Differential Equations
- Closed Form Solutions
- Non closed Form Problems
- Extensions
- DFT, Fourier Series, Fourier Transform
- Finite Difference
- Computers
- Higher PDE Dimensions
- Functional Analysis
- Misc
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.
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
Lie Method
Extensions
Matrix “Decay”
$\frac{}$
Control Problems
LQR
Inverse Problems?
DFT, Fourier Series, Fourier Transform
Bracewell book Course
Schwarz distributions
Dirac Delta
https://en.wikipedia.org/wiki/Dirac_delta_function
The identity operator as an integral transform $\int \delta(x-y) f(x) dx = f(y) $
$\delta(x) = \int_{-\inf}^{\inf} e^{ikx}dx $
Can consider limit of gausian or square function
$ \delta(x) = e^{-\frac{x^2}{\alpha}} $
from sympy import *
init_printing(use_unicode=False, wrap_line=False)
x, a = symbols('x a')
print(integrate(exp(-x**2/a), (x, -oo, oo), conds='none'))
Laplace
More appropriate for initial value problems / transients. Fourier is kind of goofy and yet it feels more sensible to consider a superposition of sine and cosine. There is also more symmettry between the forward and inverse trasnform
Related transforms
Radon Mellin Hankel
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