Special Session 8: Differential, Difference, and Integral Equations: Techniques and Applications

On the existence of solutions to the Cauchy-Dirichlet problem for a class of third-order hyperbolic operators

Annamaria Barbagallo University of Naples Federico II Italy Co-Author(s):    Vincenzo Esposito

Abstract: The talk is devoted to the analysis of the Cauchy-Dirichlet problem for a class of third-order hyperbolic differential operators in a domain of $\mathbb{R}^3$. The study begins to establish a priori estimates, both local and global, which play a crucial role in controlling the behavior of solutions. These estimates are established through careful analytical procedures that take into account the structure of the operators under consideration. By combining the obtained estimates with appropriate pseudo-differential techniques, an existence theorem is proved.

A family of Gibbs constants for symmetric Krawtchouk expansions

John Davis Baylor University USA Co-Author(s):

Abstract: We investigate the Gibbs overshoot for partial sums of symmetric Krawtchouk expansions of a centered sign function. By combining parity reduction, an exact coefficient identity on a neighboring lattice, and one-point WKB asymptotics, we obtain a continuum-limit description of the overshoot profile for degree cutoffs of the form $m\sim \delta N$, where $\delta$ is the truncation fraction. This leads to a one-parameter family of discrete Gibbs constants interpolating between the classical Fourier Gibbs constant as an upper bound and a recently numerically observed Krawtchouk Gibbs constant as a lower bound. We discuss how this framework may extend to other discrete orthogonal polynomial families.

Quasilinear eigenvalue estimate for coupled differential equations

Sougata Dhar Fairfield University USA Co-Author(s):    Jessica Kelly, Lisa Naples

Abstract: In this talk, we present an one dimensional Sobolev-type inequality and use it to obtain the estimates for eigenvalues for even ordered differential equations satisfying periodic and anti periodic boundary conditions. In the higher order periodic case, we introduce suitable moment constraints to eliminate the polynomial kernel, leading to a Sobolev-Poincare inequalities with explicit scaling. In contrast, anti-periodic conditions naturally yield coercive estimates without additional constraints. These Sobolev-type inequalities provide a unified framework for analyzing boundary value problems and serve as the main tool for deriving further spectral and stability results.

On BVP for linear fractional differential equations

Pavel Dubovski Stevens Institute of Technology USA Co-Author(s):    J.A.Slepoi

Abstract: We demonstrate that boundary value problems on segment $[a, b]$ are not necessarily overdetermined for the first order fractional differential equations and, thus, may be well posed.

Multiple positive solutions for a fractional differential system with nonlocal boundary conditions

Wenying Feng Trent University Canada Co-Author(s):    Y. Cui, S. Kang, H. Chen, L. Li

Abstract: We prove existence of three positive solutions for a class of Caputo fractional differential equation with nonlocal boundary conditions involving both the first and second-order derivatives. Our result extends the Riemann-Liouville fractional system to the Caputo equation. By converting the problem into a Caputo integral operator, we employ the Leggett-Williams fixed point theorem to establish conditions that ensure existence of three positive solutions. We illustrate the result with examples and numerical simulations that demonstrate one or more nontrivial solutions and highlight parameter regimes yielding multiplicity.

Prolongation of solutions and Lyapunov stability for Stieltjes dynamical systems

Marlene Frigon University of Montreal Canada Co-Author(s):    Marlene Frigon

Abstract: In the last years, there has been quite an interest in the theory of Stieltjes differential equations which have the important advantage of providing a unified framework to differential equations, discrete equations, dynamic equations on time scales and differential equations with impulses at fixed times. They are particularly useful for modeling evolution processes in which sudden changes and stationary periods occur. In this talk, we present Lyapunov-type results to study the stability of an equilibrium of a Stieltjes dynamical system. We utilize prolongation results to establish the global existence of the maximal solution. We present also examples and applications to population dynamics models.

Modeling Density-Dependent Emigration Between Two Competing Species of Tribolium: Part I -- Methods

Jerome Goddard II Auburn University Montgomery USA Co-Author(s):    J. T. Cronin & R. Shivaji

Abstract: Emigration is an important ecological process that can substantially affect persistence, coexistence, and spatial population dynamics. In many systems, emigration is influenced by intraspecific density, while the role of interspecific competitor density remains less clear. Motivated by a 12-generation microcosm experiment involving two competing Tribolium species, we develop a reaction-diffusion Lotka-Volterra framework for studying density-dependent emigration between habitat patches. The model incorporates patch size, permeability, and density effects on the proportion emigrating from a patch and is parameterized using independent experimental data. To account for uncertainty and biological variability, we consider distributions of parameter values and use these to obtain probabilistic predictions of long-term steady-state behavior. This talk focuses on the experimental setting, the mathematical formulation of the model, parameterization, and the computational framework used to connect the model to observed microcosm dynamics.

