Mathematical Models and Simulations, 2nd Edition

1. Introduction

In this Editorial, we are pleased to introduce a Special Issue of the scientific journal Axioms, entitled “Mathematical Models and Simulations, 2nd Edition”. This is a continuation of a previous Special Issue, entitled “Mathematical Models and Simulations”, published in Contribution 1.

Mathematical modeling is a cornerstone of scientific inquiry, providing essential tools for understanding and predicting the behavior of complex systems in physics, biology, finance, and engineering. Alongside theoretical developments, numerical simulations play a crucial role in practical applications, enabling researchers to explore and forecast the dynamics of quantities of interest.

This Special Issue brings together a collection of papers in the field of mathematical physics and applied mathematics, showcasing a diverse range of modeling approaches. These include the development of new analytical frameworks and numerical schemes, as well as innovative applications in fluid dynamics, electrodynamics, decision-making, and data-driven modeling.

The contributions encompass a broad range of mathematical and computational approaches. Several papers address deterministic systems governed by ordinary or partial differential equations, focusing on stability, convergence, and solution behavior. Others explore stochastic, fuzzy, or hybrid modeling techniques that capture uncertainty and complexity in real-world contexts. A variety of computational methods are featured, including finite-difference, finite-element, Galerkin, spline-based, and network simulation techniques, each tailored to the mathematical structure of the problem under study.

Beyond methodological innovation, many works bridge theory and application. Examples include modeling optical fusion effects via equivalent electrical networks, simulating two-dimensional viscous flows with high boundary accuracy, and developing fuzzy decision frameworks for complex business environments. Other studies extend classical analysis through modern theoretical tools such as C-pseudoresolvent families, generalized electrodynamics, and penalized likelihood estimation in spatial statistics.

This Special Issue reflects the dynamic interplay between mathematical theory and computational practice, offering valuable insights and methodologies for researchers engaged in modeling and simulation across scientific disciplines.

2. Overview of the Published Papers

This Special Issue brings together eight high-quality contributions that reflect the diversity and depth of current research in mathematical modeling and simulation. Each paper underwent a rigorous peer-review process and presents original approaches to theoretical analysis, numerical methods, and applied modeling across various domains. The selected works span topics such as differential equations, fluid dynamics, fuzzy decision-making, electrodynamics, statistical modeling, and computational schemes, offering valuable insights and methodologies for both academic and practical applications.

In Contribution 2, M. Alotaibi presents an implicit finite difference scheme for a wave equation with strong damping and discrete delay. Stability is analyzed using the Jury criterion, leading to a condition linking discretization steps with model parameters. A numerical example confirms the scheme’s decay behavior and theoretical stability.

In Contribution 3, M. Kostić investigates existence and uniqueness for abstract degenerate non-scalar Volterra equations using the vector-valued Fourier transform and (A,k,B)-regularized C-pseudoresolvent families. The theoretical results are supported by remarks and illustrative applications.

In Contribution 4, V. Sobchuk, O. Kurylko, O. Boryseiko, O. Perehuda, I. Lebedyeva, I. Ulitko, and O. Vashchilina investigate a two-dimensional periodic Stokes flow in a rectangular cavity with constant wall velocities. The velocity field is derived analytically via superposition, while particle trajectories are computed numerically. Boundary accuracy is assessed through local integration control, and fluid advection is modeled using cubic spline interpolation. The results align well with experimental data.

In Contribution 5, J. F. Sanchez-Pérez, J. Solano-Ramírez, F. Marín-García, and E. Castro model the fiber optic fusion effect using the Network Simulation Method (NSM), transforming thermal behavior into an equivalent electrical system. Simulations in NGSpice capture temperature dynamics under high-power conditions, with nondimensional analysis revealing the influence of fiber geometry and refractive index. The results confirm NSM’s effectiveness for nonlinear thermal problems in optical systems.

In Contribution 6, V. Nikolova Traneva, V. Todorov, S. Tranev Tranev, and I. Dimov introduce a confidence-interval circular intuitionistic fuzzy set (CIC-IFS) framework for franchise selection, applied to Pizza Hut’s European operations. By integrating decision-maker competence and modeling uncertainty through fuzzy aggregation scenarios, the method improves selection accuracy and aligns franchise choices with brand values.

In Contribution 7, Y. M. Alawaideh, A. Alb Lupas, B. M. Al-khamiseh, M. A. Yousif, P. Othman Mohammed, and Y. S. Hamed analyze the Hamiltonian formulation for continuous systems with second-order derivatives using Dirac’s theory, focusing on Podolsky’s generalized electrodynamics. They show that the Hamiltonian and Euler–Lagrange approaches yield equivalent results, highlighting the Hamiltonian method as a powerful alternative for modeling complex second-order systems.

In Contribution 8, G. Ibacache-Pulgar, P. Pacheco, O. Nicolis, and M. A. Uribe-Opazo extend Thin-Plate Spline Generalized Linear Models (TPS-GLMs) to evaluate local influence via maximum penalized likelihood estimators. Using Fisher Scoring and weighted backfitting, they assess sensitivity to data perturbations and detect influential observations. Applications in agronomy and environmental data illustrate the model’s flexibility in capturing nonlinear spatial effects.

In Contribution 9, R. Čiegis, O. Suboč, and R. Čiegis analyze the efficiency of finite-difference and finite-element Galerkin schemes for non-stationary hyperbolic and parabolic problems on dynamic grids. They present improved stability and convergence estimates, highlighting the impact of grid movement and recommending projection operators to enhance accuracy. Computational experiments confirm the theoretical findings.