Searcharxiv⌕ Search

arXiv subjects

Douglas R. Anderson

Publications and source records attributed to Douglas R. Anderson.

At least 19 recordsLinked to original sources
arXiv2026

Equilibrium stability for a continuous time scale with discrete uniform gaps

We investigate the equilibrium (trivial solution) stability, also known as Lyapunov stability, of a certain first-order linear complex constant coefficient dynamic equation on the time scale $\p_{α,β}$, which has continuous intervals of length $α>0$ followed by discrete gaps of length $β>0$. In particular, we establish results in the case of this specific time scale, for coefficient values in the complex plane, including where the exponential function alternates in sign. In our analysis, we employ the Lambert $W$ function. For increasing gap size $β$ relative to $α$, we prove that the complex constant coefficient undergoes a bifurcation in its parameter space. We establish interesting results for both the delta dynamic equation and the nabla dynamic equation. Lastly, we connect these results to those related to Hyers--Ulam stability of the same nabla equations.

math.DS↗
arXiv2023

Best Ulam constants for two-dimensional non-autonomous linear differential systems

This study deals with the Ulam stability of non-autonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two-dimensional non-autonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on $(-\infty,\infty)$, but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for non-autonomous systems other than periodic systems.

math.CA↗
arXiv2023

Best Ulam constants for damped linear oscillators with variable coefficients

This study uses an associated Riccati equation to study the Ulam stability of non-autonomous linear differential vector equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these non-autonomous linear differential vector equations are derived. These robust results apply to vector equations with solutions that blow up in finite time, as well as to vector equations with solutions that exist globally on $(-\infty,\infty)$. Illustrative, non-trivial examples are presented, highlighting the main results.

math.CA↗
arXiv2023

Hyers-Ulam Stability For A Type Of Discrete Hill Equation

We establish the Hyers-Ulam stability of a second-order linear Hill-type $h$-difference equation with a periodic coefficient. Using results from first-order $h$-difference equations with periodic coefficient of arbitrary order, both homogeneous and non-homogeneous, we also establish a Hyers-Ulam stability constant. Several interesting examples are provided. As a powerful application, we use the main result to prove the Hyers-Ulam stability of a certain third-order $h$-difference equation with periodic coefficients of one form.

math.CA↗
arXiv2022

Hyers-Ulam stability for differential systems with $2\times 2$ constant coefficient matrix

We explore the Hyers-Ulam stability of perturbations for a homogeneous linear differential system with $2\times 2$ constant coefficient matrix. New necessary and sufficient conditions for the linear system to be Hyers-Ulam stable are proven, and for the first time, the best (minimal) Hyers-Ulam constant for systems is found in some cases. Several examples are provided. Obtaining the best Hyers-Ulam constant for second-order constant coefficient differential equations illustrates the applicability of the strong results.

math.CA↗
arXiv2020

Hyers--Ulam stability for quantum equations

We introduce and study the Hyers--Ulam stability (HUS) of a Cayley quantum ($q$-difference) equation of first order, where the constant coefficient is allowed to range over the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation has Hyers--Ulam stability for certain values of the Cayley parameter, and we establish the best (minimal) HUS constant in terms of the coefficient only, independent of $q$ and the Cayley parameter. If the Cayley parameter equals one half, then there is no Hyers--Ulam stability for any coefficient value in the complex plane.

math.CA↗
arXiv2020

Best constant for Ulam stability of first-order h-difference equations with periodic coefficient

We establish the best (minimum) constant for Ulam stability of first-order linear $h$-difference equations with a periodic coefficient. First, we show Ulam stability and find the Ulam stability constant for a first-order linear equation with a period-two coefficient, and give several examples. In the last section we prove Ulam stability for a periodic coefficient function of arbitrary finite period. Results on the associated first-order perturbed linear equation with periodic coefficient are also included.

math.CA↗
arXiv2020

A Multi-Valued Logarithm on Time Scales

A new definition of a multi-valued logarithm on time scales is introduced for delta-differentiable functions that never vanish. This new logarithm arises naturally from the definition of the cylinder transformation that is also at the heart of the definition of exponential functions on time scales. This definition will lead to a logarithm function on arbitrary time scales with familiar and useful properties that previous definitions in the literature lacked.

