Abstract Spreading waves in electrically excited media is a basic phenomenon that underlines crucial events in many domains, from how electrical signals move through heart tissue to how nerve impulses move through biological neurons. The nonlinear dynamics of electrical excitation and recovery in cardiac tissue give rise to cardiac wave propagation, which is crucial for heart function. FitzHugh-Nagumo (FHN) equation is a prominent mathematical framework for studying this type of excitability. It has been studied in great detail, but complete set of exact solutions that capture its entire broad range of propagation characteristics is still insufficient. To address this limitation, the paper thoroughly develops new exact traveling wave solutions for the FHN equation. The new auxiliary equation method is used for the FHN equation because it can produce a broader and more consistent range of solutions compared to older methods. These solutions reveal a range of nonlinear wave patterns that describe different propagation behaviors in excitable systems, such as solitary, kink-type, and periodic forms etc. The obtained solutions are illustrated using three-dimensional surface plots, density profile plots, and a composite two-dimensional (2-D) configuration, derived from the integration of individual 2-D representations, providing an enhanced understanding of the effect of wave speed. The findings are compared with the outcomes from previously published studies conducted by other researchers. The present method exhibits efficiency, broad applicability, and simplicity in addressing nonlinear evolution equations encountered in mathematical physics and applied sciences.

In this work, we construct a new class of Appell-type polynomials generated through extended truncated and truncated exponential kernels, and we analyze their core algebraic and operational features. In particular, we establish a suitable recurrence scheme and obtain the associated multiplicative and differential operators. By confirming the quasi-monomial structure, we further deduce the governing differential equation for the proposed family. In addition, we present both a series expansion and a determinant formulation, providing complementary representations that are useful for symbolic manipulation and computation. As special cases, we introduce and study subfamilies arising from this setting, namely, extended truncated exponential versions of the Bernoulli, Euler, and Genocchi polynomials, and discuss their structural identities and operational behavior. Overall, these developments broaden the theory of special polynomials and furnish tools relevant to problems in mathematical physics and differential equations.

The equilibrium shape of a crystal is a fundamental problem in materials science and condensed matter physics. The Wulff construction, a cornerstone of crystal morphology prediction, is traditionally presented and utilized as a powerful geometric algorithm to derive equilibrium shapes from anisotropic surface energy γ(n). While its application across materials science is vast, the profound mathematical physics underpinning it, specifically its intrinsic identity as a manifestation of the Legendre transform, is often relegated to a passing remark. This work recenters the focus on this fundamental duality. We present a comprehensive, step-by-step derivation of the Wulff shape from the variational principle of surface energy minimization under a constant volume, employing the language of support functions and differential geometry. We then rigorously demonstrate that the equilibrium shape, defined by the support function h(n), and the surface energy density γ(n) are conjugate variables linked by a Legendre transformation; the Wulff shape W is precisely the zero-sublevel set of the dual function γ*(x)=supn[x·n−γ(n)]. This perspective elevates the Wulff construction from a mere graphical tool to a canonical example of convex duality in thermodynamic systems, connecting it to deeper principles in convex analysis and statistical mechanics. To bridge theory and computation, we provide a robust computational algorithm implemented in pseudocode capable of generating Wulff shapes for two-dimensional (2D) crystals with arbitrary N-fold symmetry. Finally, we discuss the relevance and extensions of the classical theory in contemporary research, including non-equilibrium growth, nanoscale effects, and machine learning approaches.

This paper examines the Landau–Ginzburg–Higgs (LGH) equation that arises in mathematical physics. We introduce vital solitary wave solution for the LGH model using the unified solver strategy. We depict the dynamic evolution of the discovered soliton wave. To statistically model the situation, a new two-parameter hyperbolic secant (TPHS) distribution is also derived based on the solution of the nonlinear LGH equation. The unit, generalized, and wrapped two-parameter hyperbolic secants are presented using the TPHS distribution. The probability density function (PDF) plots for each of the models under consideration are shown, demonstrating how adaptable their shapes are to represent various scenarios. Moreover, the hazard rate function (HRF) plots of several considered models are shown.

Context. In this paper, we describe a mathematical formalism for a [Formula: see text]-dimensional manifold with [Formula: see text]-correlators of [Formula: see text] types of targeted source objects, with cross correlations and contaminants. Methodology. In particular, we build this formalism using simple notions of mathematical physics, field theory, topology, algebra, statistics n-correlators and Fourier transform. Results. We present and discuss the applicability of this formalism in the context of cosmological scales, i.e. from astronomical scales to quantum scales, for which we give some intuitive examples, explicitly, for standard spacetime dimensions and extra dimensions. Conclusion. We conclude that this study can be used as a guide to analyse and build models for future cosmological and collider experiments. Furthermore, this study opens the road of extra dimension studies.