Lyapunov-type Inequalities for Even-Order Quasilinear Differential Equations

Jessica Kelly Christopher Newport University USA Co-Author(s):    Sougata Dhar

Abstract: In this talk, we establish Lyapunov-type inequalities for even-order quasilinear differential equations subject to Dirichlet boundary conditions, considering both constant and variable coefficient cases. We further extend these results to systems of even-order quasilinear differential equations. The inequalities presented generalize classical scalar results, sharpen existing estimates, and yield explicit lower bounds for generalized eigenvalues.

Inverse interior scattering problems for perturbed quantum Bunimovich billiards

Qi Li School of Mathematics and Statistics, Shanxi Datong University Peoples Rep of China Co-Author(s):

Abstract: This work investigates the inverse interior scattering problem for quantum Bunimovich billiards subject to small boundary perturbations. We consider billiards whose boundaries are piecewise $C^2$ perturbations of the original domain, satisfying the generalized defocusing mechanism. An incident field is generated from the center of each perturbed circular arc, and the resulting scattered field is measured on an auxiliary arc. By synthesizing the transformed field expansion method with Fourier analysis and mathematical induction, we develop a procedure to reconstruct the perturbation`s shape. We establish that, when the billiard boundary and the auxiliary arc are sufficiently close, the perturbation is uniquely determined by the imaginary part of the total field, yielding an explicit reconstruction formula.

A Purely Algebraic Unified Approach to Linear Differential and Difference Equations

Vakhtang Lomadze Javakhishvili State University Rep of Georgia Co-Author(s):

Abstract: Let $({\mathcal U},\hslash)$ be a pair consisting of a module $\mathcal U$ over the ring of proper rational functions and a nonzero element $\hslash\in {\mathcal U}$. Elements of ${\mathcal U}$ are interpreted as classical functions, with $\hslash$ representing the constant function 1; multiplication by the local parameter is viewed as the integration operator. We assume that the zero function is the only constant function that arises as an integral. Within this framework, we introduce Schwartz distributions of finite order and study linear dynamic equations, encompassing linear differential and difference equations, as well as more general linear dynamic equations on time scales. Acknowledgment: This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [FR-24-8249].

Shooting Method Using Difference Equations

Jeffrey Lyons The Citadel USA Co-Author(s):    Douglas Anderson, Richard Avery, Johnny Henderson

Abstract: We introduce a new fixed point shooting method that replaces the classical continuous shooting technique with a sequence of discrete initial value problems. Each discrete problem is easy to solve and collectively provides a high fidelity approximation to the trajectory of the corresponding continuous boundary value problem. Using these trajectories, we then construct a difference equation approximation to the original boundary value problem and establish rigorous error bounds for the method. This discrete framework offers both theoretical and computational advantages, as difference equations are often simpler to analyze and implement while retaining strong approximation properties. To demonstrate the effectiveness of the method, we apply it to a right focal boundary value problem and highlight its accuracy and efficiency.

Existence and asymptotics of global positive solutions to a nonlinear differential system of fractional order via extension of Karamata theory

Serena Matucci Department of Mathematics and Computer Sciences Italy Co-Author(s):    Pavel \v{R}eh\`ak

Abstract: In this talk some results are presented about the existence and the precise asymptotic behavior for positive global solutions of a system of two differential equation of fractional order. The aims is both to generalize some of the results known for the ordinary case to the fractional case, and to analyze the role of fractional order, putting in evidence purely fraactional phenomena. Under the assumption that the coefficients are \textit{regularly varying} functions at infinity, an extension of the Karamata integration theorem to fractional integration enables the use of a fixed-point approach to prove the existence of positive unbounded solutions satisfying given asymptotic conditions, since it provides sharp two-sided estimates for the solutions. Furthermore, conditions are found under which all solutions with nonnegative initial conditions are regularly varying at infinity with a prescribed index. We then apply these results to the special case of a scalar equation of order $1+\alpha$, highlighting both the analogies and discrepancies with the integer-order case.

Fixed point theorems in orders metric and Banach spaces based on degree of nondensifiability and applications

Abdelghani Ouahab University of Djillali Liabes Sidi Bel Abbes Algeria Co-Author(s):

Abstract: Using the concept of the degree of nondensifiability in ordered metric and Banach spaces, we establish fixed point theorems for monotone operator contractions with respect to the degree of nondensifiability and condensing mappings. Our results extend and unify several classical and recent results available in the literature. Furthermore, we apply the main results to study the existence of minimal maximal mild solutions for certain classes of semilinear differential equations.

NONSTANDARD DISCRETIZATION SCHEME IN VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS THAT PRESERVES UNIFORM ASYMPTOTIC STABILITY

Youssef N Raffoul University of Dayton USA Co-Author(s):    SVETLIN G. GEORGIEV, Halis Can Koyuncuo\u{g}lu}, Marko Kosti\` c

Abstract: We apply a nonstandard discretization scheme to continu- ous Volterra integro-differential equations and we show that under this discretization, the necessary and sufficient conditions for uniform as- ymptotic stability of continuous Volterra integro-differential equations are preserved. Our analysis is based on the notion of resolvent. An example is provided as an application to our theory.

