Equations Of Mathematical Physics Research Articles

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.

Snapping of a viscoelastic cylindrical panel under loading with small volume compressed gas

The paper considers the dynamics of the process of a flat rectangular elastic panel having an initial deflection equal to the critical deflection of the static problem of panel stability. The panel is loaded with compressed gas enclosed in a small volume container. Using this value, the load acting on the panel during the snapping process will depend on the deflection of the panel as a result of the adiabatic expansion of the compressed gas. The main goal of the work is to investigate the stability region of a cylindrical panel when loaded with compressed gas. The problem is solved using a developed methodology based on the method of separating variable equations of mathematical physics. Based on calculations, it was found that when the panel is loaded with compressed gas enclosed in a small-volume container, a stabilizing effect appears, i.e. a certain deflection interval appears where the panel does not lose stability. The presence of external pressure makes it possible to expand this area, i.e. the external environment increases the “stabilizing” effect. Also, taking into account the viscoelastic properties of the material expands this area to 10%.

“Differential Equations of Mathematical Physics and Related Problems of Mechanics”—Editorial 2021–2023

Based on the published papers in this Special Issue of the elite scientific journal Mathematics, we herein present the Editorial for “Differential Equations of Mathematical Physics and Related Problems of Mechanics”, the main topics of which are fundamental and applied research on differential equations in mathematical physics and mechanics [...]

The Maximum Principles of Differential Equations in Mathematical Physics—Elliptic Equations with Oblique Derivative Boundary Condition

The Maximum Principles of Differential Equations in Mathematical Physics—Elliptic Equations with Oblique Derivative Boundary Condition

Adequate Closed Form Wave Solutions to the Generalized KdV Equation in Mathematical Physics

Adequate Closed Form Wave Solutions to the Generalized KdV Equation in Mathematical Physics

Travelling Wave Solutions to the Space-Time Fractional ZKBBM Equation in Mathematical Physics

Travelling Wave Solutions to the Space-Time Fractional ZKBBM Equation in Mathematical Physics

Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Об одном способе ультразвукового бурения скважин для забора грунта на небесных телах

The article discusses technical and theoretical issues in the development of an ultrasonic drill for drilling soil on celestial bodies. The specific use of the drill requires low energy consumption, an efficient operating cycle and a wide temperature range during operation. One of the promising areas of development is devices that have vibration decoupling from the ground due to the transformation of ultrasonic vibrations into repeated shocks of sound frequency through a specially introduced “free” body between the ultrasonic and impact parts of the device. This allows you to maintain the functionality of the device and carry out drilling regardless of the state of the impact bit, including even its temporary jamming. The main technical solutions of both domestic and foreign authors are considered, an analysis is made and design schemes, operating conditions, capabilities and limitations are evaluated. A number of theoretical works were also analyzed and it was found that the materials presented in them do not contain detailed mathematical models necessary for simulation modeling at the stage of design and optimization of the structure. The theoretical part of the article considers a lumped mathematical model obtained on the basis of equivalent parameters obtained from a distributed description of the drill design elements in numerical calculation systems. The equations of mathematical physics, classical mechanics, elements of impact theory and similarity theory were used. A parametric study was carried out and, in particular, the dependences of the oscillation frequency of a free body on the size of the impact gap, body mass and the amplitude of oscillations of the concentrator were obtained. The considered model can be used for simulation modeling when developing a drill design.

Innovative Teaching Model Exploration of Combining Ideological and Political Education in Mathematical Physics Equations Course

Innovative Teaching Model Exploration of Combining Ideological and Political Education in Mathematical Physics Equations Course

Dynamic Behaviors and Analytical Solutions of the Damped (2+1)‐Dimensional Nonlinear Schrödinger Equation

In this paper, abundant analytical solutions of the damped (2 + 1)‐dimensional nonlinear Schrödinger equation are achieved by taking advantage of the extended systematic method. By considering various values of parameters, dynamic behaviors of bright, dark soliton, period wave, and kink solitary solutions are displayed with different amplitudes and directions in 3D graphs, lines, contour maps, and time evolution plots. Further analysis of the obtained results is studied to show the efficient effect of the parameters on the propagation and configuration of waves. In addition, computer simulations are considered to show the correlation between parameters and velocity. Our results indicate that the extended systematic method for solving differential equations is more convenient and yields richer solutions. The other mathematical physics equations that appear in nonlinear science can also be investigated by the concept of this study.

Numerical Solution of Cahn-Allen Equations by Wavelet Based Lifting Schemes

The Cahn-Allen equation is a reaction-diffusion equation of mathematical physics that describes the process of phase separation in multi-component alloy systems and order-disorder transitions. This paper presented a numerical solution of Cahn-Allen equations by Lifting scheme using different wavelet filter coefficients. The numerical results obtained using this scheme are compared with the exact solution to demonstrate the accuracy and fast convergence in less computational time than the existing scheme. Some problems are taken to show the validity and applicability of the scheme.