ABSTRACT This paper deals with Kant's elaboration of a metaphysical foundation of the principle of inertia in the Metaphysical Foundations of Natural Science. Many of Kant's contemporaries treat inertia not as an issue of mathematical physics but rather as a general feature of material objects that is addressed by metaphysics and, to some extent, by theology as well. In turn, inertia is often seen as the reason why matter is fundamentally passive, thus providing an argument against materialism. In particular, Abraham Gotthelf Kästner and Johann Samuel Traugott Gehler are considered on this score. They agree with Kant in that the principle of inertia follows from the general causal principle. Contrary to Kant, Kästner and Gehler treat inertia as a phenomenon of experience, whereas it seems a unique feature of Kant's approach to conceive of inertia as expressing the lifelessness of matter.

In this study, the nonlinear partial differential equation that governs the free vibration of a carbon nanotube composite beam is analytically investigated using the truncated M‐fractional derivative. This model is a beam supported by a nonlinear viscoelastic base and reinforced by carbon nanotubes. The exact solitary wave solutions are obtained by applying the generalized exponential rational function. Consequently, we obtain new types of solitary traveling wave solutions for this model as well as innovative soliton solutions such as bright, dark, singular, trigonometric, and rational solutions with complex structures. Furthermore, we plotted 2D, 3D, and contour graphs for several stated solutions by selecting appropriate parameter values. These results suggest that a more powerful, direct, and efficient mathematical tool for locating the exact single solutions to the nonlinear partial differential equations that arise in many natural science domains, such as mathematical biology, materials physics, mathematical physics, chemistry, and fluid mechanics, is the generalized exponential rational function technique.

Sharp coefficient inequalities and their applications for a class of analytic functions associated with an apple-shaped domain

The Klein–Gordon equation is of fundamental importance in mathematical physics, particularly due to its extensive applications in the analysis of solitonic phenomena, condensed matter systems, and the behavior of nonlinear wave dynamics. In this study, we develop a highly accurate numerical algorithm based on Taylor wavelets combined with the collocation technique, to approximate the solutions of nonlinear Klein-Gordon equations. An integration operational matrix is constructed and employed to transform the nonlinear Klein-Gordon initial–boundary value problem into an equivalent system of algebraic equations. One of the advantages of this method is that it does not require any restriction on domain discretization. This study also provides valuable insights into the underlying theoretical properties of the proposed method. To verify the reliability and accuracy of the proposed Taylor wavelet-based algorithm, a convergence analysis is performed. The method is then applied to four benchmark problems to further assess its effectiveness and computational performance. The comparison between the numerical and exact solutions demonstrates that the proposed method yields highly accurate results with minimal errors. All computations have been executed using MATLAB-2023b programming language.

Purpose. Determination of the safest variant of ventilation schemes for excavation areas with straight-flow (type 3-B) and reverse flow (type 1-M), which are used in coal mines of Ukraine to ensure gas and fire safety, based on their comparative assessment. Methodology. To achieve this goal, an analysis of scientific papers presented in the literature on the circumstances of methane explosions in coal mines of Ukraine and two main schemes for ventilation of treatment sites to prevent explosions and outbreaks in mine workings has been carried out. The equations of mathematical physics calculate the efficiency of methane removal depending on the ventilation scheme. A general analysis of these ventilation schemes has been carried out to select the safest of them. Results. Modern high-performance cleaning equipment, both domestic and foreign, used in the mines of Ukraine, provides coal production depending on the reservoir capacity with a load of 2-6 thousand tons per day. Achieving such high performance is possible only when using a pillar development system and two main schemes for ventilation of treatment sites. These are 3-B (or 2-B) type circuits, where the ventilation jet adheres to the produced space and is reinforced by air coming from the rear sight, and 1-M type circuit, in which the air jet is adjacent directly to the coal rear sight. The main difference between these schemes is the presence of supported ventilation workings behind the face when using the 3-B (2-B) scheme and their absence in the case of the 1-M scheme. Therefore, it is impossible to unequivocally determine which of them is the best in terms of manufacturability and safety. However, compliance with the requirements for degassing, current safety regulations and rules, as well as technological discipline, allows you to ensure an appropriate level of safety when using any of these schemes. Novelty. The paper provides a comparative analysis of the direct-flow ventilation scheme with refreshment (type 3-B) and the reverse flow scheme (type 1-M) in view of ensuring the safe conduct of mining operations. All the advantages and disadvantages of each of the systems are highlighted. A detailed list of methane control methods and the effectiveness of each of the considered schemes is separately given. Practical significance. In coal mines of Ukraine, ventilation in treatment workings is carried out mainly according to the schemes of type 3-B and 1-M. In the event of an accident – which, as a rule, is the result of design errors, technical defects or violations of safety rules – the consequences when using the 3-B ventilation scheme will be less severe than when applying the 1-M scheme. That is why the 3-B ventilation scheme will be considered safer and more progressive. The results obtained can be used in the design of ventilation and degassing systems for treatment sites, taking into account specific mining and geological conditions.

