paper.md

title	tags	authors	affiliations	date	bibliography
Inteq: Solve Volterra and Fredholm integral equations in Python	Python mathematics economics numerical	name affiliation Matthew W. Thomas^[corresponding author] 1	name index Matthew Wildrick Thomas, Economics PhD Candidate, Northwestern University 1	18 June 2021	paper.bib

Summary

Volterra and Fredholm integral equations describe important processes in fluid dynamics, signal processing, physics, computer graphics, actuarial science, and economics. Most of these equations do not admit closed form solutions. As a result, these equations must be approximated numerically. Due to the importance of these equations in a variety of fields, there is a requirement for efficient numerical tools which can be used across fields.

Statement of need

Inteq is a Python package that efficiently constructs numerical approximations of Fredholm and Volterra integral equations. Despite the wide interdisciplinary applicability of these integral forms, there is currently no pre-built solution for solving them. Existing solutions are typically purpose built by practitioners for specific problems and are not public.

None of the algorithms implemented in Inteq are entirely novel. However, there are several enhancements of existing methods.

Fredholm integral equations of the first kind take the form: $$ f(s)=\int_a^b K(s,y)g(y)dy $$ where you want to solve for $g$. For Fredholm integral equations of the first kind, we apply the methods of @Twomey:1963 which is an improved version of @Phillips:1962. The method can be applied using Simpsons rule, the trapezoid rule, the midpoint rule, or Gaussian quadrature depending on the properties of the functions. Volterra integral equations of the first kind take the form: $$ f(s)=\int_a^s K(s,y)g(y)dy $$ where we want to solve for $g$. We solve Fredholm integral equations using the methods in @Betto:2021 and @Linz:1969.

Inteq is a dependency of the allpy package which uses Volterra integral equations to find the equilibria of all-pay auctions. Both packages are used in @Betto:2021.

References