Physical Equations Research Articles

An Experimental Test of Viewing the Past Based on Cosmic Memory

Some past events were brought out so that they were seen experimentally after hiding in the quantum vacuum in relation to the room in which we conducted the experiment after we and the work team made a fairly small nuclear accelerator in which the light could be rotated to shed light on the optical memory represented by nano-optical circuits lurking in the quantum vacuum, and with this highlighting we were able to enlarge it. And seeing it after it was stretched. Some of the physical equations of some researchers reached in their results the multiplicity of universes, but they did not mean the existence of the past and climbing to the future, and through experiments this was made clear in the pure equivalence of movement and mass, as well as in the study of cosmic memory, and that the body is still in its place even though he left it after taking two pictures of it, one of which was before. He left and the other was later. When he did so, he did so at a speed that did not exceed a fraction of a second. He explained this physically in his research on cosmic memory, which is a clear scientific confirmation of the existence of the past. On our part, we carried out a tiring experiment on living some of the facts of the past, and the results were amazing.

The problem of the structure and the soil plastic environment interaction in the conditions of dynamic evolutionary processes

Dynamic influences are one of the important factors affecting project decisions during the construction of buildings and structures. In the case of dynamic analysis, the range of possible variations in structural response is very difficult to determine. The complexity is determined not only by the parameters of the load itself (intensity, the law of change over time), but also by the characteristics of the research object (their values and change over time), which is an interconnected system of the structure and the soil base. Dynamic factors of influence can include periodic vibration or shock loads, the effects of explosions that cause a sharp change in pressure on the structure of the building, seismic fluctuations, etc. In many cases, the load is the result of stress waves propagating in the soil base. These factors can act on the natural environment long-term, relatively short-term, short-term and instantaneous. Therefore, the development of refined approaches for assessing the stress-strain state of structures during the interaction of the structure and the soil environment under the conditions of dynamic evolutionary processes remains an actual task today. In order to effectively find a solution to the dynamics problems, it is necessary to possess the entire set of analysis tools, to clearly present the rules and limits of their application. The ability to correctly assess the engineering and geological conditions of the construction site, the properties of the soils, the joint work of these soils with the foundations and structures of the building has always been and remains the basis of reliable and effective engineering design. The given statement of the problem is a theoretical basis to build the numerical approaches to the study of the vibrational processes of heterogeneous spatial bodies during their contact interaction with an elastic-plastic medium. The paper considers the modeling aspects of the structure and the soil elastic-plastic environment interaction under the conditions of dynamic evolutionary processes. The basic geometric and physical equations of the elasticity theory for heterogeneous circular and prismatic spatial bodies are given. In the range of solving the dynamics problem, the peculiarities of the mathematical model implementation, regarding the representation of the variation of kinetic energy as part of the equations of dynamic equilibrium of linear and nonlinear mechanical systems, are considered.

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation in Mathematical Physics

A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein–Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes.

Construction of new wave structures and stability analysis for the nonlinear Klein-Gordon equation

In this study, the nonlinear Klein–Gordon equation the relativistic equivalents of the nonlinear Schrödinger equations is presented which characterizes the connection between relativistic energy-momentum in a quantized manner. A variety of exact solutions are developed by employing the extended Fan sub-equation approach and the Sardar sub-equation method. The study includes three dimensional surface plots of some derived solutions like dark and bright solitary waves, kink solitary waves, anti-kink solitary waves, periodic solitary waves and hyperbolic functions, with several solutions being novel. Furthermore, the stability analysis of the observed solutions is also established to validate the scientific computations. The 3D, 2D and contour visualizations of the wave dynamics are demonstrated using Mathematica for a suitable choice of parameters. The applied strategies are recognized as a significant mathematical tool for obtaining analytical solutions to partial differential equations in mathematical physics.

Riccati sub-equation method for solving fractional flood wave equation and fractional plasma physics equation

In the field of mathematical physics, the Riccati sub-equation method is an important tool for locating the analytical solutions of nonlinear space–time fractional equations. In this study, we provide the Riccati sub-equation method for solving the space–time fractional equation by means of the Ablowitz–Kaup–Newell–Segur (AKNS) equation and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZK-BBM) equation, which explain the effect of flood waves and plasma physics, respectively, and reach new analytical solutions, including parameters. Each equation yields fifteen different solutions, which are in the form of generalized hyperbolic functions, generalized triangular functions, and rational functions. The effect of the arising graphs of both equations is in the form of kink waves and periodic waves, which are presented in 2D and 3D graphs.

