A New Numerical Method for Fast Solution of Partial Integro-Differential Equations

TL;DR

This paper introduces a novel numerical method leveraging fast matrix multiplication for efficiently solving partial integro-differential equations, demonstrating linear computational growth compared to traditional quadratic methods.

Contribution

The paper presents a new numerical approach based on fast matrix multiplication that significantly reduces computational complexity for solving PDEs.

Findings

Linear growth in computational operations with grid size

Effective for 2D Poisson equation comparison

Potential for generalization to other differential equations

Abstract

A new method of numerical solution for partial differential equations is proposed. The method is based on a fast matrix multiplication algorithm. Two-dimensional Poison equation is used for comparison of the proposed method with conventional numerical methods. It was shown that the new method allows for linear growth in the number of elementary addition and multiplication operations with the growth of grid size, as contrasted with quadratic growth necessitated by the standard numerical methods. The proposed method can be easily generalized for any differential equations.