Using methods of mathematical physics, a comprehensive simulation of the short-range order in Fe₈₈P₁₂ and Cr₈₈C₁₂ alloys produced by electrodeposition was carried out. As the initial configuration for modeling, the crystal structure of the base metal was selected. Numerous experimental studies, including X-ray diffraction and electron microscopy analyses, have indicated that in metal-metalloid alloys, surface microstructures predominantly exhibit ellipsoidal morphologies. Based on these experimental observations, it was hypothesized that the macroscopic ellipsoidal formations observed on the alloy surfaces are composed of clusters with relatively simple geometric configurations, such as spheres or ellipsoids. The results of the simulation revealed that these clusters possess characteristic sizes not exceeding 30-50 angstroms, and their vectorial growth predominantly occurs along a single radial direction relative to the substrate surface. This anisotropic growth behavior is attributed to differences in local atomic bonding energy and diffusion kinetics, which drive the preferential alignment of cluster development. Moreover, it was established that the spatial distribution and size uniformity of the clusters significantly influence the overall mechanical and physicochemical properties of the coatings, including hardness, wear resistance, and corrosion stability. The combination of modeling outcomes with empirical data provides valuable insight into the microstructural evolution mechanisms governing electrodeposited metal-metalloid systems. These findings can serve as a basis for optimizing the electrodeposition parameters to tailor the surface structure and enhance the performance characteristics of functional coatings.

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in , highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, frequently arise in elasticity, potential theory, and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality reflects the underlying rotational and reflectional symmetries, the study constructs explicit, uniformly convergent series solutions. Through analytic continuation of integral representations, necessary and sufficient solvability criteria are established, ensuring convergence of all derivatives on compact domains. Furthermore, newly derived Green-type identities provide a systematic method to reconstruct boundary information and enforce stability constraints. This approach not only generalizes classical Laplace and biharmonic results to higher-order polyharmonic equations but also demonstrates how symmetry governs boundary data admissibility, convergence, and analytic structure, offering both theoretical insights and practical tools for elasticity, inverse problems, and mathematical physics.

The problem that this study deals with is ontogenetic growth of humans and animals. The novelty of this research is an approach to the problem how to recognise growth phenotypes in animals. The aim of this research was to analyse an analytical model of ontogenetic growth of animals with intention to recognise growth phenotypes. In this study we discuss possibility to extend results to the modelling of growth phenotypes in humans. In this study we not only analysed the model of animal growth but also offer an insight into the option to apply some methods known in mathematical physics and applied mathematics. In this research we concentrate on the modelling of growth of pigs. Pigs are known as a good model animal of humans in many aspects, including growth, obesity, digestion, and some others. In this model two aspects of ontogenetic growth were considered; the intention was to advance the biological understanding of the growth process.

Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave propagation in a circular cylinder and a cylindrical annulus. Next, using the multiplier method, conservation laws are derived for a parameterized system of constitutive equations in cylindrical coordinates involving a general expression for the Cauchy stress. Conservation laws for the Khokhlov-Zabolotskaya-Kuznetsov equation and Westervelt-type equations in various coordinate systems are also presented. To appear in Open Communications in Nonlinear Mathematical Physics. Special Issue in Honor of George W. Bluman, 2025 (27 pages, 6 tables, 54 references)

In this paper, we study the cooling of an infinitely long heated cylinder with an arbitrary initial temperature profile, modeled by the one-dimensional heat conduction equation in cylindrical coordinates. Extending the work of Chandel and Gupta, who used the multivariable $H$-function, we apply the multivariable Aleph-function, Srivastava-Daoust function, and multivariable polynomials to derive new integral and series solutions. The temperature distribution is expressed using Bessel functions and multivariable expansions, allowing for generalization, limiting cases, and asymptotic analysis. These results contribute to the analytical modeling of heat conduction and may be useful in thermal analysis and mathematical physics.