Extrapolation of Hydrodynamic Pressure in Lubricated Contacts: A Novel Multi-Case Physics-Informed Neural Network Framework

In many technical applications, understanding the behavior of tribological contacts is pivotal for enhancing efficiency and lifetime. Traditional experimental investigations into tribology are often both costly and time-consuming. A more profound insight can be achieved through elastohydrodynamic lubrication (EHL) simulation models, such as the ifas-DDS, which determines precise friction calculations in reciprocating pneumatic seals. Similar to other distributed parameter simulations, EHL simulations require a labor-intensive resolution process. Physics-informed neural networks (PINNs) offer an innovative method to expedite the computation of such complex simulations by incorporating the underlying physical equations into the neural network’s parameter optimization process. A hydrodynamic PINN framework has been developed and validated for a variant of the Reynolds equation. This paper elucidates the framework’s capacity to handle multi-case scenarios—utilizing one PINN for various simulations—and its ability to extrapolate solutions beyond a limited training domain. The outcomes demonstrate that PINNs can overcome the typical limitation of neural networks in extrapolating the solution space, showcasing a significant advancement in computational efficiency and model adaptability.

Comprehensive investigation of a novel multi-segment heat exchanger for PCM/MF based ocean thermal engine: Numerical analysis, model reduction and experimental work

Phase change material/metal foam based exchangers are widely used in thermal energy storage and ocean engineering. In this paper, a novel heat exchanger called multi-segment heat exchanger is proposed for installation on an ocean thermal engine. This heat exchanger has several concentric circular shaped thin slices inside, which divide the interior into several small areas. During the melting process, independent vortices were formed in each segment. Three aspects of research were conducted: (i) in the investigation of heat transfer enhancement mechanisms, by numerically solving the N-S equations, the reason for enhanced heat transfer in the multi-segment heat exchanger was explained using the field synergy principle theory, and the influence of dimensions and environmental parameters on the melting process was analyzed. Results show that the optimal ratio of height to radius (H/R) for each segment is 1 and the total melting time is reduced by 31–38 %; (ii) for model reduction, by simplifying the full order physical equations for conduction dominant and convection dominant processes separately, ordinary differential equations describing multi-segment heat exchangers were obtained, which can be used for rapid prediction in the engineering practice. The calculation time of the reduced order model is within 0.1 s; (iii) in experimental work, an experimental set-up was built to test the original and the multi-segment heat exchangers, respectively. The numerical work and reduced order model were validated, and results showed that the proposed heat exchanger reduced the melting time by 34 %.

Principles and Equations of Physics: a multidisciplinary laboratory

In the last two academical years, as part of the Italian Ministry plan PLS – “Piano Lauree Scientifiche” (Scientific Degree Plan), two courses for high school students and teachers consisting in some interdisciplinary and transversal online meetings have been proposed. They regarded the three principles of dynamics, the law of universal gravitation and Maxwell’s equations. “Variations” around these topics were also presented – of historical, philosophical and also of musical nature – to make the cultural setting of what has been discussed deeper and make it meaningful in the present. At the end of each course, students produced a video of few minutes with a personal reworking and rethinking of the meaning of one of the topics discussed.

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation (LGHe), which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves. The LGHe finds applications in various scientific fields, including fluid dynamics, plasma physics, biological systems, and electricity-electronics. The study adopts Lie symmetry analysis as the primary framework for exploration. This analysis involves the identification of Lie point symmetries that are admitted by the differential equation. By leveraging these Lie point symmetries, symmetry reductions are performed, leading to the discovery of group invariant solutions. To obtain explicit solutions, several mathematical methods are applied, including Kudryashov’s method, the extended Jacobi elliptic function expansion method, the power series method, and the simplest equation method. These methods yield solutions characterized by exponential, hyperbolic, and elliptic functions. The obtained solutions are visually represented through 3D, 2D, and density plots, which effectively illustrate the nature of the solutions. These plots depict various patterns, such as kink-shaped, singular kink-shaped, bell-shaped, and periodic solutions. Finally, the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a crucial role in the study of physical quantities, such as the conservation of energy and momentum, and contribute to the understanding of the underlying physics of the system.

Solitons, nonlinear wave transitions and characteristics of quasi-periodic waves for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics and plasma physics

A (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation describing many nonlinear phenomena in fluid dynamics and plasma physics is considered. The N-solitons and breathers are obtained by basing on its Hirota’s bilinear form and taking the complex conjugate condition on parameters of N-solitons. What is more, breathers can be transformed into a series of nonlinear localized waves by the mechanism of breather transformation. Then through the multi-dimensional Riemann-theta function and the bilinear method, the high-dimensional complex three-periodic wave solutions are constructed systematically, which are the generalization of one-periodic wave and two-periodic wave solutions. By a limiting procedure, the asymptotic relations between the quasi-periodic waves and solitons are strictly established. Additionally, a novel analytical method of characteristic line is introduced to analyze statistically the dynamical characteristics of the quasi-periodic waves. The analytical method employed in this paper can be further extended to investigate the other complex high-dimensional nonlinear integrable equations.