Asymptotic stability of differential equations with impulses and distributed delay

Paola Rubbioni University of Perugia Italy Co-Author(s):    T. Cardinali, S. Matucci

Abstract: Differential equations with memory and impulsive effects arise in a variety of physical and biological models. When formulated in appropriate function spaces, classes of distributed-delay models can be represented as integro-differential or functional differential equations. We establish general conditions guaranteeing existence, uniqueness, uniform asymptotic stability, and exponential stability of solutions on the half-line, even under impulsive perturbations. Applications illustrate these results in population dynamics and flexible robotic arms with integral and evanescent memory. {\small \begin{thebibliography}{999} \bibitem{cmr25} Cardinali T.; Matucci S.; Rubbioni P.; Uniform asymptotic stability of a PDE`s system arising from a flexible robotics model, Math. Methods Appl. Sci. 48 (2025), no. 11, 11242-11251 \bibitem{cmr26} Cardinali T., Matucci S., Rubbioni P., Stability of solutions in impulsive integro-differential equations with applications to fading memory systems, Commun. Nonlinear Sci. Numer. Simulat. 154 (2026) 109588 \bibitem{r21} Rubbioni, P.; Asymptotic stability of solutions for some classes of impulsive differential equations with distributed delay, Nonlinear Anal. Real World Appl. 61 (2021), 103324 \end{thebibliography} }

Passive inverse problems: stability and neural network solutions

darko volkov Worcester Polytechnic Institute USA Co-Author(s):    S. C. Hawkins, M. Ganesh, and F. Triki

Abstract: Typical PDE based nonlinear inverse problems involve a known forcing term and a feature (such as coefficient or inner geometry feature) to be recovered using overdetermined boundary data. However, this does not apply to passive inverse problems. In these problems, the forcing term in the PDE is unknown while the geometry of a feature has to be recovered. There are thus both linear and nonlinear unknowns. We focus on recovery of cracks in unbounded domains with propagating waves. We present recent results regarding the stability of the Hausdorff distance between cracks, (Triki and Volkov, 2025). Next, we examine the case where cracks are defined through a vector parameter $m$ while the forcing term for the PDE is still in an infinite dimensional space (Ganesh, Hawkins, and Volkov, 2026). We proved Lipschitz continuity of a related inverse operator if the forward operator is restricted to $m$ -dependent finite dimensional spaces. These finite dimensional spaces are spanned by $m$ -dependent singular functions which can be computed. This led us to build neural networks that can solve the crack inverse problem. The solution is computed in an efficient non-iterative way and is robust to noise.

Physics-informed stochastic models for theme park ride waiting times

Min Wang Kennesaw State University USA Co-Author(s):    Min Wang

Abstract: A stochastic model for theme-park ride waiting times is developed by modeling the waiting time as a continuous-time, discrete-state Markov process with state-dependent, time-varying transition rates. These transition rates are interpreted as a time-dependent feedback control acting on the waiting-time process, allowing us to formulate the model calibration task as a data-driven optimal control problem. To solve this problem efficiently, we construct a physics-informed neural network (PINN) that embeds the controlled Kolmogorov forward equation into its architecture. Under mild assumptions, we prove the existence of an optimal time-dependent feedback control, providing theoretical support for the learning procedure. Numerical simulations are conducted to demonstrate the effectiveness of the PINN-based solution. The framework provides an interpretable, physically consistent, and data-driven approach for modeling and forecasting ride waiting times.

Uniqueness of meromorphic functions sharing values with their difference polynomials

Huicai Xu Shanxi Datong University Peoples Rep of China Co-Author(s):

Abstract: In this work, we deal with the open question 3IM+1CM in the uniqueness theory of meromorphic functions and we prove that when a non-constant finite order meromorphic function $f$ and its difference polynomials $L(z,f)$ satisfy the condition 3IM+1CM, then $f\equiv L(z,f)$ or $f\equiv -L(z,f)$ holds.

Structure of polar director for bent-core smectic A liquid crystals in thin planar cells

Xiaodong Yan University of Connecticut USA Co-Author(s):    Alec Wendland

Abstract: We study the polar director structure in thin planar cells filled with bent-core liquid crystals in the ferroelectric smectic-A phase ($\text{SmAP}_F$). We analyze a continuum phenomenological model proposed by Gornik et al., present rigorous proof of existence and uniqueness of the equilibrium solutions, and discuss qualitative properties of nontrivial solutions. The effects of bias electric field, surface anchoring and cell thickness on the structure of polar director are also discussed. Our results agree with previous experimental and numerical simulations in the physics literature.