Equations Of Mathematical Physics Research Articles

Interesting class of quasilinear systems

With this work, I want to draw attention to a class of differential equations that encompasses a number of important equations of mathematical physics and is convenient for constructing a mathematical theory.

The extended Adomian decomposition method and its application to the rotating shallow water system for the numerical pulsrodon solutions

The rotating shallow water system is an important physical model, which has been widely used in many scientific areas, such as fluids, hydrodynamics, geophysics, oceanic and atmospheric dynamics. In this paper, we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin. By introducing some special initial values, we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings, which take the form of realistic series solutions. Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An, which fully shows the efficiency and accuracy of the proposed method. Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations. The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography, oceanic and atmospheric science.

Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного рівняння теплопровідності з експоненціально нелінійним коефіцієнтом теплопровідності

The problem of heat conduction in nonlinear media reduces to solving boundary value problems for nonlinear heat conduction equations, where the coefficients of the equation or the function of the power of heat sources depend on temperature according to some law. Among the numerical methods for solving problems for nonlinear equations of mathematical physics, one can distinguish finite difference methods, finite element methods, variational and projection methods, as well as iterative methods. Among the latter group, the two-sided approximation method is particularly attractive due to its ability to provide a convenient estimate for the error of the approximate solution and to prove the existence of a solution to the original problem. The theory of linear partially ordered spaces was developed by L. V. Kantorovich in the second half of the 1930s. Further development of this theory is associated with the works of M. A. Krasnosel’skii, H. Amann, V. I. Opojtsev, N. S. Kurpel, B. A. Shuvar, A. I. Kolosov etc. The aim of this article is to develop a two-sided approximation method based on the use of Green's functions for solving the first boundary value problem for a one-dimensional nonlinear heat conduction equation and to investigate its performance when solving test problems. To achieve this goal, the unknown function was replaced, and the boundary value problem was reduced to a Hammerstein integral equation, which was considered as a nonlinear operator equation in a partially ordered Banach space. Conditions for the existence of a unique positive solution to the problem and conditions for two-sided convergence of successive approximations to it were obtained. The developed method was implemented in software and tested on solving test problems. The results of the computational experiment are illustrated with graphical and tabular information

A novel multilevel finite element method for a generalized nonlinear Schrödinger equation

In this article, we focus on an efficient multilevel finite element method to solve the time-dependent nonlinear Schrödinger equation which is one of the most important equations of mathematical physics. For the time derivative, we adopt implicit schemes including the backward Euler method and the Crank–Nicolson method. Based on these stable implicit schemes, the proposed method requires solving a nonlinear elliptic problem at each time step. For these nonlinear elliptic equations, a multilevel mesh sequence is constructed. At each mesh level, we first derive a rough approximation by correcting the approximation of the previous mesh level in a special correction subspace. The correction subspace is composed of a coarse finite element space and an additional approximate solution derived from the previous mesh level. Next, we only need to solve a linearized elliptic equation by inserting the rough approximation into the nonlinear term. Then, we derive an accurate approximate solution by performing the aforementioned solving process on the multilevel mesh sequence until we reach the final mesh level. Owing to the special construct of the correction subspace, we derive a multilevel finite element method to solve the nonlinear Schrödinger equation for the first time, and meanwhile we also derive an optimal error estimate with linear computational complexity. Additionally, unlike the existing multilevel methods for nonlinear problems, that typically require bounded second-order derivatives of the nonlinear terms, the nonlinear term in our study requires only one-order derivatives. Numerical results are provided to support our theoretical analysis and demonstrate the efficiency of the presented method.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Exploration of different multi-peak solitons and vibrant breather type waves’ solutions of nonlinear Schrödinger equations with advanced dispersion and cubic–quintic nonlinearity, unveiling their applications

Abstract The nonlinear Schrödinger equations (NLSEs) of higher order illustrate the transmission of extremely short light pulses in fiber optics. In this manuscript, we employ the two-variable (1/G, G′/G)-expansion technique to construct bright and multi-peak solitons, periodic multi-solitons, breather type solitary waves, periodic peakon solitons, and other wave solutions of higher-order NLSE in mono-mode optical fiber and generalized NLSE with cubic–quintic nonlinearity. The two-variable (1/G, G′/G)-expansion method is a generalization of the (G′/G)-expansion method, offering a more robust mathematical tool for solving various nonlinear partial differential equations (PDEs) in mathematical physics. We also analyze the characteristics of waves conducive to the formation of bright–dark and other soliton forms within this medium. Additionally, we provide graphical representations of the obtained results to visually depict the dynamical models under consideration. Our findings highlight the potency, reliability, and versatility of the proposed technique, which holds promise for solving a wide array of similar models encountered in applied sciences and engineering.