A Solution of Fractal Dynamic Pharmacokinetics Problem

There are numerous uses for fractional differential equations in engineering, physics, and technology. A path for solving the fractal differential equations was examined, and its homogeneous form of was introduced. The notions of Riemann-Liouville fractional derivatives served as the foundation for the journey.. The solutions of the linear non-homogeneous fractal differential equations are given in detail.

A boundary value problem for the fourth-order degenerate equation of the mixed type

Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary value problems for mixed type’s equations of the fourth-order with power-law degeneration remains relevant. In this work, a boundary value problem in a rectangular domain for a degenerate equation of the fourth-order mixed-type is posed and investigated. Well-posedness of the boundary value problem for a fourth-order partial differential equation is established by proving the existence and uniqueness of the solution. Under sufficient conditions, a solution to the problem under consideration was explicitly found by the variable separation method.

Analysis of the Application for Random Walk in Computer Science, Physics and Mathematical Equations

Analysis of the Application for Random Walk in Computer Science, Physics and Mathematical Equations

The essentials of the finite element method to solve differential equations: an illustrative case in physics

Here we review the basic steps of the modern and accurate finite element method in its application to solve ordinary differential equations in physics. To exemplify this well-in-vogue numerical technique, we have chosen the second order Poisson-Boltzmann equation, which is a classic equation of colloid science. Aiming to formulate a viable, but didactic, implementation of the finite element technique, we have combined a linear basis of functions, the Galerkin weighted residuals method, the Swartz-Wendroff approximation and the Picard iteration algorithm. In summary, the finite element method transforms a differential equation into a simpler system of algebraic equations for the coefficients of the approximate solution in terms of a set of basis functions. We describe the full computational realisation of the finite element procedure and, also, we examine the corresponding Poisson-Boltzmann numerical predictions for various representative conditions.

Physics-informed Neural Networks for the Resolution of Analysis Problems in Electromagnetics

Learning from examples is the golden rule in the construction of behavioral models using neural networks (NN). When NN are trained to simulate physical equations, the tight enforcement of such laws is not guaranteed by the training process. In addition, there can be situations in which providing enough examples for a reliable training can be difficult, if not impossible. To alleviate these drawbacks of NN, recently a class of NN incorporating physical behavior has been proposed. Such NN are called “physics-informed neural networks” (PINN). In this contribution, their application to direct electromagnetic (EM) problems will be presented, and a formulation able to minimize an integral error will be introduced.

Travelling Wave Solutions for Some Time-Fractional Nonlinear Differential Equations

This study employs the powerful generalized Kudryashov method to address the challenges posed by fractional differential equations in mathematical physics. The main objective is to obtain new exact solutions for three important equations: the (3+1)-dimensional time fractional Jimbo-Miwa equation, the (3+1)-dimensional time fractional modified KdV-Zakharov-Kuznetsov equation, and the (2+1)-dimensional time fractional Drinfeld-Sokolov-Satsuma-Hirota equation. The generalized Kudryashov method is highly versatile and effective in addressing nonlinear problems, making it a pivotal component in our research. Its adaptability makes it useful in diverse scientific disciplines. The method simplifies complex equations, improving our analytical capabilities and deepening our understanding of system dynamics. Additionally, we define fractional derivatives using the conformable fractional derivative framework, providing a strong foundation for our mathematical investigations. This paper examines the effectiveness of the generalized Kudryashov method in solving complex challenges presented by fractional differential equations and aims to provide guidance for future studies.

Applications of the Simplest Equation Procedure to Some Fractional Order Differential Equations in Mathematical Physics

Applications of the Simplest Equation Procedure to Some Fractional Order Differential Equations in Mathematical Physics

Mathematical modeling and component generalization of (2+1)-dimensional Schwarz–Kortweg–de Vries model in shallow water waves

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of [Formula: see text]-dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF’s parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising.

Numerical study of two operator splitting localized radial basis function method for Allen–Cahn problem

In this article, we will explore the numerical simulation of the Allen–Cahn equation and provide effective combination methods to efficiently solve it. The Allen–Cahn equation, an equation of mathematical physics, represents a singularly perturbed reaction–diffusion phenomenon that elucidates the phase separation mechanism occurring in multi-component alloy systems. Finding a numerical solution for the Allen–Cahn equation presents a difficult challenge for computational science and engineering researchers. To solve the Allen–Cahn equation, we will use Lie–Trotter’s and Strang’s splitting techniques combined with the radial basis function partition of unity method. We will discretize the problem’s spatial domain using the classical and direct RBF-PU methods. We will also provide suitable theorems for analytical support of the presented numerical methods. We will provide several numerical examples which include examples with exact solutions to check the accuracy and efficiency of the method and examples in the field of phase transition to demonstrate the adaptability, performance, and efficiency of the proposed method.