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/
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