A Tale of a Boundary-Layer Problem

This repository holds references and slides that I used for a talk at a university seminar called Problems of Contemporary Mathematics. The ideal length of the talk is about 2 hours. It is based on the SIAM article Surprises in a Classic Boundary-Layer Problem (Clark et al., 2023) and follows the structure of this article. The slides are made with beamer and include a number of motivational figures and some figures that appeared in the article and are reimplemented in julia also in this repo. A part of the talk is dedicated to illustrating how interactive figures can help understand the problem at hand and lead to pinpointing the correct approach to arrive at the solution. For a hint where to look for the sources mentioned above, take a look at the Contents section.

Title: A Tale of a Boundary-Layer Problem

Abstract: I will demonstrate the solution with all its steps (and missteps) to the boundary layer problem $$\varepsilon y'' = yy' - y,$$ such that $$y(0) = 1\quad \text{and} \quad y(1) = -1\quad \text{and where}\quad \varepsilon \ll 1.$$ The journey towards the solution will take us on an excurse through some basics of perturbation theory and asymptotic approximation matching. I will demonstrate visualizations that will give you insight into the reasoning behind the steps taken towards the solution serving as an exhibition for the leverage in problem solving accessible to us through the computational power at our fingertips.

Contents

• Slides_Interactive/: Contains a Pluto notebook with interactive figures
• Slides/figures/sources/: Contains the source julia code used to generate the figures in the beamer presentation
• Slides/figures/: Contains the generated figures that appear in the beamer presentation
• Slides/src/: Contains all the $\LaTeX$ source code that was used to compile the slides
• Slides/build/Slides_A-Tale-of-a-Boundary-Layer-Problem/: Contains the PDF slides

Figures

Here is a showcase of some figures that were created for this project

| Comparison of the solutions (approximations) of the simple pendulum problems |Comparison of the solutions (approximations) to the projectile problem | | :---: | :---: | | ||

| Phase portrait vector field of the boundary value problem |Solutions of the Boundary layer problem in the phase portrait | | :---: | :---: | | | |

For the solution to the boundary layer problem in the $x$ and $y$ plane and other figures, take a look into the slides!

Usage

Interactive Notebook

Requirements: - julia (It is possible to run the notebook using )

Note: You have to activate the project and download the dependencies. It is as easy as running update in the julia package REPL. You can find how to do this at the Pkg.jl documentation page.

It is as easy as running

bash $ julia --project --eval "using Pluto; Pluto.run()" in the Slides_Interactive directory and selecting slides.jl in the notebook UI. You can run it online in binder that is linked (top-right corner) from static HTML version of the interactive slides.

The GLMakie figures that are more interactive can be run by re-evaluating the cells that define the figures.

$\LaTeX$ Compilation

Requirements: - tectonic (but any $\LaTeX$ compiler will do if you know what you are doing) - biber

Note: Currently there is a problem with the version of biblatex included in the tectonic bundle, so if biber fails to run, you should look over here: tectonic#35

The slides were compiled by tectonic using bash $ tectonic -X build ran in the Slides directory. You can disable / enable presenter notes by modifying the beamer package options in the preamble

| Notes | No notes| | :---: | :---: | | \documentclass[x11names]{beamer} | \documentclass[x11names, notes]{beamer} |

Figures Generation

Requirements: - julia - inkscape (used for creating PDF_TEX from the SVG files)

They are generate using CairoMakie as SVG graphics and converted to PDF_TEX using inkscape by running the julia script

bash $ julia --project --eval "include(\"generate_figures.jl\")" inside the Slides/figures/sources directory.

Note: You have to activate the project and download the dependencies. It is as easy as running update in the julia package REPL. You can find how to do this at the Pkg.jl documentation page.

License

This work is triple-licensed under Apache 2.0 and GPL 2.0 (or any later version) and CC0 1.0 Universal . You can choose between one of them if you use this work.

SPDX-License-Identifier: Apache-2.0 OR GPL-2.0-or-later OR CC0-1.0-universal