An Illuminating Physical View on the Principles of “3 Generations” Structure of Elementary Particles, and on Evolution of Universe–Energy Basic State Field of the Universe (IV)

In this paper, we review the history research on the principles of “3 generations” structure of elementary particles, including Steven Weinberg’s basic physical ideas on the Standard Model of elementary particles, and Barr and Zee’s research and so on. We propose a physical view on the “Dual-production” of quarks and leptons in the evolution of the universe, according to our exiting physical views on the systematic evolution of cosmic and the fundamental theory of EBSF. As for the three generation structure of elementary particles, we propose that the three-generation structure of particle correspond to the evolution of the universe from the plasma quantum state (for the exiting physical phase states of matter formed by the early cosmic evolution, see Y. Xiaodong, Y. Yuchen, L. Zhen and Y. Bi, Rep. Adv. Phys. Sci. 2 (2023) 2250007) to today’s celestial matter, which has undergone three (only three) important physical phase changes in the composition of material structures; that is, each generation of elementary particle structure is the physical elementary phase structure of every stage on condense and sedimentary physical celestial phase. These three stages of condense and sedimentary celestial phase can be simply likened to the three stages of gas, liquid and solid in condensed matter physics, which are essentially formed by the interweaving of cosmic matter formed in the early evolution and the exciters of current-cosmic-evolution EBSF, of course, there is also the complex role of the differentiation inherent time of universe evolution. We analyze that these views not only do not contradict our existing physical theories, but also better understand the nature of its physics and science. The three-generation structure of elementary particles corresponds to a historical epoch in the evolution of the universe, and called the one Mini-era of the domino mechanism before Weinberg’s famous picture <the First Three Minutes>. Although our analysis is somewhat phenomenological, it is still necessary to establish the evolution equation from the plasma state of the early universe to the current physical phase of cosmic celestial matter. More cosmic observation and more experiment data are needed in order to establish the precise quantification of the mathematical physics equations on the evolution process. But we believe that it has a major breakthrough in physical thought, and for the first time, human beings were able to rationally explain the three-generation structure of elementary particles, able to reveal the mystery of the generations of elementary particles in physics.

Reform of Mathematical Physics Equation Teaching Methods Based on Information-based Teaching Platform

Reform of Mathematical Physics Equation Teaching Methods Based on Information-based Teaching Platform

Approximate Solutions of the Fractional Clannish Random Walker’s Parabolic Equation with the Residual Power Series Method

One of the prominent nonlinear partial differential equations in mathematical physics is the Clannish Random Walker’s Parabolic (CRWP) equation. This study uses Residual Power Series Method (RPSM) to solve the time fractional CRWP equation. In this equation, the fractional derivatives are considered in Caputo’s sense. The effectiveness of RPSM is illustrated with graphical results. The series solutions are utilized to represent the approximate solutions. Besides, the approximate solutions found by the suggested method ensure good accuracy when compared with the exact solution. Moreover, RPSM efficiently analyzes complex problems that emerge in the related mathematical and scientific fields.

New Exact Traveling Wave Solutions of the Non-linear (2+1)-dimensional Klein-gordon Equation

In this article, we discuss the nonlinear (2+1)-dimensional Klein-Gordon equation with an additional term. The functional variable method is used to construct exact solutions of the quadratic and cubic nonlinear (2+1)-dimensional Klein-Gordon equation. The exact solutions of these equations including soliton and periodic wave solutions are obtained. The advantage of the used method beyond other existing methods is that it provides more new exact solutions. Some selected solutions of the equations are presented graphically by Matlab program. This method is efficient and it can be successfully used to obtain another nonlinear wave equations in mathematical physics and engineering.

Simulation of a new process of mixing liquid metal in CCM mold with rotating cooling jacket with vertical ribs

The article proposes a new technology of filling the CCM mold with liquid metal and mixing it. The original patented device consists of a closed bottom nozzle and a rotating jacket. Experimental studies of liquid metal flow in a mold are long, complex and time-consuming, therefore, in the work was used mathematical modeling by numerical method. The objects of research are the hydrodynamic and thermal flows of liquid metal during the new process of steel casting into a CCM mold of rectangular cross-section, and the result is a spatial mathematical model that describes the flows and temperatures of liquid metal in the mold. To simulate the processes occurring during the metal flow in the mold, the authors used a specially crea­ted software package. The theoretical calculations are based on the fundamental equations of hydrodynamics, the equations of mathematical physics (equation of thermal conductivity taking into account mass transfer) and a proven numerical method. The area under study is divided into elements of finite dimensions, for each element a formulated system of equations is written in a difference form. The result is the velocity and temperature fields of the metal flow in the mold volume. According to the developed numerical schemes and algorithms, a calculation program was compiled. The paper considers an example of calculating the steel casting into a mold of rectangular cross-section and flow diagrams of liquid metal over various mold sections. Vector flows of liquid metal in various mold sections are clearly presented for different rotary speed of the rotating jacket. The authors identified the areas of intense turbulence and presented the results of the problem numerical solution in graphical form by diagrams of the velocity fields of liquid metal flows and their temperature over various mold sections.

Exact solutions of the two-dimensional nonlinear Schrodinger equation

The nonlinear Schrödinger equation as a classical model in physics is gradually becoming an urgent problem that needs to be studied. Obtaining an exact solution of the corresponding model can not only provide theoretical support for the experiment but also provide a basis for solving practical problems. In this work, we study the propagation of waves of the two-dimensional nonlinear Schrödinger equa- tion in nonlinear media with dispersion processes. The tan-cot function method is applied. This method is effective in solving nonlinear equations of mathematical physics. Various solutions in the form of periodic waves are obtained. Graphs of the solutions arepresented.