Solution of the modified Helmholtz equation using mixed boundary conditions in an equilateral triangle

The modified Helmholtz equation qxx+qyy−4β2q=0, is one of the basic equations of classical mathematical physics. In this paper we have obtained the solution of the boundary-value problems for the modified Helmholtz equation in an equilateral triangle. An additional mixed boundary condition related to the symmetry of the solution is taken into consideration. We have analysed the Global relation and only used the algebraic techniques to obtain the explicit solution of modified Helmholtz equation bypassing the Riemann Hilbert approach. This solution is applied to the problem of diffusion-limited coalescence, A+A⇌A.

Revealing the Essence of Differential and Integral Calculus based on Scientific Philosophy

As it is known in the way where the basic equations of mathematical analysis and mathematical physics were taken as a basis it was possible to come to the main results of the Cantor set theory. However, contradictions and paradoxes were inherent to the results obtained in this field. In the way where the basic equations of theoretical physics, as well as empirical and probabilistic physics were taken as a basis, it was possible to arrive at the basic results of quantum theory. The results obtained in this area proved to be very useful in describing the experimental data. This was a sign of their truth. Therefore, there is reason to come to the following conclusion. That taking as a basis the ideas and results obtained in the field of quantum theory, there is a possibility to come to the disclosure of the essence of differential and integral calculus. As well as the results obtained in the field of new set theory. The main purpose of this work is an attempt to solve this problem.

On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

Examine the soliton solutions and characteristics analysis of the nonlinear evolution equations

The consistent and feasible enhanced (G′/G) -expansion technique is applied to construct an assortment of fresh and universal soliton solutions to the two important nonlinear evolution equations. One of them is the Benjamin-Ono equation, which is a partial differential equation that describes the propagation of weakly nonlinear and weakly dispersive long waves in deep stratified fluids in one dimension. Another equation is the modified Benjamin-Bona-Mahony equation, which is a variant of the Benjamin-Bona-Mahony equation and is used to examine Rossby waves in rotating fluids as well as drift waves in plasma. In this study, we investigated the aforementioned equations and obtained sufficient soliton solutions, such as bell soliton, periodic and kink-shape soliton, singular and singular-kink soliton, and so on. All the solutions achieved have physical explanations, and plotting some 3D figures reveals the diverse physical configurations and characteristics. We also include a comparison of other’s solutions in the literature to our own, which validates our solutions. The solutions indicate that the mentioned method is adaptable, improved, and effective for other nonlinear evolution equations in mathematical physics.

Electric network and Hirota type 4-simplex maps

Bazhanov-Stroganov (4-simplex) maps are set-theoretical solutions to the 4-simplex equation, namely the fourth member of the family of n-simplex equations, which are fundamental equations of mathematical physics. In this paper, we develop a method for constructing Bazhanov-Stroganov maps as extensions of tetrahedron maps which are set-theoretical solutions to the Zamolodchikov tetrahedron (3-simplex) equation. We employ this method to construct birarional Bazhanov-Stroganov maps which boil down to the famous electric network and Hirota tetrahedron maps at a certain limit.

Numerical Approach Based on the Haar Wavelet Collocation Method for Solving a Coupled System with the Caputo–Fabrizio Fractional Derivative

In the present paper, we consider an effective computational method to analyze a coupled dynamical system with Caputo–Fabrizio fractional derivative. The method is based on expanding the approximate solution into a symmetry Haar wavelet basis. The Haar wavelet coefficients are obtained by using the collocation points to solve an algebraic system of equations in mathematical physics. The error analysis of this method is characterized by a good convergence rate. Finally, some numerical examples are presented to prove the accuracy and effectiveness of this method.

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.