The sparse representation related with fractional heat equations

TL;DR

This paper presents a novel sparse series method using POAFD for efficiently solving fractional heat equations, demonstrating fast convergence and effectiveness in numerical approximations.

Contribution

Introduces a new sparse approximation technique with POAFD for fractional heat equations, improving convergence and computational efficiency.

Findings

Fast convergence of the series solutions.

Effective numerical approximations demonstrated.

Enhanced efficiency over traditional methods.

Abstract

This study introduces pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). The method, as the first step, expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as the second step, makes use the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show effectiveness and efficiency of the proposed series solutions.