On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

TL;DR

This paper derives explicit solutions and analyzes the asymptotic behavior of a class of complex functional difference equations with variable coefficients, with applications to subdiffusion boundary value problems.

Contribution

It introduces a method to explicitly solve a specific class of functional difference equations and describes their asymptotic properties, extending the theory and applications.

Findings

Explicit solutions constructed under certain conditions.

Asymptotic behavior characterized as |z|→∞.

Applications to subdiffusion boundary value problems.

Abstract

For $\nu\in[0,1]$ and a complex parameter $\sigma,$ $Re\, \sigma>0,$ we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane $z\in\mathbb{C}$: \[ (a_{1}\sigma+a_{2}\sigma^{\nu})\mathcal{Y}(z+\beta,\sigma)-\Omega(z)\mathcal{Y}(z,\sigma)=\mathbb F(z,\sigma), \quad\beta\in\mathbb{R},\, \beta\neq 0, \] where $\Omega(z)$ and $\mathbb{F}(z)$ are given complex functions, while $a_{1}$ and $a_{2}$ are given real non-negative numbers. Under suitable conditions on the given functions and parameters, we construct explicit solutions of the equation and describe their asymptotic behavior as $|z|\to +\infty$. Some applications to the theory of functional difference equations and to the theory of boundary value problems governed by subdiffusion in nonsmooth domains are then discussed.