Foundations

16 pages, 1869 KB

Open AccessArticle

Chebfun in Numerical Analytic Continuation of Solutions to Second Order BVPs on Unbounded Domains

by Călin-Ioan Gheorghiu and Eduard S. Grigoriciuc

Foundations 2026, 6(1), 4; https://doi.org/10.3390/foundations6010004 - 3 Feb 2026

Viewed by 104

The well-known shooting algorithm has produced important results in solving various linear as well as nonlinear BVPs, defined on unbounded intervals, but has become obsolete. The main difficulty lies in the numerical handling of the domain’s infiniteness. This paper presents a three-step strategy that significantly improves the traditional truncation algorithm. It consists of Chebyshev collocation, implemented as Chebfun, in conjunction with rational AAA interpolation and analytic continuation. Furthermore, and more importantly, this approach enables us to provide a thorough analysis of both possible errors in dealing with and the hidden singularities of some BVPs of real interest. A singular second-order eigenvalue problem and a fourth-order nonlinear degenerate parabolic equation, all defined on the real axis, are considered. For the latter, Chebfun provides properties-preserving solutions. Travelling wave solutions are also studied. They are highly nonlinear BVPs. The problem arises from the analysis of thin viscous film flows down an inclined plane under the competing stress due to the surface tension gradients and gravity, a long-standing concern of ours. By extending the solutions to these problems in the complex plane, we observe that the complex poles do not influence their behaviour. On the other hand, the real ones involve singularities and indicate how long solutions can be extended through continuity. Full article

(This article belongs to the Section Mathematical Sciences)

► Show Figures

Figure 1

32 pages, 491 KB

Open AccessArticle

Complexity Assessments for Decidable Fragments of Set Theory. IV: A Quadratic Reduction from Constraints over Nested Sets to Boolean Formulae

by Domenico Cantone, Andrea De Domenico, Pietro Maugeri and Eugenio G. Omodeo

Foundations 2026, 6(1), 3; https://doi.org/10.3390/foundations6010003 - 30 Jan 2026

Viewed by 107

Abstract

As a contribution to automated set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms

x = y ∖ z

x \neq y ∖ z

, and

z = \{x\}

, where

x, y, z

stand for [...] Read more.

As a contribution to automated set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms

x = y ∖ z

x \neq y ∖ z

, and

z = \{x\}

, where

x, y, z

stand for variables ranging over the von Neumann universe of sets, into quantifier-free Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness, and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms

x = y ∖ z

and

x \neq y ∖ z

and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability preservation, the proposed translation has quadratic algorithmic time complexity and bridges two languages, both of which are known to have an

NP

-complete satisfiability problem. Full article

(This article belongs to the Section Mathematical Sciences)

► Show Figures

Figure 1

35 pages, 2106 KB

Open AccessArticle

A Novel Method That Is Based on Differential Evolution Suitable for Large-Scale Optimization Problems

by Glykeria Kyrou, Vasileios Charilogis and Ioannis G. Tsoulos

Foundations 2026, 6(1), 2; https://doi.org/10.3390/foundations6010002 - 23 Jan 2026

Viewed by 200

Abstract

Global optimization represents a fundamental challenge in computer science and engineering, as it aims to identify high-quality solutions to problems spanning from moderate to extremely high dimensionality. The Differential Evolution (DE) algorithm is a population-based algorithm like Genetic Algorithms (GAs) and uses similar operators such as crossover, mutation and selection. The proposed method introduces a set of methodological enhancements designed to increase both the robustness and the computational efficiency of the classical DE framework. Specifically, an adaptive termination criterion is incorporated, enabling early stopping based on statistical measures of convergence and population stagnation. Furthermore, a population sampling strategy based on k-means clustering is employed to enhance exploration and improve the redistribution of individuals in high-dimensional search spaces. This mechanism enables structured population renewal and effectively mitigates premature convergence. The enhanced algorithm was evaluated on standard large-scale numerical optimization benchmarks and compared with established global optimization methods. The experimental results indicate substantial improvements in convergence speed, scalability and solution stability. Full article

(This article belongs to the Section Mathematical Sciences)

► Show Figures

Figure 1

43 pages, 1907 KB

Open AccessArticle

Analysis of Telegraph Equation for Propagating Waves with Dispersion and Attenuation

by Hyoung-In Lee, Sang-Hyeon Kim, Tae-Yeon Kim and Hee-Jeong Moon

Foundations 2026, 6(1), 1; https://doi.org/10.3390/foundations6010001 - 6 Jan 2026

Viewed by 218

Abstract

The structural vibration of industrial droplet dispensers can be modeled by telegraph-like equations to a good approximation. We reinterpret the telegraph equation from the standpoint of an electric–circuit system consisting of an inductor and a resistor, which is in interaction with an environment, say, a substrate. This interaction takes place through a capacitor and a shunt resistor. Such interactions serve as leakage. We have performed an analytical investigation of the frequency dispersion of telegraph equations over an unbounded one-dimensional domain. By varying newly identified key parameters, we have not only recovered the well-known characteristics but also discovered crossover phenomena regarding phase and group velocities. We have examined frequency responses of the electric circuit underlying telegraph equations, thereby confirming the role as low-pass filters. By identifying a set of physically meaningful reduced cases, we have laid the foundations on which we could further explore wave propagations over a finite domain with appropriate side conditions. Full article

(This article belongs to the Section Mathematical Sciences)

► Show Figures

Graphical abstract

Journal Menu

Journal Browser

Foundations, Volume 6, Issue 1 (March 2026) – 4 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI