Questions tagged [differential-equations]
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37
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What is the origin of the "Japanese bracket"?
In discussions of Sobolev spaces one often sees the Japanese bracket, $$\langle x \rangle = (1+|x|^2)^{1/2},$$ as useful shorthand.
I was not easily able to find information about this term.
(1) What ...
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1
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Who invented the exponential ansatz for linear differential equations with constant coefficients?
Who invented using $e^{\lambda t}$ as ansatz for solving linear differential equations with constant coefficients?
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Who wrote down the first differential equation?
I am just curious who was the first person to write down a differential equation? And what was this differential equation?
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What are the early applications of differential equations to social sciences?
I am reading Karatzas and Shreeve (Brownian Motion and Stochastic Calculus) and on page 128, at the beginning of chapter 3 on Stochastic Integrals, one reads:
I can't think of any social problem that ...
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Who did first use the Method of Characteristics?
Which mathematician did introduce the Method of Characteristics for solving linear Partial differential equations?
I some papers I saw that Saint Venant, Lagrange, Cauchy or Riemann were attributed ...
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source of "logistic growth"?
I've been trying to find the source of the name of the DE modelling population growth known as logistic growth, for some time: why "Logistic" ? So far all my attempts to research it have hit dead ends ...
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What were the early uses of differential equations for modeling chemical reactions?
What are some of the original examples of uses of differential equations for modeling and analyzing chemical reactions, particularly those relevant to biochemistry, involving proteins and enzymes? ...
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2
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Did Newton find the trajectory of a body moving in uniform gravity under the quadratic resistance law (the ballistic problem)?
I'm very confused by contradicting accounts of a supposed solution by Newton to the problem of finding the trajectory of a projectile moving under uniform gravity against resistance that is ...
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Who came up with the link between the spectrum of an operator and the poles of a meromorphic function?
From Dieudonné's "History of Functional Analysis" I learned that Picard in 1893 gave a characterization of an eigenvalue of the Laplacian as the simple pole of a meromorphic function.
Is there an ...
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History of hypergeometric equation
It is known that Gauss studied hypergeometric equation
$$x(1-x) \dfrac {d^2y}{dx^2}+(c-(a+b+1)x)\dfrac {dy}{dx}-aby=0$$
I would like to know something about history of this equation:
1) If $a=b=c=0$...
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Notations for Laplacian: $\nabla^2$ vs. $\Delta$
For a (sufficiently smooth) function $f\colon \Bbb R^n\to\Bbb R$, the Laplacian of $f$ is defined to be $\sum_{j=1}^n \frac{\partial^2 f}{\partial x_j^2}$. There are two notations for the Laplacian ...
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What is the origin of the term "Ordinary Differential Equation"?
Who first used the term "Ordinary Differential Equation (ODE)"? Is it known why the word "ordinary" is used here? What makes an ODE "ordinary"?
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How did Peano prove his existence theorem without Ascoli's theorem?
In modern proofs of the Peano Existence Theorem for ordinary differential equations, Ascoli's theorem is used. Ascoli's theorem came after Peano's proof. Did Peano prove a form of Ascoli's theorem in ...
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What is the etymology of "phase space" of a dynamical system?
The state space of a dynamical system is often called a "phase space". What is the etymology of this?
(Note that I'm not asking about the history of the concept, but rather about the history of the ...
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Earliest Instances of a Slope/Direction Field for a First-Order ODE
Background
When first encountering slope fields in calculus or elementary differential equations, students often ask "What is the purpose?"
A concise answer is that slope fields provide a way to ...
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Who was the first person in history to calculate the limit $\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n$?
At the beginning of a calculus course, we encounter two famous limits. They are
$$\lim_{x\to0}\frac{\sin x}{x}\qquad\text{and}\qquad\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n.$$
I'm not ...
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Origin of the terminology “trace operator” related to boundary-value problems for PDEs
Important results in the theory of PDEs regarding boundary-value problems are trace and extension theorems. Since the trace operator (not to be confused with the trace from linear algebra) essentially ...
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What kinds of differential equations were solved on the early differential analyzer?
Harold Locke Hazen and Vannevar Bush created the Differential Analyzer at MIT between 1928 and 1931 to solve differential equations, according to the Wikipedia Article on the topic. I've read this ...