math.CA↗
arXiv2018

On the nature of the conformable derivative and its applications to physics

The purpose of this work is to show that the Khalil and Katagampoula conformable derivatives are equivalent to the simple change of variables $x$ $\rightarrow $ $x^{α}/α,$ where $α$ is the order of the derivative operator, when applied to differential functions. Although this means no \textquotedblleft new mathematics\textquotedblright\ is obtained by working with these derivatives, it is a second purpose of this work to argue that there is still significant value in exploring the mathematics and physical applications of these derivatives. This work considers linear differential equations, self-adjointness, Sturm-Liouville systems, and integral transforms. A third purpose of this work is to contribute to the physical interpretation when these derivatives are applied to physics and engineering. Quantum mechanics serves as the primary backdrop for this development.

math-ph↗
arXiv2018

Hyers--Ulam Stability for Discrete Time Scale with Two Step Sizes

We clarify the Hyers--Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, in the case of a specific time scale with two alternating step sizes, where the exponential function changes sign. In particular, in the case of HUS, we discuss the HUS constant, and whether a minimal constant can be found.

math.CA↗
arXiv2016

Second-Order Self-Adjoint Differential Equations Using a Conformable Proportional Derivative

In this study, linear second-order conformable differential equations using a proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. Defining a Wronskian, we establish a Lagrange identity and Abel's formula. Several reduction-of-order theorems are given. Solutions of the conformable second-order self-adjoint equation are then shown to be related to corresponding solutions of a first-order Riccati equation and a related quadratic functional and a conformable Picone identity. The first part of the study is concluded with a comprehensive roundabout theorem relating key equivalences among all these results. Subsequently, we establish a Lyapunov inequality, factorizations of the second-order equation, and conclude with a section on boundary value problems and Green's functions.

math.CA↗
arXiv2014

Positive Green's functions for some fractional-order boundary value problems

We use the newly introduced conformable fractional derivative, which is different from the Caputo and Riemann-Liouville fractional derivatives, to reformulate several common boundary value problems, including those with conjugate, right-focal, and Lidstone conditions. With the fractional differential equation and fractional boundary conditions established, we find the corresponding Green's functions and prove their positivity under appropriate assumptions.

math.CA↗
arXiv2014

Fractional-order boundary value problem with Sturm-Liouville boundary conditions

Using the new conformable fractional derivative, which differs from the Riemann-Liouville and Caputo fractional derivatives, we reformulate the second-order conjugate boundary value problem in this new setting. Utilizing the corresponding positive fractional Green's function, we apply a functional compression-expansion fixed point theorem to prove the existence of a positive solution. We then compare our results favorably to those based on the Riemann-Liouville fractional derivative.

math.CA↗
arXiv2013

Hyers-Ulam stability of certain singular linear differential equations

We establish the Hyers-Ulam stability of certain linear first-order differential equations with singularities. We then extend these results to higher-order singular linear differential equations that can be written with these first-order factors. An example of our results is given for a second-order singular linear differential equation that is not be covered by the current literature in this area.

math.CA↗
arXiv2012

Hyers-Ulam stability of higher-order Cauchy-Euler dynamic equations on time scales

We establish the stability of higher-order linear non-homogeneous Cauchy-Euler dynamic equations on time scales in the sense of Hyers and Ulam. That is, if an approximate solution of a higher-order Cauchy-Euler equation exists, then there exists an exact solution to that dynamic equation that is close to the approximate one.

math.CA↗
arXiv2012

Existence of three solutions for a first-order problem with nonlinear non-local boundary conditions

Conditions for the existence of at least three positive solutions to the nonlinear first-order problem with a nonlinear nonlocal boundary condition given by && y'(t) - p(t)y(t) = \sum_{i=1}^m f_i\big(t,y(t)\big), \quad t\in[0,1], && λy(0) = y(1) + \sum_{j=1}^n Φ_j(τ_j,y(τ_j)), \quad τ_j\in[0,1], are discussed, for sufficiently large $λ>1$. The Leggett-Williams fixed point theorem is utilized.

math.CA↗