The Diffusion Difference Equation

TL;DR

This paper introduces a new difference equation as a discrete analogue of the diffusion differential equation, analyzing its spectral properties and eigenfunctions, and providing asymptotic formulas for solutions.

Contribution

It presents a novel diffusion difference equation, explores its spectral characteristics, and derives asymptotic formulas for eigenfunctions, advancing discrete diffusion analysis.

Findings

The diffusion difference operator is self-adjoint.

Eigenvalues are simple and real.

Eigenfunctions are orthogonal and asymptotic formulas are established.

Abstract

In this work, we introduce a new difference equation which is discrete analogue of Diffusion differential equation and analyze some essential spectral properties, Diffusion difference operator is self-adjoint, eigenvalues of this problem are simple and real, eigenfunctions corresponding to distinct eigenvalues, of this problem are orthogonal. Also, some useful sum representation for the linearly independent solutions of Diffusion difference equation with Dirichlet boundary conditions has been acquired and by means of this result, asymptotic formula for eigenfunction is analyzed and these results are proved.