Tìm phần mềm FOSS (trên Linux/Windows) hỗ trợ mô phỏng hệ thống phức tạp qua sơ đồ hồi tiếp và phương trình vi phân, có thể phân tích quy mô tương đối của các quá trình tác động lẫn nhau (ví dụ: động lực dân số, nhiệt truyền). Đã dùng Antix Linux, ưu tiên giải pháp trực quan thay vì R/Excel. #PhầnMềmMở #ToánHọcỨngDụng #HệThốngPhứcTạp #MôPhỏngKhoaHọc #FOSS #DifferentialEquations #SystemsModeling
https://www.reddit.com/r/opensource/comments/1pifowd/graphical_feedback_loopdifferential_equations/
Equations Of The Mixed Type by A.V. Bitsadze
The theory of equations of mixed type originated in the fundamental researches of the Italian mathematician Francesco Tricomi, which were published in the twenties of this century. Owing to the importance of its applications, the discussion of problems concerned with equations of mixed type has become, in the last ten years, one of the central problems in the theory of partial differential equations.
The present work is not meant to be a summary of all results in this field, especially since the number of results increases with great speed; nevertheless, the reader of this monograph will obtain an idea of the present state of the theory of equations of mixed type.
This book was developed from a series of lectures dealing with certain fundamental questions in the theory of equations of mixed type, which the Author delivered in scientific establishments in the Chinese People’s Republic at the end of 1957 and the beginning of 1958.
Translated by P. Zador
Translation edited by I. N. Sneddon
You can get the book here and here.
CONTENTS
Foreword ix
Introduction xi
1. General remarks on linear partial differential equations of mixed type 1
1. Equation of the second order with two independent variables 1
2. The theory of Cibrario 3
3. Systems of two first order equations 12
4. Linear systems of partial differential equations of the second order with two independent variables 17
2. The study of the solutions of second order hyperbolic equations with initial conditions given along the lines of parabolicity 20
1. The Riemann function for a second order hyperbolic linear equation 20
2. A class of hyperbolic systems of second order linear equations 27
3. The Cauchy problem for hyperbolic equations with given initial conditions on the line of parabolic degeneracy 32
4. Generalizations 41
3. The study of the solutions of second order elliptic equations for a domain, the boundary of which includes a segment of the curve of parabolic degeneracy 44
1. The linear elliptic partial differential equation of the second order 44
2. Elliptic systems of second order 49
3. The Dirichlet problem for second order elliptic equations in a domain, the boundary of which includes a segment of the curve of parabolic degeneracy 58
4. Some other problems and generalizations 66
4. The problem of Tricomi 71
1. The statement of the problem of Tricomi 72
2. The extremal principle and the uniqueness of the solution of problem T 74
3. Solution of problem T by means of the method of integral equations 78
4. Continuation. The proof for the existence of a solution of the integral equations obtained in the preceding paragraph 90
5. Other methods for solving problem T 94
6. Examples and generalizations 103
5. Other mixed problems 112
1. The mixed problem M 112
2. The proof of the uniqueness of solution for problem M 113
3. Concerning the existence of the solution of problem M 117
4. The general mixed problem 124
5. The problem of Frankl 135
6. Short indication of some important generalizations and applications 141
References 151
Index 157
#1964 #differentialEquations #mathematics #solutionsToDifferentialEquations #sovietLiterature #tricomiProblem
how long it will take for a can of pop to explode in the freezer
#differentialequations #calculus #math #pop #chemistry
https://youtube.com/shorts/kXYBQo66Fw8?si=UlWXj1a5AdNjvfXJ
On Pi Day 2025, as you might recall, I introduced you (more or less) to 3Blue1Brown. Also known as Grant Sanderson.
If there's any better source of animated math presentations, than Mr. Sanderson, I'm unaware of it.
Key word: "animated." In addition, he is a warm, enthusiastic teacher. And of course he knows his stuff well—how else could he have made the truly fantastic animations?
I haven't watched them all yet. But so far, I especially like 2016's BUT WHAT IS THE RIEMANN ZETA FUNCTION? VISUALIZING ANALYTIC CONTINUATION and 2019's DIFFERENTIAL EQUATIONS, A TOURIST'S GUIDE | DE1.
I may never again be able to ponder some ideas they convey, without seeing those animations in my head. They're that perfect.
So give those two videos a try, if you're comfortable enough with the one I had shared on Pi Day...if now you want a couple which are more challenging. Maybe unforgettable, too.
#3Blue1Brown
#RiemannZetaFunction
#DifferentialEquations
#learning
https://mindly.social/@setsly/114161481212443838
Polynomial normal forms for ODEs near a center-saddle equilibrium point, now in Journal of Differential Equations from our colleague A. Delshams and his collaborator P. Zgliczyński.
Check it out here to learn more:
https://www.sciencedirect.com/science/article/abs/pii/S0022039625002955?via%3Dihub
A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).
