Mathematical Physics Research Articles

Using Desmos for visualizing Fourier series in mathematical a physics course

The Fourier series has been used widely to model various physical phenomena. It becomes a core concept taught in mathematical physics courses. To help students understanding the concept and interpretation of the Fourier series, we propose the utilization of the Desmos application in the classroom. Desmos has a feature of a graphing calculator that can be used for visualizing a function without programming. With an appropriate teaching approach, the Fourier series can be understood more easily.

Special Generalized Trigonometric and Hyperbolic Function Solutions of the Space-Time Fractional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation

In mathematical physics, the nonlinear dispersive modified Benjamin-Bono-Mahony equation is an essential nonlinear evolution equation. This paper employs the Sardar sub-equation approach to derive the analytic solutions of the space-time fractional nonlinear dispersive modified Benjamin-Bona-Mahony equation with a conformable derivative. These solutions, obtained using the proposed approach, are presented in specialized generalized hyperbolic and trigonometric forms. The soliton solutions are also obtained using the presented approach. Additionally, some of these solutions are illustrated in both two- and three-dimensional graphs for visualization. Finally, the suggested approach is novel, easy, and productive for obtaining analytic solutions and is applicable to numerous nonlinear fractional partial differential equations.

On the special norm and completeness of spaces of continuous functions of several variables with Lipschitz–Hölder type constraints

Let $X_0\subseteq\mathbb R^n$ be a nonempty open set and $X_0\subseteq X\subseteq\overline X_0$. We admit that the set $X_0$ is unbounded and/or has a countable number of connected components. In this paper, we study some spaces of functions $f\colon X\to\mathbb R$ endowed with a special norm $\|\cdot\|$. The definition of the norm involves an $n$-dimensional vector $(\Delta x)^{-1}\Delta f$, which is an analogue of the relation $\frac{\Delta f}{\Delta x}$ generating the concept of the derivative of a function of one variable. The vector $(\Delta x)^{-1}\Delta f$ can be associated with the vector $\mathrm{grad}\,f(\cdot)$. The invertible matrix $\Delta x$ of order $n$ consists of special increments of the argument $x\in \mathbb R^n$, and the vector $\Delta f$ consists of special increments of the function $f$. A number of properties of the vector $(\Delta x)^{-1}\Delta f$ is proved, and an exact formula for its Euclidean norm is obtained. We prove the completeness with respect to a special norm $\|\cdot\|$ of the space $\mathcal G(X)$ consisting of continuous bounded functions $f\colon X\to\mathbb R$ and having additional restrictions of the Lipschitz–Hölder type. Such functions play an important role in solving mathematical physics problems. A number of important subspaces of the space $\mathcal G(X)$ is investigated. It is proved that two of them are Banach, and one of them, for $n=1$ and under certain conditions, is the closure of the space of piecewise linear functions $f\colon X\to\mathbb R$.

NEW APPLICATIONS OF THE FRACTIONAL DERIVATIVE TO EXTRACT ABUNDANT SOLITON SOLUTIONS OF THE TIME FRACTIONAL ZAKHAROV–KUZNETSOV–BENJAMIN–BONA–MAHONY EQUATION IN MATHEMATICAL PHYSICS

This paper explores the time fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation as a framework for various phenomena, including water wave mechanics, shallow water waves, quantum mechanics, ion-acoustic waves in plasma, and electro-hydro-dynamical models for local electric fields and signal processing waves are transmitted over optical cables. Extended Direct Algebraic Technique (EDAT) and Improved Generalized Tanh-Coth Technique are used to find new accurate traveling-wave solutions with appropriate physical free parameter values. The fractional traveling-wave transformation is used to convert the equation into a nonlinear ordinary differential equation, where the fractional derivative is assessed in a conformable manner. Trigonometric and hyperbolic functions are the forms in which the solutions can be obtained. The suggested techniques can achieve periodic, mixed dark-bright soliton, bright soliton, dark soliton, M-shaped, Compacton soliton, bell-type soliton, smooth mixed dark-bright soliton and W-shaped soliton. Some of the obtained solutions are graphically represented as 3D and contour plots. Meanwhile, the impacts of the fractional parameter are shown in 2D plots. The above techniques are effective and reliable, and it may be utilized as a substitute to develop new solutions for many fractional differential equation types employed in mathematical physics.

