Differential Equations Tutorial

cds.diffeq solves initial-value problems with classical schemes plus an adaptive RK45 integrator.

1. Single ODE

Solve dy/dt = -y, y(0) = 1 whose exact value at t=1 is e^-1 ≈ 0.3679:

from cds.diffeq import euler_method, midpoint_method, rk4, rk45

f = lambda t, y: -y
print(euler_method(f, t0=0.0, y0=1.0, t_end=1.0, n=100).value)
print(midpoint_method(f, t0=0.0, y0=1.0, t_end=1.0, n=100).value)
print(rk4(f, t0=0.0, y0=1.0, t_end=1.0, n=10).value)        # very accurate
print(rk45(f, t0=0.0, y0=1.0, t_end=1.0, rtol=1e-6).value)  # adaptive

2. System of ODEs

from cds.diffeq import solve_system

def lotka(t, state):
    x, y = state
    return [1.1*x - 0.4*x*y, 0.1*x*y - 0.4*y]   # Lotka-Volterra

sol = solve_system(lotka, t0=0.0, y0=[10.0, 5.0], t_end=15.0, n=300)
print(sol.value)    # final [prey, predator]
print(len(sol.ts))  # trajectory points

Run the full demo with python examples/diffeq_demo.py.