Background:
The standard pendulum period of \(2\pi\sqrt{L/g}\) or frequency \(\sqrt{g/L}\) holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.
In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by \((x, y)=R(\theta-\sin\theta, -1+\cos\theta)\), where \(\theta=0\) corresponds to the cusp. Consider a pendulum of length \(L=4R\) hanging from the cusp, and let \(\alpha\) be the angle the string makes with the vertical, as shown (in the proof).
#Pendulum #Cycloid #Period #Frequency #SHM #TimePeriod #CycloidalPendulum #Lagrange #Cusp #Energy #KineticEnergy #PotentialEnergy #Lagrangian #Length #Math #Maths #Physics #Mechanics #ClassicalMechanics #Amplitude #CircularFrequency #Motion #Vibration #HarmonicMotion #Parameter #ParemeterizedEquation #GoverningEquations #Equation #Equations #DifferentialEquations #Calculus
Splash-free urinals: Design through physics and differential equations
https://academic.oup.com/pnasnexus/article/4/4/pgaf087/8098745?login=false
#HackerNews #SplashFreeUrinals #DesignThroughPhysics #DifferentialEquations #Innovation #Invention
I could watch forever ... #LorenzAttractor #ButterflyEffect #JavaScript #p5js #DifferentialEquations #math #visualization #computergraphics #dynamicsystems
L'Hôpital's rule is when you're trying to calculate the derivative of a complex function. Not only you fail, but the function beats you up, steals your lunch money, and sends you to the hospital.
#math #calculus #DifferentialEquations #LHôpitalsRule #AbjectFailure #MathJoke
WHY VARIATION OF PARAMETERS WORKS
This is more conceptual than a proof, but I find it comforting.
Consider the equation:
y' - y = xe^x
The solution will consist of a Homogeneous Solution and a Particular Solution; add the two for the complete solution.
The Homogeneous Solution is the solution that, when you run it through the left side of the equation, it always goes to zero. So you need the Homogeneous Solution for the same reason you need the "+ C" in antiderivatives: even if it's kind of the boring throwaway part of the solution, it is still part of the complete solution.
But Variation of Parameters finds another use for the Homogeneous Solution. Let us suppose that the Particular Solution is the Homogeneous Solution times a function "u". Well, if you feed the Particular Solution through the left side of the equation, the Homogeneous Solution part will tend to go away, leaving "u". So then you can multiply "u" by the Homogeneous Solution, and you've got your Particular Solution.
It kind of reminds me of how Taylor Series work. In a Taylor Series, you have a function that you can think of as secretly containing a multitude of polynomial terms, and the trick is finding a way to torture the function into confessing the coefficients on each polynomial term. In the case of Taylor it's done by iteratively differentiating and then setting "x" to zero, thus leaving a constant that is the coefficient for a given polynomial. (There's also that "n!" term but that's just details.)
Or Fourier Series: a periodic function secretly contains a multitude of sine and cosine terms, and again you find a way to torture it into confessing the coefficients on each sine / cosine. In that case the torture technique involves integration.
And in the case of Variation of Parameters, the torture technique is, we know what part of the particular solution gets burned away by the left side of the equation; the charred skeleton that remains is the other part of the particular solution.
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i wonder if it's possible to make a video game that would teach kids the intuition behind the "butterfly effect" by making each subsequent level's initial conditions determined by the outcome of the previous level, in such a way that the gameplay would guide you towards understanding that tiny decisions made on the first level will have huge consequences on level two, etc. #chaos #butterflyEffect #chaosTheory #deterministic #systems #nonlinear #differentialEquations #bifurcation cc @JamesGleick
Convection–diffusion equation
The convection-diffusion equation is a more general version of the scalar transport equation. It is a combination of the diffusion and convection (advection) equations. It describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection.
\[\dfrac{\partial c}{\partial t} = \mathbf{\nabla} \cdot (D \mathbf{\nabla} c - \mathbf{v} c) + R\]
\[\dfrac{\partial c}{\partial t} = \underbrace{\mathbf{\nabla} \cdot (D \mathbf{\nabla} c)}_{\text{diffusion}}-\overbrace{\underbrace{\mathbf{\nabla}\cdot (\mathbf{v} c)}_{\text{advection}}}^\text{convection} + \overbrace{\underbrace{R}_\text{destruction}}^\text{creation}\]
\(\mathbf{\nabla} \cdot (D \mathbf{\nabla} c)\) is the contribution of diffusion.
\(- \mathbf{\nabla}\cdot (\mathbf{v} c)\) is the contribution of convection or advection.
\(R\) describes the creation or destruction of the quantity.
where
\(c\) is the variable of interest.
\(D\) is the diffusivity.
\(\mathbf{v}\) is the velocity field, and
\(R\) is the sources or sinks of the quantity \(c\).
