@diff_eq

Tweets on ordinary and partial differential equations from @JohnDCook

johndcook.com/blog/partial-differential-equations/
Joined March 2013
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297Media
Differential equation modeling johndcook.com/blog/partial-d…
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We regard Butcher's work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results. -- Alain Connes (Fields medal 1982)
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Ten life lessons from differential equations johndcook.com/blog/2015/07/2…

Life lessons from differential equations

Some problems simply have no solution. Some problems have no simple solution. Some problems have many solutions. ...

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'The more I teach differential equations, the less I understand the mystery of differential equations.' -- Gian-Carlo Rota
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Eliminating terms from higher order ODEs johndcook.com/blog/2022/11/1…

Eliminating terms from higher-order differential equations

Change of variables to eliminate terms involving (n-1)st derivatives from an nth order differential equation

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Numerical solution of differential-algebraic equations web.mscsoftware.com/support/… (DASSL)
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'Classical mathematics concentrated on linear equations for a sound pragmatic reason: it could not solve anything else.' -- Ian Stewart
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The differential equation implicit in The Pit and the Pendulum by Edgar Allan Poe. johndcook.com/blog/2026/10/1…
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2x2 grid giving an overview of numerical differential equation solvers johndcook.com/blog/2020/06/1…

Numerical ODE solver landscape overview

Many methods for solving differential equations numerically can be classified in a 2 x 2 grid: Runge-Kutta or multistep, explicit or implicit.

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Discretized wave equations terrytao.wordpress.com/2014/…

Discretised wave equations

The wave equation is usually expressed in the form $latex \displaystyle \partial_{tt} u – \Delta u = 0&fg=000000$ where $latex {u \colon {\bf R} \times {\bf R}^d \rightarrow {\bf C}}&…

terrytao.wordpress.com
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I got an email just now from PDF Expert, an inexpensive alternative to Adobe software. At first glance I thought it said PDE expert, a flashback to an earlier career.
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Differential equation on a donut johndcook.com/blog/2025/10/0…

Differential equation on a doughnut

A system of three first-order differential equations whose solutions live on a torus (doughnut).

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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.

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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

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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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