Equations Of Mathematical Physics Research Articles

New Approaches on the Theory of Planetary Sciences: Applications of Non-Classical Equations of Mathematical Physics for Plasma Models of Jupiter’s Magnetosphere

In this paper, considered non-classical equations of mathematical physics are applied in the fields of astronomy and astrophysics in the case of plasma models of Jupiter’s magnetosphere. It is known that non-classical equations of mathematical physics have applications in gas dynamics, aerodynamics, hydrodynamics, and magneto-hydrodynamics. According to comparisons and observation results of Pioner-10, 11, and Voyager 1-2, considered mathematical models of Jupiter’s magnetosphere, which is cold plasma, as searches of Jupiter’s Io. At first, the mathematical justification of the physical process of Io concerning plasma was described by a non-classical equation of the Keldysh type. For this reason, using MHD equations for the derivation of the model equations of cold plasma and hot plasma on Jupiter’s magnetosphere. In the region tail of Jupiter given analyses of basic model equations of the Jupiter magnetosphere for the equilibrium between magnetic force, pressure gradient, and centrifugal force in the presence of plasma rotations. Additionally, based on the basic theoretical and observational results, the role of the Alfven Mach number with a constant Euler potential parameter in the region tail of Jupiter’s magnetosphere proves the justification of the steady magneto-hydrodynamic equilibrium. As agreed previously in the results of observation Voyager 1,2. Therefore, in the magnetosphere, Jupiter’s hot and cold plasma describe the same class equation of Keldysh-Tricomi types. In this case, the exact solution is obtained by integrals, which are first expressed as analytical formulas. Theoretical aspects of the model hot and cold plasma on the tail magnetosphere contain concepts of reconnection, which connects lost mass from Jupiter’s Io. Such an effect reconnection coronal problem as Parker’s also occurs by lost temperature and energy dissipation. Lorentz force, supported by means of solar wind, changes cold plasma to hot plasma in cases where a magnetic disk acts as a balancing mechanical equilibrium to retain cold-hot plasma. For motivation, both mathematical and physical, we used some figures, a table, and an appendix. Note that considered approaches to the theory of planetary sciences at first time applicable for Jupiter.

Analytical insights into the behavior of finite amplitude waves in plasma fluid dynamics

This study introduces innovative analytical solutions for the [Formula: see text]-dimensional nonlinear Jaulent–Miodek ([Formula: see text]) equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of [Formula: see text] waves. Within the realm of nonlinear fluid dynamics, the [Formula: see text] equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the [Formula: see text] equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the [Formula: see text] equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the [Formula: see text]-dimensional [Formula: see text] equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field.

Finite-dimensional perturbation of the Dirichlet boundary value problem for the biharmonic equation

Abstract The biharmonic equation is one of the important equations of mathematical physics, describing the behaviour of harmonic functions in higher-dimensional spaces. The main purpose of this study was to construct a finite-dimensional perturbation for the Dirichlet boundary value problem associated with the biharmonic equation. The methodological basis for this study was an integrated approach that includes mathematical analysis, algebraic methods, operator theory, and the theorem on the existence and uniqueness of a solution for a boundary value. The main tool is a finite-dimensional perturbation, which allows for examining the properties and behaviour of boundary value problems in as much detail as possible. In the study, descriptions of correctly solvable internal boundary value problems for a biharmonic equation in non-simply connected domains were considered in detail. The study is also devoted to the search for solutions and the analytical representation of resolvents of boundary value problems for a biharmonic equation in multi-connected domains. Within the framework of the study, theorems and their consequences were proved, and a finite-dimensional perturbation was constructed for the Dirichlet boundary value problem. Analytical representations of resolvents of boundary value problems for a biharmonic equation in multi-connected domains were also obtained. The examination of a finite-dimensional perturbation of the Dirichlet boundary value problem for a biharmonic equation has expanded the understanding of the properties of this equation in various contexts.

The kink-antikink single waves in dispersion systems by generalized PHI-four equation in mathematical physics

An essential aspect of mathematical physics is the PHI-four equation, which is a specific version of the Klein–Gordon equation that predicts particle physics phenomena. The present paper addresses numerical approaches to generalized PHI-four equation based on Laplace Adomian Decomposition Technique (LADT) which is governed by coupling of Laplace transform and Adomian decomposition method to determine the kink-antikink single waves in dispersion systems arises in mathematical physics. The nonlinear terms in the PHI-four equation are handled using the accelerated polynomial i.e., Adomian polynomial. The approach is extremely interesting computationally and is straightforward to execute. The accuracy and robustness of the current scheme are demonstrated by four test problems. To demonstrate the efficacy of our suggested approach, the current result is contrasted with both the analytical solution and existing solutions in literature. Stability and convergence analysis are well developed to justify the applicability of the current approach.

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.

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.

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 [...]

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

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

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

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.

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.