volterra-equation-solvers

Collocation-method solvers for Volterra integral and integro-differential equations (VIEs/VIDEs), based on:

Brunner H. Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press; 2004.

Solvers

Function	Equation
solve_VIE_1	\(g(t) = \int_0^t K(t-s)y(s)ds\)
solve_VIE_2	\(y(t) = g(t) + \int_0^t K(t-s)y(s)ds\)
solve_VIDE	\(y'(t) = a(t)y(t) + g(t) + \int_0^t K(t-s)y(s)ds\)

Quick install

pip install git+https://github.com/trout314/volterra-equation-solvers

Mathematical derivations

The docs/ directory contains worked derivations of the analytic solutions used in the test suite:

File	Contents
scalar_solutions.pdf	Derivations for all six scalar test cases (VIE-1, VIE-2, VIDE)
coupled_vector_solutions.pdf	Derivations for the coupled 2×2 vector test cases, constructed via a similarity transform

LaTeX source files are provided alongside the PDFs.

Quick example

import numpy as np
from volterra_equation_solvers import solve_VIE_2

# y(t) = sin(t) satisfies this VIE-2
time_step = 0.05
times = np.arange(0, 9.05, time_step)
kernel = np.exp(-times)
g = np.sin(times) - 0.5 * (np.exp(-times) + np.sin(times) - np.cos(times))

soln = solve_VIE_2(kernel_values=kernel, g_values=g, time_step=time_step)