New Applications of the Fractional Derivatives to Extract Abundant Soliton Solutions of the Fractional Estevez-Mansfield-Clarkson Equation in Mathematical Physics

New Applications of the Fractional Derivatives to Extract Abundant Soliton Solutions of the Fractional Estevez-Mansfield-Clarkson Equation in Mathematical Physics

Explicit representations of the norms of the Laguerre-Sobolev and Jacobi-Sobolev polynomials

Abstract This paper deals with discrete Sobolev orthogonal polynomials with respect to inner products built upon the classical Laguerre and Jacobi measures on the intervals $$ [0,\infty ) $$ [ 0 , ∞ ) and $$ [-1,1] $$ [ - 1 , 1 ] , respectively. In addition, they are equipped with point masses at a finite endpoint of the interval involving the underlying functions and their derivatives of first or higher order. One of the intrinsic features of these polynomials are their $$ L^2 $$ L 2 -norms in the corresponding inner product spaces. Their knowledge is essential to orthonormalize the polynomials and thus indispensable to treat the corresponding Fourier-Sobolev series and other topics, notably in approximation theory, spectral theory or mathematical physics. Proceeding from an appropriate representation of the Sobolev polynomials which reflect the influence of the point masses, we explicitly establish their squared norm in an efficient form. In each case, the value differs from the familiar squared norm of the Laguerre or Jacobi polynomials by a factor which itself is a product of two essentially identical terms. Surprisingly, each of these factors turns out to be the quotient of the leading coefficients of the Sobolev polynomial and its classical counterpart. Obviously, our results enable to determine the asymptotic behavior of the norms of the orthogonal polynomials considered for large n.

Kojève’s planetary approach to Greek philosophy

Abstract This article discusses the second volume of Kojève’s Essai d’une historie raisonnée de la philosophie paienne dedicated to Plato and Aristotle. In particular it examines the strange emphasis and interpretation that Kojève gives of the central role played by astral theology in the philosophy of both Plato and Aristotle. The astral theology of the classical Greek philosophers was given pride of place by Werner Jaeger’s reconstruction of Aristotle’s development in the 1920s. The article discusses Kojève’s motivations, centrally that of distinguishing philosophy from mathematical physics as the seat of true knowledge, for considering Plato and Aristotle as “theologians.” It then advances a hypothesis about the relation between Kojève’s theological interpretation of Plato and Aristotle and its relation to his conception of Hegel’s philosophy.

SPECIAL HYPERBOLIC TYPE APPROXIMATION AND FOURIER METHOD FOR SOLVING 3-D STATIONARY DIFFUSION PROBLEM

In this paper, we explore the conservative averaging methods (CAM) used to solve three-dimensional boundary-value problems of second order. We investigate various interpretations of CAM to address these problems. We consider a special type of hyperbolic approximation - spline interpolation, that estimates middle integral values of piecewise smooth functions. By employing these splines, along with parabolic-type splines, we can reduce multidimensional mathematical physics problems in three dimensions to problems involving two dimensions for one coordinate. This approach also enables us to further simplify the two-dimensional problems into one-dimensional problems, where analytical solutions can be obtained. Additionally, we numerically solve the corresponding problem with homogeneous boundary conditions of the first kind using the Fourier series method, and we compare these numerical results with the analytical solutions.

Spectrum of Rota–Baxter operators

Rota–Baxter operators have been studied since 1960, and there are lots of their applications in mathematical physics, number theory, and noncommutative geometry. We state a surprisingly general property of such operators: the spectrum of every Rota–Baxter operator of weight [Formula: see text] on a finite-dimensional unital (not necessarily associative) algebra is a subset of [Formula: see text]. We even extend this result to the case of infinite-dimensional algebraic algebras (when [Formula: see text]). Based on these results, we define a new invariant of an algebra: the Rota–Baxter [Formula: see text]-index [Formula: see text] of an algebra [Formula: see text] as the infimum of the degrees of minimal polynomials of all Rota–Baxter operators of weight [Formula: see text] on [Formula: see text]. We compute the Rota–Baxter [Formula: see text]-index for the matrix algebra [Formula: see text], [Formula: see text]: it is shown that [Formula: see text].