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History behind the heat equation
Who was the first mathematician/physicist to derive the heat equation $u_t=\Delta u$ and when? Was it already known to explain most diffusion phenomena? How much time passed between the first ...
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Source of claim that Leibniz discovered separation of variables for ODEs in 1691?
Claims I'm evaluating
I've read in multiple sources that Leibniz formulated separation of variables for ODEs in 1691. A couple example sources are below.
Mathematical Thought from Ancient to Modern ...
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Does the “O” in the google doodle for Olga Ladyzhenskaya have anything to do with her work?
Ladyzhenskaya is famous for fluid dynamics and partial differential equations, both of which are beyond my pay grade. And she worked on the Navier-Stokes equations. Does this circle with the arrows ...
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History of PDE's in the 19th Century
I've been asked to write an essay on whether the work on PDE's in the 19th century belonged to applied or pure mathematics. Does anyone know of any useful sources I could use?
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What does the Fourier transform have to do with heat?
For example the current version of the Fourier analysis article on Wikipedia says the study is:
[…] named after Joseph Fourier, who showed that representing a function as a sum of trigonometric ...
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Who is Wanner from Rosenrock-Wanner (ROW) methods?
I've spent some time with a search engine trying to find out about Wanner, a person whose surname is mentioned in the name of Rosenbrock-Wanner (ROW) methods primarily used for iteratively solving ...
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From where the so-named "elastica problem" is coming from?
In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here).
The problem is presented as a system of ODEs,
$$
x' = \cos (\phi)
$$
$$
y' = \sin (\phi)
$$...
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Poincare's last geometric theorem
Which problem in celestial mechanics led Poincare to his conjecture about fixed points of area preserving maps of the annulus to itself? I believe he was working on some differential equations. What ...
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Asymptotically Periodic Potentials
Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials? I heard it was Pierre Louis Lions, ...
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1
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D'Alembertian symbol $\Box$
The D'Alembertian is a generalization of the Laplacian operator to a space of arbitrary dimension and metric.
Where does the D'Alembertian symbol $\Box$ come from?
According to Wikipedia it has to ...
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Why was Indicial equations named so?
In ODE, in Frobenius method, there's an equation called "Indicial Equation." Is there any particular contextual/historical reason that it is named so?
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The origin of $∂^2=0$ and $d^2=0$
I know that formula $∂^2=0$ and $d^2=0$ very important in the homology and cohomology theory. And I understand that this formula was generated from the process of finding a solution to the partial ...
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The original Dirac equation
In the original 1928 paper (pdf) the Dirac equation appears on page 615 in equation (9) as
$$
[p_0+\rho_1\left(\boldsymbol{\sigma},\boldsymbol{p}\right)+\rho_3mc]\psi=0\qquad(1)
$$
Using the ...
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Why are partial derivatives necessary when deriving the equation for a vibrating string?
AFAIK, partial derivatives made it to the forefront as a result of coming up with an equation for a vibrating string i.e., 1D wave equation.
From purely a physical phenomena to math mapping POV: Why ...
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Motivation of Monge-Ampere equation
I have a question about Monge-Ampere equation.
I read a book The shape of inner space by Shing-Tung Yau.
Chapter 5 in this book, Complex-Monge-Ampere equation's explaination was so interesting. In the ...
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History of PDE's in the 19th Century (part 2)
This is a follow up to this question: History of PDE's in the 19th Century
The question I have been given to answer is:
The history of partial differential equations in the 19th Century belongs ...
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How did Newton write his equations?
Once, after a lecture, my professor of differential equations said, that Newton did not use derivatives in his work as we do today. He told us that Newton rather used some series expansions for his ...
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Invariance principle for stability in the sense of Lyapunov
On Wikipedia this article about the invariance principle and article states that
The general result was independently discovered by J.P. LaSalle (then
at RIAS) and N.N. Krasovskii, who published ...
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The Original Proofs of The Stable Manifold Theorem
The book "Differential Equations and Dynamical Systems" by Lawrence Perko says that the first proofs of the Stable Manifold Theorem are from Hadamard in 1901, Perron in 1928, and Liapunov and Perron. ...
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