Ordinary Differential Equations

y' = f(x,y(x))

y'' = f(x,y(x),y'(x))

...

x may be a vector
no partial derivatives!

Goal

Develop a purely semantic account of approximate solutions to differential equations.

Subgoals

Leave the numbers to the computers
Use only high school calculus

Sub-subgoals

Choices should be as natural as possible
Choices should preserve as much information as possible

Anti-Goals

Magic numbers

Butcher tableau for Runge-Kutta 4

Non-compositionality

type Step s = (s,TimeDelta)->s
run :: s -> Step s -> TimeDelta -> [(TimeDelta,s)]

steps don't compose.
Step /= run Step

Intuitions

A differential equation is an instantaneous recurrence relation
Like recurrence relations we have existance and uniqueness proofs
Like recurrence relations these proofs derive from fix-point theorems
C.f. "Calculus in coinductive form," Dusko Pavlovic and Martin H. Escardo.

Tools

class Analytic a where
    integ :: a -> a
    derive :: a -> a

class Sample a where
   sampleAt :: a -> Double -> Double

newtype Poly = Poly [Double]
instance Num Poly where...
instance Fractional Poly where...
instance Analytic Poly where...
instance Sample Poly where...
c.f. Doug McIlroy

Simple (est?) example of a differential

f(x) = f'(x)

f(x) = C * e^x

Provide an initial value

First fundamental theorem of calculus:

when...

$\displaystyle F = \int f(x)\,\mathrm{d}x$

then...

$\displaystyle{\mathrm{d}F\over\mathrm{d}x} = f(x)$

i.e. F'(x) = f(x)
i.e. diff . integrate = id

Second Fundamental Theorem of Calculus

$\displaystyle\int_a^b f(x) dx = F(b) - F(a)$

where is an indefinite integral (antiderivative) of .

A Little Algebra...

Flip

$\displaystyle F(b) = \int_a^b f(x)\,\mathrm{d}x + F(a)$
Distinguish two Fs

$\displaystyle F(b) = \int_a^b f(x)\,\mathrm{d}x+ F_0(a)$

Expand...

$\displaystyle F(t_n) =\int_{t_{n-1}}^{t_n} f(x)\,\mathrm{d}x + F_{n-1}(t_{n-1}) + F_{n-2}(t_{n-2}) ... F_0(t_0)$

is of type Stream!

Streams are straightforward

type TimeDelta = Double
data Stream a = Stream TimeDelta a (Stream a)
instance Sample a => Sample (Stream a) where ...
instance Num a => Num (Stream a) where...
instance Fractional a => Fractional (Stream a) where...
instance (Analytic a, Sample a) => Analytic (Stream a) where..
instance Sample a => Sample (Stream a) where...

As is lifting operations to streams.

applWith f a b = go a b
  where go sx@(Stream dtx x nx) sy@(Stream dty y ny) =
     case compare dtx dty of
          EQ -> Stream dtx (f x y) $ go nx ny
          GT -> Stream dty (f x y) $ go (sdrop dty sx) ny
          LT -> Stream dtx (f x y) $ go nx (sdrop dtx sy)

c.f. Conal Elliot

Reformulate

y' = f(x,y(x))

Pointwise functions lift functorially into linear operators

$\displaystyle lift :: (a -> a) -> (\mathbb{R} -> a) -> (\mathbb{R} -> a)$

$lift(f) = \lam x -> f(x,y(x))$

y' = lift(f)(y)

Extrapolate...

Given F_n and taking F_n -> f_n , approximate $F_{n+1}$

$op(F_{n-1}) = f_{n-1}$

$h = t_{n} - t_{n-1}$

$\displaystyle F_{n} = {\int_{t_{n-1}}^{t_n} f(x)\,\mathrm{d}x} + F_{n-1}(t_{n-1})$

$\displaystyle F_{n} = {\int_{t_{n-1}}^{t_n} (op(F_{n-1})) }\, \mathrm{d}x} + F_{n-1}(t_{n-1})$

$\displaystyle F_{n} = {\int_{0}^{h} op(F_{n-1})\circ (+h) }\, \mathrm{d}x} + F_{n-1}(t_{n-1})$

If is on polynomials and is polynomial then every term is polynomial!
is (approximately) a coinductively defined polynomial.

Streams of Polynomials

type PolySpline = Stream Poly
instance {Num, Fractional, Analytic} PolySpline where...

instance Num PolySpline where
   fromInteger = pure . fromInteger
   negate = fmap negate
   (+) = applWith (+)
    ...

And we can run it

expFun = splice h initExpFun $ 
           (integ expFun' .+ (initExpFun @> h)) 
           `extrapForward` h
    where 
      initExpFun = pure 1
      expFun' = expFun
      h = 0.01

-- expFun @> 1 = 2.71814
-- exp 1 = 2.71828
-- by comparison, rk3 at h of 0.01 = 2.69120

Wait, what?

We implemented a straightforward representation of analytic functions.
We did a little math.
A solver for differential equations emerged.

Is it any good?