Pseudospectra of the heat operator Pencil

TL;DR

This paper analyzes the pseudospectra of a non-self-adjoint heat operator pencil modeled as a 2x2 unbounded block matrix, providing spectral enclosures and visualizations to understand its behavior.

Contribution

It introduces a novel analysis of the pseudospectra of the heat operator pencil with mixed boundary conditions, using a block operator matrix framework.

Findings

Spectral and pseudospectral enclosures are established for the heat operator pencil.

Discretized plots illustrate the spectral properties and pseudospectra.

The analysis enhances understanding of non-self-adjoint operator behavior in heat equations.

Abstract

This article undertakes an analysis of the one-dimensional heat equation, wherein the Dirichlet condition is applied at the left end and Neumann condition at the right end. The heat equation is restructured as a non-self-adjoint $2\times 2$ unbounded block operator matrix pencil. The spectral, pseudospectral, and $(n,\epsilon)$-pseudospectral enclosures of the $2\times 2$ unbounded block operator matrix pencil are explored to scrutinize the heat operator pencil. The plots of the discretized equation are depicted to illustrate the observations.