Special Session 140: Recent advances in wavelet analysis, PDEs and dynamical systems – part III

An Accurate Numerical Model for Solving Elliptic-Type PDEs Directly

Emmanuel O Adeyefa Federal university Oye-Ekiti Nigeria Co-Author(s):    E. O. OMOLE

Abstract: This paper presents a new accurate elliptic partial differential equation solver (AEPDES). The AEPDES has its nodes within the eight-step interval and targets elliptic partial differential equations on the two-dimensional domain. A collocation approach is adopted to develop this method, while a Hermite polynomial is employed as the basis function. The interpolation points are carefully selected at the two desired points, and at all the suitably preferred grid and off-grid points. By uniting the resulting equations and evaluating them at selected points,, the classical AEPDES is obtained. Investigating the numerical properties of the AEPDES, it was confirmed that the AEPDES is zero-stable and consistent. The accuracy and efficiency of the AEPDES were established by solving varying elliptic partial differential equations

ANALYSIS AND MODELLING OF FASHION TRENDS

Hyeong-Ohk Bae Ajou University Korea Co-Author(s):    Hyeong-Ohk Bae

Abstract: We suggest a system of differential equations that models the formulation and evolution of a trend cycle through the consideration of underlying dynamics between the trend participants. Our model captures the five stages of a trend cycle, namely, the onset, rise, peak, decline, and obsolescence. It also provides a unified mathematical criterion/condition to characterize the fad, fashion and classic. Such a model is based on an elasticity in non-Newtonian fluid. We prove that the solution of our model can capture various trend cycles. Numerical simulations are provided to show the expressive power of our model.

Non local dynamical boundary condition on irregular structures

Raffaela Capitanelli Sapienza University ROMA Italy Co-Author(s):

Abstract: Recently non-local boundary value problems have been investigated together with the associated probabilistic representation in terms of sticky Brownian motions (see [CD], [D1], [D2]). In the present talk, I present some results on non-local dynamic boundary conditions on irregular structures like snowflake type fractals. [CD] Raffaela Capitanelli, Mirko D`Ovidio. On the Laplace equation with non-local dynamical boundary conditions. Evolution Equations and Control Theory, 2026, 16: 1-15. [D1] M. D`Ovidio,Fractional Boundary Value Problems, Fract. Calc. Appl. Anal., 1 25, (2022), 29-59. [D2] M. D`Ovidio, Fractional Boundary Value Problems and Elastic Sticky Brownian Motions, Fract. Calc. Appl. Anal., 27 (2024) 2162-2202.

Fixed-point theorems for semilinear operator equations and applications

Wenying Feng Trent University Canada Co-Author(s):

Abstract: We study extensions of basic fixed-point theorems to semilinear operator equations. Using an approach that moves from concrete algorithms to abstract operator formulations, we establish a class of generalized existence and multiplicity results for fixed points on order intervals in partially ordered Banach spaces. The results are obtained under conditions that relax or replace the usual compactness and cone assumptions and are designed to accommodate semilinear operators arising in applications. In particular, the theorems recover and extend classical results and provide verifiable conditions for a broader class of nonlinear operators.

Prime subsets, chaoticity and IFSs

Emanuel Guariglia Kean University Italy Co-Author(s):

Abstract: In this talk, we deal with the link between prime subsets, fractal sets and IFSs. More precisely, numerical simulations suggest that different prime subsets (e.g., Chen primes, Ramanujan primes) exhibit fractal-like behavior. Therefore we introduce an IFS to describe prime subsets. Finally, we show that our results has an application in dynamical systems, cryptography and signal processing.

Ill-Posed Boundary Value Problems for Multidimensional PDEs of Keldysh Type

Tsvetan Hristov Sofia University "St. Kliment Ohridski" Bulgaria Co-Author(s):

Abstract: From the classical works of Keldysh and Fichera, it is well known that BVPs for multidimensional linear second-order equations with a non-negative characteristic form are well understood in the sense that boundary conditions should not be imposed on the whole boundary. In this talk analogous statements of multidimensional BVPs for weakly hyperbolic equations of Keldysh type are studied. These equations are considered in a domain bonded by two characteristic surfaces and a ball in the hyperplane of the parabolic degeneration. A data is prescribed only on a part of the characteristic boundary, while the parabolic part of the boundary is free of data and the normal derivative of the solution can have singularity on it. In the frame of the classical solvability these problems are not Fredholm, since they have infinite-dimensional co-kernels. Alternatively, a notion of a generalized solution with possible singularity is given. Results on the existence and uniqueness of such solution are obtained under special condition on the lower-order terms. Further, orthogonality conditions on the smooth right-hand side functions are presented, which are necessary and sufficient for the existence of generalized solutions with fixed order of singularity.

Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation

Soyeun Jung Kongju National University Korea Co-Author(s):

Abstract: In this talk, we rigorously establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation. Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time. This result provides a clear transition from spectral to nonlinear instability in a genuinely two-dimensional setting, where the Bloch parameter ranges over an unbounded domain.

Inverse Problems for Nonlinear Epidemiological Models: Parameters Identifications and Validation of SEIRS-type Models

Veneta Koleva Sofia University St. Kliment Ohridski Bulgaria Co-Author(s):    Veneta Koleva

Abstract: Inverse epidemiological problems involving nonlinear ordinary differential equations are generally ill-posed. For this reason, it is necessary to determine parameters, or combinations of parameters, that can be uniquely identified from available real-world data. This talk addresses inverse problems for a time-dependent SEIRS model incorporating vaccination, hospitalization, and vital dynamics. First, a mathematical analysis is performed to establish essential biological and analytical properties of the model, including non-negativity, boundedness, and the existence and uniqueness of solutions. The core contribution of this work lies in the formulation and solution of time-discrete inverse problems aimed at identifying unknown, time-dependent model parameters from reported epidemiological data. This inverse methodology enables reliable calibration of the model and provides insight into the time evolution of the epidemic. Numerical experiments using actual COVID-19 data from the USA, Italy, and Bulgaria demonstrate the robustness and practical applicability of the proposed approach. Furthermore, the model is used to compute key epidemiological indicators, such as time-dependent basic and effective reproduction numbers. Provided that suitable epidemiological data are available, the model and its associated simulation tools can be applied to data from any country, making the framework broadly applicable for epidemic analysis and decision support.

Trajectorial version of the $W_h$-gradient flow for nonlinear Fokker-Planck equations

Zhenxin Liu Dalian University of Technology Peoples Rep of China Co-Author(s):    Xuewei Wang

Abstract: In this talk, we will introduce a trajectorial approach to the gradient flow of nonlinear Fokker-Planck equations. We first give the definitions of the generalized entropy and the modified Wasserstein metric $W_h$, which is adapted to the nonlinear setting. Then we establish the trajectorial version of the relative entropy dissipation identity by McKean-Vlasov SDEs. Averaging the energy dissipation of trajectories yields the free energy dissipation of nonlinear Fokker - Planck equations. Furthermore, leveraging properties of the tangent space of $(\mathcal{P}_2({\mathbb R }^d), W_h)$, we derive the $W_h$-gradient flow. As an illustrative example, we analyze the Fermi-Dirac-Fokker-Planck equation. We conclude with two questions motivated by numerical observations.

Optimal Regularity for the Scalar and N Membranes Alt-Phillips Free Boundary Problem

Kunyi (Mark) Ma Columbia University USA Co-Author(s):

Abstract: It is known that the classical Obstacle and Alt-Caffarelli Problems can be regarded as two endpoint cases for the family of free boundary problems, known as the Alt-Phillips Problem. Their minimizers are non-negative, subharmonic, and solves an elliptic-type equation in the positive set with an a priori unknown positive-zero set interface, which we call the `Free Boundary'. In proof of the Optimal Regularity, the classical approaches involve showing the optimal growth rate near the boundary, then combining with the interior equation to gain full regularity. In this work, we discuss a novel dichotomy argument initiated by De Silva - Savin. The key idea is to say, for the size of the minimizer u sufficiently large at unit scale, either the size of u shrinks by 1/2, or u is sufficiently close to its harmonic replacement, as one goes to the next scale. We also discuss a generalization to the vectorial N-Membrane Alt-Phillips Free Boundary Problem, and study how the same argument transfers. In particular, we point out the key difference between the Alt-Phillips with parameter in (0, 1), and the endpoint cases.

From bilinear forms to nonlinear PDEs: Soliton theory

Wen-Xiu Ma University of South Florida USA Co-Author(s):

Abstract: Solitons offer fundamental insight into the dynamics of nonlinear systems and arise in a wide range of applications across physics and mathematics. The Hirota bilinear method provides a powerful framework for analyzing such solutions in integrable PDEs, enabling the systematic construction of multi-soliton solutions by transforming nonlinear PDEs into bilinear form. In this talk, we examine criteria for the existence of $N$-soliton solutions, illustrated through examples in both (1+1)- and (2+1)-dimensional settings. We also explore generalized bilinear equations and their corresponding soliton solutions.

