Axioms and Methods for Handling Differential Equations and Inverse Problems

1. Special Issue Summary

Modeling real-life problems requires a variety of differential equations that often cause significant challenges for researchers. In the “handling” of these mathematical models, various axioms, mathematical methods, and techniques are able to transform often very complex mathematical objects into a better-behaving representation.

This Special Issue aimed to provide to researchers a platform to present new trends, recent advancements and future research directions. Contributions that addressed key challenges in collecting axioms, mathematical methods, and procedures that are effective for handling differential equations even in cases where classical methods have limited or no applications were invited and accepted.

2. Summary of the Special Issue Research Works

Within this Special Issue, five papers are published, covering various aspects of mathematical models, various axioms, mathematical methods, and techniques. In the following, the published papers will be summarized in brief.

In [Contribution 1], Stavroulakis presents a comprehensive survey of significant oscillation conditions for all solutions of first-order linear differential equations with retarded arguments. The conditions are organized in chronological order, with particular emphasis on cases where classical or well-known oscillation criteria fail to apply. The study also highlights key advancements and underscores the importance of these refined oscillation conditions in extending the applicability of oscillation theory.

In [Contribution 2], Savović et al. compared numerical results obtained using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN) for three test problems involving various initial conditions and Dirichlet boundary conditions. These results were benchmarked against analytical solutions. While both numerical approaches demonstrated strong agreement with the analytical solutions, EFDM exhibited slightly higher accuracy. Given the close correspondence among the EFDM, PINN, and analytical results, both numerical methods are deemed competitive and recommended for use. The findings are particularly relevant for simulating a range of nonlinear physical phenomena, including flood waves in rivers, chromatography, gas dynamics, and traffic flow. Moreover, the solution techniques presented in this study offer a foundation for the development of numerical models targeting this class of nonlinear partial differential equations, providing valuable guidance for current and future model developers working across diverse application domains.

In [Contribution 3], Savović et al. presented a study employing a novel physics-informed neural network (PINN) approach, the standard explicit finite difference method (EFDM), and an unconditionally positivity-preserving finite difference method to solve the one-dimensional Sine-Gordon equation (SGE). Two test problems with known analytical solutions were used to evaluate the performance of these techniques. All three approaches demonstrated strong agreement with the analytical solutions; however, EFDM showed the closest alignment. Given the consistent accuracy of the numerical results obtained by EFDM, the positivity-preserving FDM, and the PINN method, all three are recommended as competitive and reliable tools. The solution techniques explored in this work hold significant potential for current and future modeling efforts in various nonlinear wave phenomena, such as soliton propagation in optical fibers.

In [Contribution 4], Pyatkov and Soldatov investigated inverse problems involving the identification of lower-order coefficients in a second-order parabolic equation. The unknown coefficients are represented as finite series with time-dependent terms. Both the general nonlinear case and the linear case were considered. Overdetermination conditions are formulated as integrals of the solution over the boundary of the domain, weighted appropriately. The authors focused on proving existence and uniqueness theorems, as well as deriving stability estimates for the solutions to these inverse problems. The problem is reformulated as an operator equation, which is analyzed using the contraction mapping principle. The solution is shown to lie in an appropriate Sobolev space and possesses all generalized derivatives appearing in the equation, each integrable to a certain power. The proof method is constructive, offering a foundation for the development of new numerical algorithms to solve such inverse problems.

In [Contribution 5], Ene et al. conducted an analytical investigation of the Shimizu-Morioka dynamical system using the Optimal Auxiliary Functions Method (OAFM). This system exhibits chaotic behavior, relevant to various physical applications, particularly in chaos synchronization—an important phenomenon observed in numerous real-world processes. The authors derived semi-analytical solutions to the system and performed a comparative analysis between these results and numerical solutions. The comparison demonstrated the accuracy and computational efficiency of the OAFM. The method’s selection is further justified by its favorable performance relative to an iterative approach requiring 7–10 iterations. Additionally, the study explores the influence of physical parameters on the system’s damped oscillations and periodic behaviors, offering insights into the dynamic characteristics of the solutions.

The aim of this Special Issue is to provide a broad and timely contribution to the existing body of literature on differential equations and inverse problems. The featured methodologies are expected to be both valuable and engaging, attracting interest from both the scientific community and industry practitioners. The novel strategies presented herein are intended to inspire researchers across mathematics and related multidisciplinary fields, encouraging further investigation into advanced approaches for addressing differential equations and inverse problems. Future research may focus on refining and extending these techniques to enhance their effectiveness and applicability in increasingly complex real-world scenarios.