AwesomeEquations

Dual(a, b)

Dual number with value a and derivative b.

A dual number $x = a + b \epsilon$ is defined such that $\epsilon^2 = 0$ and $f(a + b \epsilon) = f(a) + f'(a) b \epsilon$ for all $f$. The derivative of a function is thus given by $f'(a) = y$ when $f(a + \epsilon) = x + y \epsilon$.

finite_diff(f, x; h = sqrt(eps(typeof(x))))

Compute the derivative of f at x using a central finite difference (second order).

forward_diff(f, x)

Compute the derivative of f at x using forward mode automatic differention (with dual numbers).

forward_euler(f, p, u₀, T, Δt = 0.1)

Solve ODE $du/dt = f(u, p, t)$ with forward Euler, where $p$ are parameters.

rk4(f, p, u₀, T, Δt = 0.1)

Solve ODE $du/dt = f(u, p, t)$ with RK4, where $p$ are parameters.