A quantum graph FFT with applications to partial differential equations on networks

TL;DR

This paper extends the FFT to functions on finite graphs with edges as finite intervals, enabling spectral methods to solve time-dependent PDEs on network-like domains.

Contribution

It introduces a quantum graph FFT and spectral methods for PDEs on networks, bridging graph theory and numerical PDE solutions.

Findings

Developed a quantum graph FFT algorithm.

Applied spectral methods to PDEs on network domains.

Demonstrated effectiveness for time-dependent PDEs.

Abstract

The Fast Fourier Transform is extended to functions on finite graphs whose edges are identified with intervals of finite length. Spectral and pseudospectral methods are developed to solve a wide variety of time dependent partial differential equations on domains which are modeled as networks of one dimensional segments joined at nodes.