We survey the recent developments on quantum invariants of 3-manifolds and links: Z^ and FL. They are q-series invariants originated from mathematical physics, inspired by the categorification of a numerical quantum invariant—the Witten–Reshetikhin–Turaev (WRT) invariant—of 3-manifolds. They exhibit rich features, for example, quantum modularity, infinite-dimensional Verma module structures, and knot–quiver correspondence. Furthermore, they have connections to the 3d-3d correspondence and other topological invariants. We also provide a review of an extension of the above series invariants to Lie superalgebras.

In this paper, we investigate Dirichlet and Schwarz-type boundary value problems for both the inhomogeneous Cauchy--Riemann equation and higher-order polyanalytic equations in a nonstandard domain, specifically a triangular region formed by the intersection of three circular disks in the complex plane. Such domains introduce additional geometric complexity, which requires careful analytical treatment. By constructing appropriate kernel functions tailored to the geometry of the domain, we develop integral operator techniques that allow us to derive explicit solution formulas for the given boundary conditions. In addition, we establish necessary and sufficient conditions for the solvability of these problems, depending on the compatibility of boundary data and the properties of the inhomogeneous terms. Our approach generalizes classical methods used for standard domains, extending their applicability to more intricate geometric settings. The results presented in this work contribute to the broader theory of boundary value problems for complex partial differential equations and offer new tools for addressing similar problems in applied mathematical physics and complex analysis.

Abstract This study investigates the nonlinear Klein–Gordon (𝕂𝔾) equation, a fundamental relativistic wave equation that governs the propagation of scalar fields across various physical contexts, including quantum field theory, solid-state physics, and nonlinear optics. The 𝕂𝔾 equation is closely related to other nonlinear evolution equations, such as the sine-Gordon and ϕ 4 equations. Using analytical techniques, specifically the improved Kudryashov (𝕀𝕂ud) and modified rational (𝕄ℝat) methods, we construct exact traveling wave solutions to the nonlinear 𝕂𝔾 equation. To validate the accuracy of the obtained solutions, we employ He’s variational iteration (ℍ𝕍𝕀) method as a numerical scheme, demonstrating excellent agreement between analytical and numerical results, thus ensuring the applicability of the derived solutions. The research reveals a rich variety of localized and periodic wave solutions, elucidating the intricate dynamics governed by the nonlinear 𝕂𝔾 equation. These solutions have significant implications for understanding various physical phenomena, including soliton propagation, nonlinear wave interactions, and scalar field behavior in diverse systems. The novelty of this work lies on the systematic application of the 𝕀𝕂ud and 𝕄ℝat methods to the nonlinear 𝕂𝔾 equation, providing new insights into its analytical treatment and corroborating analytical solutions with numerical results. This research contributes to the field of nonlinear mathematical physics, particularly, the study of nonlinear evolution equations and their applications in quantum field theory, condensed matter physics, and nonlinear optics.

In this short communication, we announce that the Fourier-Laplace transform of the ‘typical’ function (i.e. generic in the sense of Baire’s category theorem) in the Schwartz class of the real half-line, being analytic in the lower half of the complex plane, has natural boundary the axis of the real numbers. Motivation originates from mathematical physics and partial differential equations literatures but the result is of independent interest also for the fields of complex and harmonic analysis as well as of integral transforms and special functions.

The sun is a vital component of our natural environment, and kinetic equations are important mathematical models that show how quickly a star’s chemical composition changes. Taking inspiration from these facts, we develop and solve a novel fractional kinetic equation by calculating the Laplace transform of hypergeometric functions in the complex coefficient parameter. This was a challenging task because the function cannot be integrated concerning the coefficient parameters using classical methods due to the infinite number of singular points of the gamma function involved in it. We achieved it using the distributional representation of the generalized hypergeometric function. Moreover, on the one hand, the role of the delta function is vital to represent the electromotive forces, and on the other, the solution of differential equations of engineering and mathematical physics led to a class of hypergeometric functions. This article is the confluence of both. Therefore, innovative characteristics concerning the Fox-Wright and several related important functions are applied for the simplification of the obtained outcomes. A popular class of fractional transforms involving generalized hypergeometric functions are evaluated using the delta function, and as a distribution, numerous additional features of this function are described.