maths.laplace_transformation

Laplace Transform — Numerical Implementation.

Computes the numerical Laplace Transform using the trapezoidal integration rule. Supports real-valued, non-negative Laplace parameters only.

Reference: https://en.wikipedia.org/wiki/Laplace_transform

Functions

laplace_transform(→ float)

Calculate the numerical Laplace Transform of a function given its values over time.

Module Contents

maths.laplace_transformation.laplace_transform(function_values: numpy.ndarray, s_value: float, delta_t: float) → float

Calculate the numerical Laplace Transform of a function given its values over time.

This implementation supports only real-valued, non-negative Laplace parameters s.

Args:

function_values: A numpy array of the function values f(t). s_value: The real-valued Laplace parameter s. Must be non-negative. delta_t: The time step between samples.

Returns:

The approximate real-valued value of the Laplace transform at s_value.

Example: For f(t) = 1, the Laplace transform L{1} = 1/s. If s = 2, L{1} should be 0.5.

>>> t = np.linspace(0, 50, 10000)
>>> f_t = np.ones_like(t) # f(t) = 1
>>> res = laplace_transform(f_t, s_value=2.0, delta_t=50/10000)
>>> abs(res - 0.5) < 1e-3
True

Example: For f(t) = e^(-t), the Laplace transform L{e^-t} = 1/(s+1). If s = 1, L{e^-t} should be 0.5.

>>> t = np.linspace(0, 50, 10000)
>>> f_t = np.exp(-t)
>>> res = laplace_transform(f_t, s_value=1.0, delta_t=50/10000)
>>> abs(res - 0.5) < 1e-3
True