Exponential dichotomy of difference equations and applications to evolution equations on the half-line

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

Download Full-text

On Exponential Dichotomy of Variational Difference Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2009/324273 ◽

2009 ◽

Vol 2009 ◽

pp. 1-18 ◽

Cited By ~ 6

Author(s):

Bogdan Sasu

Keyword(s):

Difference Equations ◽

Exponential Dichotomy ◽

New Method ◽

Skew Product ◽

Skew Product Flows

We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew-product flows.

Download Full-text

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

10.1007/978-3-0348-0297-0 ◽

2012 ◽

Keyword(s):

System Theory ◽

Spectral Theory ◽

Difference Equations ◽

Evolution Equations ◽

Mathematical System Theory ◽

Mathematical System

Download Full-text

A Coupled System of Fractional Difference Equations with Nonlocal Fractional Sum Boundary Conditions on the Discrete Half-Line

Mathematics ◽

10.3390/math7030256 ◽

2019 ◽

Vol 7 (3) ◽

pp. 256 ◽

Cited By ~ 3

Author(s):

Jarunee Soontharanon ◽

Saowaluck Chasreechai ◽

Thanin Sitthiwirattham

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Boundary Conditions ◽

Point Theorem ◽

Difference Equations ◽

Existence Result ◽

Coupled System ◽

Half Line ◽

Fractional Difference ◽

Fractional Difference Equations

In this article, we propose a coupled system of fractional difference equations with nonlocal fractional sum boundary conditions on the discrete half-line and study its existence result by using Schauder’s fixed point theorem. An example is provided to illustrate the results.

Download Full-text