Pade-Partial Fraction Approximation

This package is an implementation of the Pade-partial fraction approximation for the inversion of the Laplace transform as proposed in Computation of the Distribution of the Absorption Time of the Drifted Diffusion with Stochastic Resetting and Mixed Boundary Conditions by Turin, Magalang, Aguilar, Colombani, Sanchez-Taltavull, and Gatto¹

Installation

Install the package from PyPI:

pip install ppf_approx

Usage

The main function of this package is the ppf function, with the following syntax:

ppf(m_ord, n_ord, v, D, x0, r, tval, correction = True)

Here is a description of each argument:

• m_ord : Integer. Numerator order of the Pade approximation.
• n_ord : Integer. Denominator order of the Pade approximation.
• v : Float. Drift constant.
• D : Float. Diffusion constant. Must be positive.
• x0 : Float. Initial position. Must be from $0 \leq x_0 < 1$
• r : Float. Resetting constant. Must be non-negative.
• tval : Array. Range of time values at which the PPF will be performed. Single values are accepted.
• correction : Boolean, optional. Determines whether the PPF will perform the correction steps or not. The default is True.

The function can be used to generate whole distributions $t \in [0, \infty]$, segments $t \in [t_1, t_2]$, or single values of time $t$. The outputs will always be in arrays. This is an example of the usage of the ppf function:

import numpy as np
import ppf_approx

v = -0.01
D = 0.0001
x0 = 0.8
r = (1/3)*(1/365)
tvals = np.linspace(0,200, 10)

app_fptvals = ppf(2, 3, v, D, x0, r, tvals)

Sample output:

>>> app_fptvals
array([0., 0., 0.01115898, 0.01866108, 0.01345554, 0.00637309, 0.00200653, 0.00027748, 0., 0.])

Auxiliary Functions

This package also contains auxiliary functions which are expressions for the bounded and biased Brownian motion with stochastic resetting that were derived in the reference¹:

1. lt_fptd computes for the Laplace-transformed FPT distribution.
2. mfpt computes for the analytical mean FPT from the Laplace-transformed FPT distribution.

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