Preface

The International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE) is an interdisciplinary scientific event which aims to promote the knowledge and the development of high-quality research in mathematical fields that have to do with the applications of other scientific fields and the modern technological trends that appear in them, these fields being those of Physics, Chemistry, Biology, Medicine, Economics, Sociology, Environmental sciences etc. The 13th IC-MSQUARE took place in Kalamata, Greece between September 30 and October 3, 2024. In addition, (because in previous years the conference, due to the pandemic, took place only with online presentations), the possibility of participation via the internet was also adopted this time. Thus, apart from the talks and workshops held on site, we also had a large collection of pre-recorded presentations that were available to the participants through the conference website. The Conference was attended by more than 320 participants and hosted about 295 oral and virtual presentations while counted more than 700 pre-registered authors. The 13th IC-MSQUARE consisted of different and diverging workshops and thus covered various research fields where Mathematical Modeling is used, such as Theoretical and Mathematical Physics, Neutrino Physics, Non-Integrable Systems, Dynamical Systems, Computational Nanoscience, Biological Physics, Computational Biomechanics, Complex Networks, Stochastic Modeling, Fractional Statistics, DNA Dynamics, Macroeconomics, Social Modeling etc. International Scientific Committee and Organizing Committee are available in this PDF.

Novel Nonlinear Dynamical Solutions to the (2 + 1)‐Dimensional Variable Coefficients Equation Arise in Oceanography

ABSTRACTThis article examines the (2 + 1)‐dimensional variable coefficient KdV‐type equation that arises in oceanography. Oceanography is the science of studying the oceans. It is an earth science that addresses many different subjects, such as the dynamics of ecosystems, ocean currents, waves, and geophysical fluid dynamics. Fluid dynamics is a nonlinear science that studies the flow of liquids and gases in physics, physical chemistry, and engineering. The assistance of the Hirota bilinear method secures different forms of multiple solitons and M‐lump waves, such as one‐, two‐, and three‐soliton, by using different functions of variable coefficients. Moreover, the long‐wave technique is under consideration when analyzing hybrid solutions, namely mixed soliton‐lump, two‐soliton‐lump, and soliton‐two‐lump solutions. The results of this exploration exhibit the physical characteristics of solutions and collision‐related aspects within the variety of nonlinear systems. The physical significance of the extracted solutions is meticulously analyzed by presenting a variety of profiles that display the behavior of the solutions for specific parameter values. Our results indicate their potential use in future endeavors to discover unique and assorted solutions for nonlinear evolution equations encountered in engineering and mathematical physics. The reported results in this study are the first of their kind to be reported in the literature. These results may be helpful in explaining the physical behavior of various nonlinear physical models.

Traveling wave solutions of the two-dimensional Konopelchenko-Dubrovsky equation

Abstract In this study, the (2+1)-dimensional system of Konopelchenko-Dubrovsky is studied. This equation extends the Korteweg-de Vries(KdV) equation, that describes the evolution of small amplitude dispersion waves generated in shallow water. The tan-cot method is applied to receive the traveling wave solutions. It is presented that the tan-cot method is a useful mathematical tool for obtaining analytical solution to the equations of mathematical physics. In the figures, the graphs of the obtained solutions are shown

Some remarks on the geometrization of physics and the topological theories of quantum fields

Abstract This article aims at examining the role of geometrical and topological concepts in the recent developments of theoretical physics, especially in gauge theory and string theory, as well as at showing the great significance of these concepts for a deeper understanding of the dynamics of physics and the mathematical structure of spacetime. Contemporary developments in theoretical physics suggest that another “revolution”, after those of general relativity and quantum mechanics, may be in progress, through which a new kind of geometry may enter physics, and spacetime itself can be reinterpreted as an approximate, derived concept. One of the main purposes of this paper is to show the deep significance of spacetime geometry and topology in influencing the laws which are supposed to govern the behavior of matter, and further to support the thesis that matter itself might be built from geometry, in the sense that particles of matter as well as the other forces of nature emerges in a similar way that gravity emerges from geometry. This article is an attempt to show that it is very unsatisfactory try to develop physics on a rigorous basis to give formal justification of the experimental conclusions; this is the traditional (or “instrumental”) role of mathematical physics since, in fact, the Newton’s time. Contrary to this conception, we try to understand the deeper meaning of the physics-mathematics connection. Rather than to view mathematics as a tool to establish physical theories, or physics as a way of pointing to mathematical truths, we can try to dig more deeply into the relationship between them. This investigation may not only have theoretical and philosophical interest, but it could lead to a better understanding of and even to new insights into the geometrical and topological structures acting at the frontier between mathematics and physics.

