Ciro Santilli @cirosantilli 34

 Incoming links: Differential equation

Applications of Lie groups to differential equations  Updated 2024-12-15  +Created 1970-01-01

Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.

It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.

www.researchgate.net/profile/Michael_Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf Lie-Groups as a Tool for Solving Differential Equations by Michael Frewer. Slides with good examples.

 Read the full article

Complete basis  Updated 2024-12-15  +Created 1970-01-01

 View more

Finding a complete basis such that each vector solves a given differential equation is the basic method of solving partial differential equation through separation of variables.

The first example of this you must see is solving partial differential equations with the Fourier series.

Notable examples:

Fourier series for the heat equation as shown at Fourier basis is complete for $L^{2}$ and solving partial differential equations with the Fourier series
Hermite functions for the quantum harmonic oscillator
Legendre polynomials for Laplace's equation in spherical coordinates
Bessel function for the 2D wave equation on a circular domain in polar coordinates

 Read the full article

Control theory  Updated 2024-12-15  +Created 1970-01-01

 View more

This basically adds one more ingredient to partial differential equations: a function that we can select.

And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?

It's quite fun from a mathematics point of view!

Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.

 Read the full article

Dirac equation vs quantum electrodynamics  Updated 2024-12-15  +Created 1970-01-01

 View more

TODO: in high level terms, why is QED more general than just solving the Dirac equation, and therefore explaining quantum electrodynamics experiments?

Also, is it just a bunch of differential equation (like the Dirac equation itself), or does it have some other more complicated mathematical formulation, as seems to be the case? Why do we need something more complicated than

Advanced quantum mechanics by Freeman Dyson (1951) mentions:

A Relativistic Quantum Theory of a Finite Number of Particles is Impossible.

Bibliography:

physics.stackexchange.com/questions/101307/dirac-equation-in-qft-vs-relativistic-qm
physics.stackexchange.com/questions/44188/what-is-the-relativistic-particle-in-a-box/44309#44309 says:
By several reasons explained in textbooks, the Dirac equation is not a valid wavefunction equation. You can solve it and find solutions, but those solutions cannot be interpreted as wavefunctions for a particle
physics.stackexchange.com/questions/64206/why-is-the-dirac-equation-not-used-for-calculations
www.physicsforums.com/threads/is-diracs-equation-still-useful-after-qed-is-developed.663994/

 Read the full article

Existence and uniqueness  Updated 2024-12-15  +Created 1970-01-01

 View more

Existence and uniqueness results are fundamental in mathematics because we often define objects by their properties, and then start calling them "the object", which is fantastically convenient.

But calling something "the object" only makes sense if there exists exactly one, and only one, object that satisfies the properties.

One particular context where these come up very explicitly is in solutions to differential equations, e.g. existence and uniqueness of solutions of partial differential equations.

 Read the full article

Exponential function differential equation  Updated 2024-12-15  +Created 1970-01-01

 View more

The differential equation that is solved by the exponential function:

y^{'} (x) = y (x)

(1)

with initial condition:

y (0) = 1

(2)

TODO find better name for it, "linear homogenous differential equation of degree one" almost fully constrainst it except for the exponent constant and initial value.

 Read the full article

L^{2}

 Updated 2024-12-15  +Created 1970-01-01

 View more

L^{p}

for

p = = 2

L^{2}

is by far the most important of

L^{p}

because it is quantum mechanics states live, because the total probability of being in any state has to be 1!

L^{2}

has some crucially important properties that other

L^{p}

don't (TODO confirm and make those more precise):

it is the only $L^{p}$ that is Hilbert space because it is the only one where an inner product compatible with the metric can be defined:
- math.stackexchange.com/questions/2005632/l2-is-the-only-hilbert-space-parallelogram-law-and-particular-ft-gt
- www.quora.com/Why-is-L2-a-Hilbert-space-but-not-Lp-or-higher-where-p-2
Fourier basis is complete for $L^{2}$ , which is great for solving differential equation

 Read the full article

Legendre polynomials  Updated 2024-12-15  +Created 1970-01-01

 View more

Show up when solving the Laplace's equation on spherical coordinates by separation of variables, which leads to the differential equation shown at: en.wikipedia.org/w/index.php?title=Legendre_polynomials&oldid=1018881414#Definition_via_differential_equation.

 Read the full article

SymPy  Updated 2024-12-15  +Created 1970-01-01

 View more

This is the dream cheating software every student should know about.

It also has serious applications obviously. www.sympy.org/scipy-2017-codegen-tutorial/ mentions code generation capabilities, which sounds super cool!

The code in this section was tested on sympy==1.8 and Python 3.9.5.

Let's start with some basics. fractions:

from sympy import *
sympify(2)/3 + sympify(1)/2

outputs:

7/6

Note that this is an exact value, it does not get converted to floating-point numbers where precision could be lost!

We can also do everything with symbols:

from sympy import *
x, y = symbols('x y')
expr = x/3 + y/2
print(expr)

outputs:

x/3 + y/2

We can now evaluate that expression object at any time:

expr.subs({x: 1, y: 2})

outputs:

4/3

How about a square root?

x = sqrt(2)
print(x)

outputs:

sqrt(2)

so we understand that the value was kept without simplification. And of course:

sqrt(2)**2

outputs 2. Also:

sqrt(-1)

outputs:

I is the imaginary unit. We can use that symbol directly as well, e.g.:

I*I

gives:

-1

Let's do some trigonometry:

cos(pi)

gives:

-1

and:

cos(pi/4)

gives:

sqrt(2)/2

The exponential also works:

exp(I*pi)

gives;

-1

Now for some calculus. To find the derivative of the natural logarithm:

from sympy import *
x = symbols('x')
diff(ln(x), x)

outputs:

1/x

Just read that. One over x. Beauty.

Let's do some more. Let's solve a simple differential equation:

y''(t) - 2y'(t) + y(t) = sin(t)

Doing:

from sympy import *
x = symbols('x')
f, g = symbols('f g', cls=Function)
diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x)**4)
print(dsolve(diffeq, f(x)))

outputs:

Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)

which means:

f (x) = C_{1} + C_{2} x e^{x} + c o s (x) / 2

(1)

To be fair though, it can't do anything crazy, it likely just goes over known patterns that it has solvers for, e.g. if we change it to:

diffeq = Eq(f(x).diff(x, x)**2 + f(x), 0)

it just blows up:

NotImplementedError: solve: Cannot solve f(x) + Derivative(f(x), (x, 2))**2

Sad.

Let's try some polynomial equations:

from sympy import *
x, a, b, c = symbols('x a b c d e f')
eq = Eq(a*x**2 + b*x + c, 0)
sol = solveset(eq, x)
print(sol)

which outputs:

FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))

which is a not amazingly nice version of the quadratic formula. Let's evaluate with some specific constants after the fact:

sol.subs({a: 1, b: 2, c: 3})

which outputs

FiniteSet(-1 + sqrt(2)*I, -1 - sqrt(2)*I)

Let's see if it handles the quartic equation:

x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)

Something comes out. It takes up the entire terminal. Naughty. And now let's try to mess with it:

x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(f*x**5 + e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)

and this time it spits out something more magic:

ConditionSet(x, Eq(a + b*x + c*x**2 + d*x**3 + e*x**4 + f*x**5, 0), Complexes)

Oh well.

Let's try some linear algebra.

m = Matrix([[1, 2], [3, 4]])

Let's invert it:

m**-1

outputs:

Matrix([
[ -2,    1],
[3/2, -1/2]])

 Read the full article