See also:

  • Linear Algebra

Closed Form Solutions

Differential Equations are neat. They are the mathematical underpinning of a lot of physics.

Newtonian Mechanics is to canonical instance of differential equations. This may be the mechanics of point particles, or of rigid bodies.

There is a certain set of closed form solutions which are the bones we hang our intuition around

By differential equation, I tend to mean 1-d, and initial value problem.

Integrals

$\frac{dy}{dx} = f(x)$ is indeed a differential equation, one so “trivial” in some sense that we tend to not view it as such.

The solution is $y(x) = \int_0^x f(x) + C$.

Decay

$\frac{dy}{dx} = -ky$

The solution to the equations is $y(x) = Ce^{-kx}$. You can confirm this works by differentiating the right hand side.

We may choose to think of exponential explosion, for which the sign of $k$ is different as a different problem or not. Conceptually, it can be very different.

Where can this show up?

  • RC circuits. If you attach a resistor to a charged capaictor, the charge will leak through the resistor.
  • Particle Decay
  • Cooling & Heating.
import numpy as np print(np.array([1,2,3]))

Forced Decay

The method of “integrations factors”

Primeval Green’s Functions / Impulse Function

Simple Harmonic Motion

There’s a lot here. This is really the bread and butter.

Coupled form $\frac{dp}{dt} = -kx $ $\frac{dx}{dt} = p/m$

Pippard Book Waves Book

Forced Simple Harmonic Motion

Simple Harmonic Motion with Drag

Non closed Form Problems

Many methods can be built on the assumption that the result can be approximated by a power series

Iterative Methods

For “short times” $t$ is a small parameter and it is reasonable to expect you can approximate the solution well

Perturbation Methods

Numerical parameter continuation

Fixed Points

Finding fixed points is a question that is much easier than solving a differential equation. It takes out the differential stuff and now you just need to solve a possibly nonlinear set of equations. Linearization around a fixed point can be useful. Tells you about stability.

Chaos

Attractors

Koopman Operators

Lie Method

Extensions

Matrix “Decay”

$\frac{}$

Control Problems

LQR

Inverse Problems?

DFT, Fourier Series, Fourier Transform

Finite Difference

For many of the closed form problems in the continuum differential equation case, there are analagous closed forms for the finite difference case. It is not altogether apparent to me why the continuum case is so much more emphasized, but there is a sense that the results are cleaner or simpler is some sense, while confusingly finite difference really is much more mundane foundationally speaking.

Summations

Growth Equation

Computers

Symbolic

State Machine Analogs

Symbolic Solution of Diff Eqs

See sympy docs

Pattern matching

Lie Algebra methods

Numeric

Numbers

Finite Difference

Validated Numerics

One can consider blobs (sets) of functions in function space. If you work with only a constrained class of blobs, these blobs can be finitely representable in a computer. It’s similar to how we can work in finite subsets or intervals of the reals. You can’t describe every possible set of reals in these terms. You can also over approximate sets, and perform approximations of union, intersection, complement in principled ways. This is really interesting.

Partial differential equations often come from the combination of a constitutive relation with a conservation law.

Conservations Laws:

  • Conservation of mass
  • conservation of energy
  • conservation of momentum
  • Force Balance (for statics)
  • Conservation of charge
  • Conservation of number (similar to Mass)

Constitutive Relations

  • Ohm’s Law $V = IR$
  • Hooke’s Law $F = kx$
  • Fick’s Law
  • Viscous Drag
  • Capicitance $Q = CV$
  • Inductance $\Phi = LI$
  • Fourier’s Law. Heat flow is proportional to temperature difference $J = k \Delta T$

The constitutive relations are most familiar or intuitive to us in kind of bulk form. We can imagine connecting together lots of little resistors for example to have a continuous conductive medium.

1-D Laplace

$frac{d^2 \phi }{dx^2} = 0$ $ \phi(0) = a$ $ \phi(1) = b$

Particle in a Box

Periodic Systems and Bands

A periodic system is separable into it’s periodic and cell parts. $H = I \otimes H_1 - C \otimes H_2 - C^{-1} \otimes H_3 $

Crystal momentum.

Method of Matching coefficients

Higher PDE Dimensions

Poisson Free Space (Coulomb’s Law)

Separation of Variables

Laplace Box

Analytic Functions

Circle

Morse and Feshbach

Algebraic multigrid

https://github.com/pyamg/pyamg

Fast Multi pole method

FFT

Domain Decomposition

Finite Elements / Galerkin

Boundary Element Method