Analyzing breather waves and soliton solutions to the P-type (3+1)-dimensional evolution equation arising in mathematical physics

This study investigates the Painlevé integrable generalized P-type [Formula: see text]-dimensional evolution equation. We apply the [Formula: see text]–expansion method and Nucci’s reduction approach to derive diverse traveling wave solutions of the generalized P-type [Formula: see text]-dimensional evolution equation. These techniques, along with wave transformation and homogeneous balancing methods, allow us to obtain concise and general traveling wave solutions. The obtained solutions include rational, hyperbolic trigonometric, and circular trigonometric functions. We plot two- and three-dimensional graphs for the obtained solutions to get a deeper comprehension of the dynamical properties of the P-type model. The plots show a variety of wave profiles, including singular, bright, dark, bright-dark solitons, and breather waves. These solutions will improve the exact wave types found in the P-type model’s currently available literature. They might also be useful in explaining the potential physical mechanisms behind the wave phenomenon in plasma physics and nonlinear wave theory.

On the modeling of vibrations of deformable elements of apparatus structures under vibration disturbing loads

The task of reducing the level of vibration of radio-electronic devices (REA) is an urgent task in mechanical engineering of aircraft industry. The purpose of the study is to investigate the vibrations of plate-like elements with attached masses under the influence of vibration loads. All deformable elements are viscoelastic. The viscoelastic properties obey the hereditary Boltzmann-Volterra integral relation. Linear oscillations of the considered mechanical system are investigated. For reduction of impulse perturbations of a radio-electronic unit with attached masses, a method and an algorithm for solving the problem are developed. The method of complex amplitudes, the methods of mathematical physics equations, the Gauss method, the Mueller method and the Godunov orthogonal run method were used in developing a method for solving the problem. An algorithm for determining the resonance frequency and amplitude of displacements of the considered mechanical system was proposed. Application of the proposed mathematical model taking into account viscous properties of the elements allows to reduce the total impulse loads of the REU up to 25%. It is established that the use of rubber shock absorbers reduces the amplitudes of vibrations of the equipment up to 30%. It is also established that the use of dissipative and inhomogeneous design allows maximal reduction (up to 40–50%) of resonant amplitudes of REU in low-frequency ranges.

New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis

Abstract To comprehend nonlinear dynamics, one must have access to soliton solutions, which faithfully portray the actions of numerous physical systems and nonlinear equations. Notable nonlinear equations in relativistic physics, quantum field theory, nonlinear optics, dispersive wave phenomena, contemporary industrial applications, and plasma physics include the Klein–Gordon and Sharma–Tasso–Olver equations, which shed light on wave behavior and interactions. This study introduces a powerful approach to uncovering some novel soliton solutions for these equations, namely, the new generalized ( G ′ / G ) ({G}^{^{\prime} }\left/G) -expansion method. The derived soliton solutions are articulated in terms of rational, trigonometric, and hyperbolic functions, each embodying the physical implications of the equations through meticulously specified parameters. The resulting solutions encompass several waveforms, including sharp solitons, singular periodic solitons, flat kink solitons, and singular kink solitons. The results indicate that the employed method is both robust and very effective for the analysis of nonlinear evolution equations (NLEEs). It is compatible with computer algebra systems, facilitating the generation of more generalized wave solutions. The strength and versatility of the new generalized ( G ′ / G ) ({G}^{^{\prime} }\left/G) -expansion method suggest its potential for further research, particularly in exploring exact solutions for other NLEEs. The approach represents a significant expansion in the methodologies available for handling nonlinear wave equations, opening new avenues for theoretical and applied investigations in nonlinear science. Furthermore, the bifurcation analysis is carried out, which reveals the comprehension and precise representation of the dynamics of these two nonlinear partial differential equations. It offers the information required to build a comprehensive and significant phase portrait, including insights into solution behaviors, stability changes, and parameter dependencies.

Phase plane analysis and novel soliton solutions for the space-time fractional Boussinesq equation using two robust techniques

The Boussinesq equation is essential for studying the behavior of shallow water waves, surface waves in oceans and rivers, and the propagation of long waves in nonlinear systems. Its fractional form allows for a more accurate representation of wave dynamics by incorporating the effects of nonlocal interactions and memory. In this paper, we focus on obtaining exact traveling wave solutions for the space-time fractional Boussinesq equation using two well-established methods: the modified Sardar sub-equation method and the new extended direct algebraic method, both implemented with Atangana’s beta derivative. By applying these methods, we derive a variety of soliton solutions, including kink, anti-kink, periodic, dark, bright, and singular solitary waves. These solutions are presented in different mathematical forms, such as rational, hyperbolic, trigonometric, and exponential functions. This study not only provides new solutions but also enhances the understanding of wave propagation in fractional models, demonstrating the efficiency and applicability of the chosen methods. A comparative analysis of the methods and results is presented, along with an examination of the impact of fractional derivatives by adjusting their values. The study also includes 2D and 3D plots that illustrate the temporal behavior of the solutions. This study demonstrates that the methods employed are applicable to other nonlinear models in mathematical physics. A detailed analysis of the model’s behavior is conducted, focusing on bifurcation, chaos, and stability. Phase portrait analysis at critical points reveals shifts in the system’s dynamics, and introducing an external periodic force generates chaotic patterns. The solutions provided offer new insights into shallow water wave models, presenting effective tools for in-depth investigation of wave dynamics. All solutions are verified through MATHEMATICA and MATLAB simulations, ensuring their accuracy and reliability.

