Open Math Notes

Usually a course on partial differential equations (PDEs) starts with the theory of first order PDEs, which turns out to be quite time consuming for a teacher and difficult for students due to dependence of the proofs on geometry of Monge curves. In this article we present a simpler theory of first order PDEs using only the characteristic curves in the space of independent variables. In addition we discuss existence and uniqueness first with examples and then prove rigorously It has new ideas.

Phoolan Prasad · Indian Institute of Science · Date posted: January 9, 2023

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These are a series of lecture slides from a course I taught at the University of Alabama in the Fall of 2022.

The course text was "Fundamentals of Differential Equations" 9e, by Kent Nagle, Edward Saff & Richard Snider. These slides were used as a lecture supplement with additional explantation, examples and problems.

The following sections were covered:
1.1-1.4, 2.2, 2.3, 2.6, 3.2, 3.3, 4.1-4.6, 4.9, 5.2, 7.2-7.6.

Overleaf link (for LaTeX): https://www.overleaf.com/read/gbnbzrbbyyqz

Brandon Sweeting · The University of Alabama · Date posted: January 9, 2023 · Date revised: January 11, 2023

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Much enhanced notes by the second author of a course given by the first author in 1996

Kenneth Ribet · University of California, Berkeley · William Stein · CoCalc · Date posted: November 16, 2022

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We study explicit and implicit methods for initial value problems.
For general Runge-Kutta methods, Butcher's theory of rooted trees
is presented in detail, proving Butcher's theorem on conditions for high orders of consistency and convergence. Further topics include
collocation methods, stability theory, stiff equations, asymptotic error expansions, extrapolation methods, and stepsize control.

Peter Philip · LMU Munich · Date posted: October 31, 2022

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Vector fields and differential forms have very different properties. However a given volume element allows vector fields to be converted into twisted differential (n-1)-forms via contraction. Also a metric tensor allows vector fields to be converted into 1-forms. The ultimate theory works with an arbitrary metric tensor in n dimensions. The exposition includes insightful pictures along with extensive computations.

William Faris · University of Arizona · Date posted: October 10, 2022

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