https://en.wikipedia.org/wiki/Differential_equation
A differential equation is an equation that relates one or more functions and their derivatives.
An ordinary differential equation (ODE) is an equation containing an unknown function y(t) of 'one' real or complex variable t, its derivatives, and some given functions of t.
Python code euler1.py - Euler's forward method euler2.py - Heun's method euler3.py - Midpoint method
dy/dt = f(t,y), t in [t_a, t_b]. y(t) is an unknown function. Initial condition y(t_a) = y_a. This is ODE Initial Value Problem (IVP). Euler's forward method from Taylor series t[0] = t_a, y[0] = y_a, t[i+1] = t[i] + h, y[i+1] = y[i] + f(t[i], y[i]) * h. # dy = f(t,y) * dt Example: dy/dt = 3(1+t)-y, y(1) = 4, t in [1, 2]. [pattern: dy/dt + P(t)y = Q(t)] Dokładne rozwiązanie metodą uzmienniania stałych. dy/dt = -P(t)y, y = C(t)*exp(-\int P dt), dC/dt = Q(t)*exp(+\int P dt) dy/dt = -y, y = C(t)*exp(-t), dC/dt = 3(1+t)*exp(t), Exact solution y = 3t + exp(1-t).
Improvements of Euler's method
(1) Heun's method
y0[i+1] = y[i] + f(t[i], y[i]) * h # a predictor equation
An estimated slope at the end of the interval: f(t[i+1], y0[i+1]).
The average slope is used to extrapolate linearly from y[i] to y[i+1].
Predictor: y0[i+1] = y[i] + f(t[i], y[i]) * h
Corrector: y[i+1] = y[i] + \frac{ f(t[i], y[i]) + f(t[i+1], y0[i+1]) }{ 2 } * h
(2) Midpoint method
t[i+1/2] = t[i] + h/2
y[i+1/2] = y[i] + f(t[i], y[i]) * h/2
A slope at the midpoint: f(t[i+1/2], y[i+1/2])
y[i+1] = y[i] + f(t[i+1/2], y[i+1/2]) * h.