fellow of the Institute of Advanced Study in Guildford, UK for the 2021-2022 academic year. <video> Properties of the Fourier Transform - Partial Differential Equations | Lecture 36 Description: Now that we have been introduced to the Fourier transform and how it helps to understand the heat equation, we turn to analyzing some of its key properties. In particular, we investigate how the Fourier transform acts on derivatives of functions by applying it to the heat equation. The result is an ordinary differential equation in Fourier space which can be solved using separation of variables. We further explore the convolution theorem and Parseval's identity, showcasing their importance in the context of partial differential equations. Previous discussion of the Fourier transform for diffusion problems: https://www.youtube.com/watch?v=V5w5BUSiK7o Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Boundary Reflections in the Wave Equation - Partial Differential Equations | Lecture 41 Description: The previous lecture introduced D'Alembert's solution to the wave equation which decomposes the solution into leftward and rightward travelling components. But what happens if the wave equation is not on an infinite line, but a boundary? In this lecture we study what happens when one of these waves comes up against a boundary and derive mathematically how it can reflect off it. Of course, different boundary conditions can lead to different phenomena and so the viewer is left to experiment with different boundary conditions. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> BOOK ANNOUNCEMENT: Data-Driven Methods for Dynamic Systems Description: As experimental data sets have grown and computational power has increased, new tools have been developed that have the power to model new systems and fundamentally alter how current systems are analyzed. This book brings together modern computational tools to provide an accurate understanding of dynamic data. The techniques build on pencil-and-paper mathematical techniques that go back decades and sometimes even centuries. The result is an introduction to state-of-the-art methods that complement, rather than replace, traditional analysis of time-dependent systems. Get the book here: https://epubs.siam.org/doi/10.1137/1.9781611978162 Scripts and notebooks to reproduce all examples: https://github.com/jbramburger/DataDrivenDynSyst This book provides readers with: - methods not found in other texts as well as novel ones developed just for this book; - an example-driven presentation that provides background material and descriptions of methods without getting bogged down in technicalities; - examples that demonstrate the applicability of a method and introduce the features and drawbacks of their application; and - a code repository in the online supplementary material that can be used to reproduce every example and that can be repurposed to fit a variety of applications not found in the book. More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Characteristics of the Wave Equation - Partial Differential Equations | Lecture 40 Description: In this lecture we return to the wave equation, but analyze it from a different perspective. We use the method of characteristics to arrive at D'Alembert's solution to the wave equation which decomposes the solution into a leftward and rightward propagating component. We arrive at this solution using the "factoring" of the wave equation from the previous lecture into two first-order PDEs. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Method of Characteristics - Partial Differential Equations | Lecture 39 Description: In this lecture we show that the wave equation can be decomposed into two first-order linear partial differential equations. We further solve these resulting first-order PDEs using the method of characteristics. It is shown that the solutions simply propagate the initial condition left or right in space as time goes on. We then generalize the method to nonlinear systems where now the speed of translation can be nonlinear and the initial profile can be expanded, contracted, or both in time and space. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> The Fourier Transform - Partial Differential Equations | Lecture 34 Description: In the previous lecture we solved the heat equation on an infinite line to see that the solution is written as an integral over all wave numbers. In this lecture we investigate this integral operator, termed the Fourier transform. We derive the Fourier transform as the limit of a Fourier series as the domain becomes unbounded. As an example we show that Gaussians in physical space are transformed to Gaussians in Fourier/frequency space, and vice-versa. This property will be used in the next lecture when identifying the fundamental solution of the heat equation. Previous discussion of the Fourier transform for diffusion problems: https://www.youtube.com/watch?v=V5w5BUSiK7o Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Heat Equation on a Semi-Infinite Line - Partial Differential Equations | Lecture 37 Description: We have now spent some time solving the heat equation on the entire real line. In this lecture we consider it on a semi-infinite line (the positive reals), meaning that a boundary condition must be imposed at the left endpoint x = 0. With Dirichlet boundary conditions the result is a solution written as a superposition of sines. We show that this corresponds to the Fourier Sine Transform, while a Fourier Cosine Transform can be defined similarly. Like the Fourier series we saw before, many important properties of the Fourier Transform carry over to these sine and cosine transforms and so we only briefly review these properties here. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Shock Waves - Partial Differential Equations | Lecture 44 Description: In the previous lecture we looked at characteristics that cause solutions to PDEs to expand. In this lecture we look at the opposite case: contraction. Characteristics that cause the solution to contract or collapse on itself leads to non-unique characteristics and results in so-called shock waves. Here we introduce the basics of shock waves and provide the Rankine-Huginot condition which determines the speed at which shocks propagate. We further apply the theory to our traffic flow equation to model a green light turning red, causing cars to stop and wait at the light. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Modelling Traffic with PDEs - Partial Differential Equations | Lecture 42 Description: In this lecture we continue our study of quasi-linear PDEs and show how they can be used to model traffic flow. Throughout this lecture we derive a model of traffic on a stretch of road with no exits or entrances except at the endpoints. In particular, the model is amenable to the method of characteristics that has been the focus on the previous lectures. Thus, through this derived model, we are further able to motivate the study of quasi-linear PDEs independent of the wave equation. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Eigenfunction Expansions and Resonance - Partial Differential Equations | Lecture 32 Description: With our theory of solutions to the Helmholtz equation we are able to apply it to solving inhomogeneous heat and wave equations in higher spatial dimensions. In this lecture we perform an eigenfunction expansion on the PDE model for a vibrating membrane to obtain an explicit series solution. In particular, we focus on the case of a time-periodic forcing applied to the membrane and demonstrate that resonance can take place when the forcing frequency matches one of the natural frequencies of vibration of the membrane. In the case of resonance we show that oscillations do not remain bounded, but grow in time, thus leading to blow-up of solutions. Recall variation of parameters for ODEs: https://www.youtube.com/watch?v=dvdjtyMkGUA Learn more about mechanical vibrations and resonance in ODEs: https://www.youtube.com/watch?v=dQ6_aaFlVTo Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Fanlike Characteristics - Partial Differential Equations | Lecture 43 Description: In this lecture we investigate quasi-linear PDEs with nonlinear characteristics. In particular, we look at what happens when characteristics are moving components of initial conditions with discontinuities away from each other at different speeds. The result can be fanlike characteristics, where the initial condition with a jump discontinuity is made continuous as time goes on by characteristics that are spreading, or fanning, out. We apply the theory of fanlike characteristics to our traffic model from the previous video to model traffic flow when a red light turns green. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> The Eikonal Equation - Partial Differential Equations | Lecture 45 Description: This is the final lecture in this series on partial differential equations! Congratulations on making it this far! In this final lecture we introduce a non-linear PDE called the Eikonal equation. We give a brief derivation of the Eikonal equation from the 2D wave equation and show that in some cases it can be solved using the method of characteristics. However, the main goal of this lecture is to introduce the viewer to the wider world of PDEs wherein even simple equations need to be approximated or simplified further to produce related PDEs that can be even more complicated than we started with. Unlike most of the models we have worked with in this lecture series, the Eikonal equation is genuinely nonlinear and there are no general methods for solving it in complete generality. Such a situation is what one encounters at the frontiers of mathematical research into PDEs and so this lecture provides a possible first taste of this for the viewer. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> The Heat Equation on an Infinite Line - Partial Differential Equations | Lecture 33 Description: For the duration of this lecture series we have looked at PDEs on finite (bounded) domains. Mathematically, we typically study very large domains as being infinite to exclude finite-size effects. However, this infinite domains lead to new issues that have to be carefully dealt with. In this lecture we derive the solution to the heat equation on an infinite line using separation of variables. We see that there are now uncountably many eigenvalues of the associated Sturm-Liouville problem, leading to a solution written as the superposition of a continuum of eigenfunctions. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Scattering and the Schrodinger Equation - Partial Differential Equations | Lecture 38 Description: In this lecture we encounter a new PDE: the Schrodinger equation. The Schrodinger equation is a well-studied PDE coming from quantum physics and in this lecture we look at some of its important solutions. In particular, we show that separation of variables results in solutions that oscillate and time and hold fixed profiles based that are solutions of a Sturm-Liouville problem on an infinite line. We focus on special solutions that hold the same shape while travelling left and right as time goes on. The theory is complimented with an example that is explicitly solvable. Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video> <video> Fourier Transform and the Heat Equation - Partial Differential Equations | Lecture 35 Description: In the previous lecture we learned about the Fourier transform. In this lecture we will now apply this knowledge to the heat equation on an infinite line. In particular, we find the Guassian arising naturally in numerous contexts throughout our study: in the influence function, the fundamental solution to the heat equation, and self-similar solutions. Thus, the importance of the Gaussian and Fourier transform cannot be understated for the heat equation. Previous discussion of the Fourier transform for diffusion problems: https://www.youtube.com/watch?v=V5w5BUSiK7o Lectures series on differential equations: https://www.youtube.com/watch?v=UDUH58w4-5w&list=PLXsDp0z6VWFTkAhFApYsNfqgRvZrPTeom More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. </video>

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I am an assistant professor in applied mathematics at Concordia University with a passion for teaching. I am also actively researching at the interface of dynamical systems, machine learning, and computation. I have broad interests in the application of dynamical systems theory, including pattern formation, chaos theory, and data analysis. I received my PhD in applied mathematics from the University of Ottawa in 2017 under the supervision of Benoit Dionne and Victor LeBlanc. I have also held postdoctoral fellowships at Brown University, the University of Victoria, and the University of Washington. I was named a fellow of the Institute of Advanced Study in Guildford, UK for the 2021-2022 academic year.

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