@diff_eq

Tweets on ordinary and partial differential equations from @JohnDCook

johndcook.com/blog/partial-differential-equations/
Joined March 2013
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terrytao.wordpress.com/2014/…

Noether’s theorem, and the conservation laws for the Euler equations

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous jus…

terrytao.wordpress.com
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Nonlinear Poisson equation: -Δ u = f(u)
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Noether's theorem, and the conservation laws for the Euler equations
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Difference equations and differential equations johndcook.com/blog/2022/06/1…

Difference equations and differential equations

Example showing how similar the process is for solving difference and differential equations.

johndcook.com
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History of the finite element method link.springer.com/article/10…

Eighty Years of the Finite Element Method: Birth, Evolution, and Future

Archives of Computational Methods in Engineering - This document presents comprehensive historical accounts on the developments of finite element methods (FEM) since 1941, with a specific emphasis...

link.springer.com
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Wave equation: u_tt - Δu = 0
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'The history of transistors is the history of solving Schrödinger's equation in various materials.' -- Leon Lederman
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Operator calculus johndcook.com/blog/2022/10/1…

Operational calculus for solving differential equations

A justification for useful tricks that seem unjustified, and a worked-out example

johndcook.com
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Laplace's equation was first studied by Laplace around 1780, Stigler's law of eponymy not withstanding.
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Scalar reaction-diffusion equation: u_t - Δu = f(u)
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Solving the wave equation on a disk johndcook.com/blog/2021/06/1…

Solving the wave equation on a disk

Modeling a vibrating circular membrane, like a drum head, using the wave equation in polar coordinates.

johndcook.com
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Lorenz Attractor animation lkforge.com/tools/physics/lo…

Lorenz Attractor — Interactive 3D Strange Attractor | LK Forge

Spin the 3D butterfly, turn the ρ dial, and split a twin trajectory to watch the butterfly effect diverge on a log plot. Free, no sign-up, embeddable.

lkforge.com
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Schrödinger's equation: i u_t + Δu = 0
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Heat (or diffusion) equation: u_t - Δu = 0
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Very brief introduction to delay differential equations johndcook.com/blog/2024/10/2…

Delay differential equations

Delay differential equations relate a function and its derivatives at the current point in time with their values in the past.

johndcook.com
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Laplace's equation: Δu = 0.
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In R^n, Δ u is the sum of the second derivatives of u with respect to each variable.
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Power series solutions for DIFFERENTIAL equations lead to DIFFERENCE equations for the coefficients.
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An atlas of periodic solutions to the three-body problem threebodyorbits.com/

Three Body Orbits — 3,942 periodic three-body orbits, visualized

An atlas of 3,942 periodic three-body orbits, precomputed to machine precision and animated one period at a time. Explore the map, open any orbit, rank them.

threebodyorbits.com
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