Exploring the interaction between lump, stripe and double-stripe, and periodic wave solutions of the Konopelchenko–Dubrovsky–Kaup–Kupershmidt system

Abstract The Konopelchenko–Dubrovsky–Kaup–Kupershmidt system has significant implications in various fields, including fluid mechanics, ocean dynamics, and plasma physics. This system includes various widely recognized nonlinear evolution equations as special cases. In this study, we present a systematic approach to identifying novel wave solutions to the system, examining various combinations of interactions between lumps, stripes, double stripes, and periodic waves. By strategically selecting the arbitrary free parameters, we incorporate various 2D and 3D profiles to clearly demonstrate the dynamic behaviors of the mixed localized wave structures. These graphical representations indicate that some of the identified solutions exhibit periodic wave propagation along a straight line at specific angles to the spatial axes, maintaining constant wavelengths, amplitudes, and velocities. The solutions proposed in this study provide valuable insights into the mechanisms of wave propagation across diverse physical domains. Furthermore, the approach outlined herein is versatile and can be applied to various nonlinear differential equations in mathematical physics.

Visualization of the impact of noise of the closed-form solitary wave solutions for the stochastic Zhiber–Shabat model

This paper investigates the stochastic Zhiber–Shabat problem under the multiplicative time noise analytically. The Sardar subequation approach is used to explored the solitons and solitary wave solutions under noise. Different families of solutions are gained in the form of dark soliton, singular soliton, bright soliton, complex dark–bright soliton and mixed form solitons along with mixed trigonometric functions. In mathematical physics, the solutions describe a wide range of physical phenomena including chemical kinetics, nonlinear optics, plasma physics, and quantum field theory. The composition of the results would also be displayed in 3D for a variety of parameters using the most modern scientific tools. Graphical representation of the solution’s physical behavior demonstrates that the obtained solutions are generated as periodic soliton wave structures with noise term effects that are singular, dark, bright and complex bright dark. Using 3D, 2D, and contour plots with different noise strengths, we illustrate the characteristics of the nonlinear model. It has been shown that choosing a suitable set of parameters optimizes the visual representation of the physical structures.

Determination of analytical dependencies of distributed forces in a deformable wheel – deformable surface contact zone

The object of this study is the contact interaction of two deformable bodies of inconsistent geometric shape, in particular, change in the stress-strain state of the wheel and the supporting surface. The significance of this topic arises from growing demands for vehicle mobility in difficult terrains, the necessity of minimizing environmental impact, and the need to optimize the design of mobile machinery components. A major challenge lies in developing a suitable analytical solution to define the stress-strain state variations within the contact zone between the wheel and soil or other surfaces. This study employs an approach grounded in the fundamental principles of mathematical physics applied to elasticity theory problems. This enabled the derivation of analytical equations that describe the absolute deformations of both the surface and the wheel (tire), along with the contact pressure distribution. The pressure distribution within the contact zone was determined using the properties of surface integrals of the second kind. Concentrated forces, when related to the contact area, were equated to the integral value of this surface integral. The values of these distributed forces were then incorporated into the transformed Boussinesq and Cerruti potential equations. The resulting analytical relationships can be utilized to determine the relative deformations of the contacting bodies and the stress distribution within them. Crucially, these relationships also serve as a basis for deriving equations that define the contact zone boundaries and the rolling resistance coefficient for deformable bodies. These derived relationships are general and presented in a form applicable to loads on both driving and passive (driven) wheels. This proposed model offers substantially improved analytical accuracy over existing empirical methods. Moreover, these analytical dependencies help circumvent the computationally intensive calculations typically required by FEM (Finite Element Method) or DEM (Discrete Element Method) simulations for every unique loading scenario and material property set.