ON SOME THERMOELASTIC SHEAR BRESSE SYSTEMS

Salim Messaoudi University of Sharjah United Arab Emirates Co-Author(s):

Abstract: In this talk we present three one-dimensional linear thermoelastic shear Bresse systems, where the heat effect is given by Fourier's law. First, for one problem, we discuss the well-posedness, using the semigroup approach and the stability, using the multipliter method. Secondly, we shade some light on the others.

Deterministic and stochastic analysis of eco-epidemic models, focusing on fear, refuge, and selective predation dynamics

Samares Pal University of Kalyani India Co-Author(s):    Sasanka Shekhar Maity

Abstract: In this investigation, we delve into the dynamics of an eco-epidemic model, considering the intertwined influences of fear, refuge-seeking behavior, and alternative food sources for predators with selective predation. We extend our model to incorporate the impact of fluctuating environmental noise on system dynamics. The deterministic model undergoes thorough scrutiny to ensure the positivity and boundedness of solutions, with equilibria derived and their stability properties meticulously examined. Furthermore, we explore the potential for Hopf bifurcation within the system dynamics. In the stochastic counterpart, we prioritize discussions on the existence of a globally positive solution. Through simulations, we unveil the stabilizing effect of the fear factor on susceptible prey reproduction, juxtaposed against the destabilizing roles of prey refuge behavior and disease prevalence intensity. Notably, when disease prevalence intensity is too low, the infection can be eradicated from the eco-system. Our deterministic analysis reveals a complex interplay of factors: the system destabilizes initially but then stabilizes as the fear factor suppressing disease prevalence intensifies, or as predators exhibit a stronger preference for infected prey over susceptible ones, or as predators are provided with more alternative food sources.

Hyperbolic Cahn-Hilliard Equations with Dynamic Boundary Conditions

Sema Yayla Hacettepe University Turkey Co-Author(s):

Abstract: In this talk, we study the hyperbolic Cahn-Hilliard equation endowed with dynamic boundary conditions, derived from principles of mass conservation and energy minimization. We analyze the well-posedness and long-time behavior of the model in the two-dimensional setting, employing the framework of global attractors. This work is part of the TUBITAK 3501 project (Grant No. 123F453), titled Investigation of the Long-Time Behavior of Hyperbolic Cahn-Hilliard Equations, and is funded by The Scientific and Technological Research Council of Turkiye (TUBITAK).

On Parabolic Equations with Singular Coefficients and Initial-Boundary Data

Alibek Yeskermessuly Altynsarin University Kazakhstan Co-Author(s):    Michael Ruzhansky

Abstract: We study a linear second-order parabolic equation in divergence form with drift and potential terms on a bounded domain with initial data and non-homogeneous Dirichlet boundary conditions. The analysis focuses on situations where the coefficients and data may exhibit singular or distributional behavior. First, assuming bounded measurable diffusion, drift, and potential coefficients, we prove existence, uniqueness, and a priori estimates for weak solutions in the natural energy space associated with the divergence-form operator. The proof is based on energy methods and Galerkin approximations. We then extend the framework to allow singular coefficients and data, including distributional potentials, source terms, initial values, and boundary conditions. In this setting we introduce a notion of very weak solutions obtained via regularization of the coefficients and data. Existence and uniqueness of such solutions are established under suitable ellipticity and positivity assumptions. Finally, we prove a consistency result showing that, in the regular case, representatives of the very weak solution converge in the energy space to the classical weak solution.

An Efficient Laguerre Minimum Action Method for Quasi-Potential and Transition Path Computing

Haijun Yu Academy of Mathematics and Systems Science Peoples Rep of China Co-Author(s):    Shenghe Huang, Yishuang Yue and Haijun Yu

Abstract: Minimum action methods provide a powerful framework for analyzing rare transitions in stochastic dynamical systems, but their practical performance is often limited by time truncation and parameter sensitivity in infinite-horizon problems. We propose an efficient Laguerre spectral minimum action method for computing quasi-potentials associated with fixed points of dynamical systems. Based on the large deviation framework, the method computes minimum action transition paths by formulating the problem on a semi-infinite time interval and discretizing the temporal direction using Laguerre functions. An appropriate time-scaling strategy is incorporated to enhance accuracy and convergence of the Laguerre spectral approximation. To efficiently handle nonlinear terms, we develop an improved procedure for evaluating Laguerre functions at Gauss--Radau quadrature points, which enables stable double-precision computations with a large number of Laguerre modes. Numerical analysis and experiments are presented to illustrate the accuracy and efficiency of the proposed method.