#Convection #Diffusion #Transport #Advection #Equation #ConvectionDiffusionEquation #DifferentialEquations #AdvectionEquation #DiffusionEquation #TransportEquation #ConvectionEquation
New lectures on undergraduate differential equations:
18. Mechanical vibrations, part 2: free damped motion (Notes on Diffy Qs, 2.4)
https://youtu.be/Z6BR9aDPdy4
19. Nonhomogeneous equations, part 1: undetermined coefficients (Notes on Diffy Qs, 2.5)
https://youtu.be/jRPuCCQqGXU
Based on the free book: Notes on Diffy Qs https://www.jirka.org/diffyqs
The entire playlist is at:
https://www.youtube.com/playlist?list=PLRfQb6m35rf5E7QllafnyOXD0tHHI_N9E
#math #maths #mathematics #differentialequations #diffyqs
#OER
A new lecture on undergraduate differential equations:
17. Mechanical vibrations, part 1: free undamped motion (Notes on Diffy Qs, 2.4)
https://youtu.be/RYti_vJ8sQU
Based on the free book: Notes on Diffy Qs https://www.jirka.org/diffyqs
The entire playlist is at:
https://www.youtube.com/playlist?list=PLRfQb6m35rf5E7QllafnyOXD0tHHI_N9E
#math #maths #mathematics #differentialequations #diffyqs
#OER
A new lecture on undergraduate differential equations:
16. Higher order linear ODEs (Notes on Diffy Qs, 2.3)
https://youtu.be/AzshfIf-qCA
Based on the free book: Notes on Diffy Qs https://www.jirka.org/diffyqs
The entire playlist is at:
https://www.youtube.com/playlist?list=PLRfQb6m35rf5E7QllafnyOXD0tHHI_N9E
#math #maths #mathematics #differentialequations #diffyqs
#OER
Whenever I walk to/from home, I have to walk up/down an inclined street; I noticed that the asphalt floor has different curvatures depending on how near it is of a bend, and I try to find a less steep incline while walking.
This got me inspiration for the few questions below. Any simple explanations, and related links, are welcome.
Given a #differentiable surface within R^3, and two distinct points in it, there are infinitely many differentiable paths from one point to another, remaining on the surface. At each point of the #path, one can find the path's local #curvature. Then:
- Find a path that minimizes the supreme of the curvature. In other words, find the "flattest" path.
- Find a path that minimizes the variation of the curvature. In other words, find a path that "most resembles" a circle arc.
Are these tasks always possible within the given conditions? Are any stronger conditions needed? Are there cases with an #analytic solution, or are they possible only with numerical approximations?
#Analysis #DifferentialGeometry #Calculus #DifferentialEquations #NumericalMethods
from "Present State of the Lanchester Theory of Combat" by Ladislav Dolanský (1964)
https://pubsonline.informs.org/doi/10.1287/opre.12.2.344?utm_source=dlvr.it&utm_medium=mastodon
LINEAR TRANSPORT EQUATION
The linear transport equation (LTE) models the variation of the concentration of a substance flowing at constant speed and direction. It's one of the simplest partial differential equations (PDEs) and one of the few that admits an analytic solution.
Given \(\mathbf{c}\in\mathbb{R}^n\) and \(g:\mathbb{R}^n\to\mathbb{R}\), the following Cauchy problem models a substance flowing at constant speed in the direction \(\mathbf{c}\).
\[\begin{cases}
u_t+\mathbf{c}\cdot\nabla u=0,\ \mathbf{x}\in\mathbb{R}^n,\ t\in\mathbb{R}\\
u(\mathbf{x},0)=g(\mathbf{x}),\ \mathbf{x}\in\mathbb{R}^n
\end{cases}\]
If \(g\) is continuously differentiable, then \(\exists u:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) solution of the Cauchy problem, and it is given by
\[u(\mathbf{x},t)=g(\mathbf{x}-\mathbf{c}t)\]
#LinearTransportEquation #LinearTransport #Cauchy #CauchyProblem #PDE #PDEs #CauchyModel #Math #Maths #Mathematics #Linear #LinearPDE #TransportEquation #DifferentialEquations
I am excited to see "Splitting methods for differential equations" by Sergio Blanes, Fernando Casas and Ander Murua on arXiv: https://arxiv.org/abs/2401.01722
This is a review article to be published in the excellent Acta Numerica. It discusses numerical methods for solving differential equations which can be split in several parts that are easier to solve. In formulas, the (ordinary or partial) differential equation is 𝑑𝑢/𝑑𝑡 = 𝑓(𝑢) and the splitting is 𝑓(𝑢) = 𝑓₁(𝑢) + 𝑓₂(𝑢).
For people that don't do numerical analysis or computational mathematics, it may be helpful to think of the Lie–Trotter product formula
\[ e^{A+B} = \lim_{n\to\infty} (e^{A/n}e^{B/n})^n. \]
This is the simplest splitting method. Part of the game is to find formulas that converge faster.
