Differential Equations In Mathematical Physics Research Articles

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.

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.

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

“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

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.

“Computational Mathematics and Mathematical Physics”—Editorial I (2021–2023)

Based on the papers published in the Special Issue of the scientific journal Axioms, here we present the Editorial Article “Computational Mathematics and Mathematical Physics”, the main topics of which include both fundamental and applied research in computational mathematics and differential equations of mathematical physics [...]

Blow-up criteria of the simplified Ericksen–Leslie system

In this paper, we establish scaling invariant blow-up criteria for a classical solution to the simplified Ericksen–Leslie system in terms of the positive part of the second eigenvalue of the strain matrix and orientation field in mixed-norm Lebesgue spaces. Our result may be also regarded as an extension or improvement of the corresponding results obatined by Neustupa and Penel (Trends in Partial Differential Equations of Mathematical Physics, pp. 197–212, 2005), Miller (Arch. Ration. Mech. Anal. 235(1):99–139, 2020) and Huang and Wang (Commun. Partial Differ. Equ. 37(5):875–884, 2012).

Diverse optical solitons to nonlinear perturbed Schrödinger equation with quadratic-cubic nonlinearity via two efficient approaches

In this study, the nonlinear perturbed Schrödinger equation(NPSE) with nonlinear terms as quadratic-cubic law nonlinearity media with the beta derivative is investigated and this investigated model is considered an icon in the field of optical fibers where it describes the wave function or state function of the optical system. Numerous solutions are extracted by engaging two novel schemes known as generalized -expansion method (GEEM) and rational extended sinh-Gordon equation expansion method (REShGEEM) in distinct forms such as bright, dark, singular and combinations of these solutions. In addition, plane wave, periodic and exponential solutions are also recovered. Using the computer application Wolfram Mathematica 11, the dynamic structure and physical characterization of some observed solutions are vividly depicted by sketching different plots. Comparing our new results with well-known literature is also given which justifies the novelty of this work. Obtained results will hold a significant place in the field of nonlinear optical fibers and suggest that the proposed methods are very influential and effective tools for solving more complex nonlinear partial differential equations in mathematical physics, engineering and nonlinear optics.

Closed form solutions of Gerdjikov–Ivanov equation in nonlinear fiber optics involving the beta derivatives

In this paper, the closed form solutions of Gerdjikov–Ivanov equation with the beta derivatives are studied. This equation has quintic nonlinearity coefficients and group velocity dispersion, which shows the pulse behaviors in nonlinear fiber optics. It also has lots of significant applications in photonic crystal fibers. To this end, abundant families of closed form solutions in single or combined forms such as bright, singular, dark-singular are obtained by two systematic methods. Moreover, chaotic solutions are emerge as well. In order to understand dynamic behaviors of the begotten results, some computer simulations of the solutions are presented via some graphs with different values of the parameter beta. The results show that the approaches in this paper are efficient and direct methods to solve various fractional differential equations in mathematical physics.

EXACT SOLUTIONS OF THE LAKSHMANAN – PORSEZIAN – DANIEL EQUATION

In this paper, the Lakshmanan-Porsezian-Daniel (LPD) equation is considered. This equation is integrable and admits Lax pair. The LPD equation is the generalization of the nonlinear Schrodinger (NLS) equation and described by Ablowitz-Kaup-Newell-Segur (AKNS) system. Using the sine-cosine method and the hyperbolic tangent method a variety of new exact solutions are obtained. These methods are effective tools for searching exact solutions of nonlinear partial differential equations in mathematical physics. The obtained solutions are found to be important for the explanation of some practical physical problems.

Computational wave solutions of some nonlinear evolution equations

This paper aims to realize some of the vital nonlinear evolution of partial differential equations in mathematical physics. We implement an extended simple equation (ESE) method to get exact and solitary traveling wave solutions of the Calogero-Degasperis (CD) equation, potential Kadomtsev–Petviashvili (pKP) equation, generalized Korteweg–de Vries (gKdV) equation, and the (2 + 1)–dimensional Zoomeron (Z) equation. We obtain a new form of solutions for each model as mentioned above. This method is considered a new, robust, practical, and direct method. As we see, the remarkable ability of this method to apply on the fractional and integer order of nonlinear partial differential equations.

Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

Closed form solutions of nonlinear space‐time fractional Drinfel'd‐Sokolov‐Wilson equation via reliable methods

In current study, the closed form solutions for nonlinear space‐time fractional Drinfel'd‐Sokolov‐Wilson (FDSW) equation are obtained in the presence of the beta derivatives and using reliable methods. Applying the (G′/G)‐expansion method, the hyperbolic, trigonometric, and rational function solutions of space‐time FDSW equation are derived. It is interesting to propose the generalized complex method to construct the closed form solutions of nonlinear fractional differential equations (NFDEs), and it is a good example to apply for searching exact solutions of space‐time FDSW equation. Computer simulations of solutions obtained via these two reliable methods are also presented by some figures with different values of the parameter beta. The results reveal that these two methods are efficient and direct approaches for solving various fractional differential equations in mathematical physics.

Dynamics of media with stationary parameters and telegraph equations

The paper proposes an approach for constructing exact solutions of differential equations of mathematical physics, in particular, the telegraph equation. The method is based on the theory of finite-dimensional dynamics of systems of evolutionary differential equations. This theory is a natural extension of the theory of dynamical systems to partial differential equations. It allows one to construct exact solutions of partial differential equations even in the case when equations do not have symmetry algebras sufficient for integration.

Analysis of Transient Thermal Distribution in a Convective–Radiative Moving Rod Using Two-Dimensional Differential Transform Method with Multivariate Pade Approximant

The transient temperature distribution through a convective-radiative moving rod with temperature-dependent internal heat generation and non-linearly varying temperature-dependent thermal conductivity is elaborated in this investigation. Symmetries are intrinsic and fundamental features of the differential equations of mathematical physics. The governing energy equation subjected to corresponding initial and boundary conditions is non-dimensionalized into a non-linear partial differential equation (PDE) with the assistance of relevant non-dimensional terms. Then the resultant non-dimensionalized PDE is solved analytically using the two-dimensional differential transform method (2D DTM) and multivariate Pade approximant. The consequential impact of non-dimensional parameters such as heat generation, radiative, temperature ratio, and conductive parameters on dimensionless transient temperature profiles has been scrutinized through graphical elucidation. Furthermore, these graphs indicate the deviations in transient thermal profile for both finite difference method (FDM) and 2D DTM-multivariate Pade approximant by considering the forced convective and nucleate boiling heat transfer mode. The results reveal that the transient temperature profile of the moving rod upsurges with the change in time, and it improves for heat generation parameter. It enriches for the rise in the magnitude of Peclet number but drops significantly for greater values of the convective-radiative and convective-conductive parameters.

On the modulation instability analysis and deeper properties of the cubic nonlinear Schrödinger’s equation with repulsive [formula omitted]-potential

This projected work applies the generalized exponential rational function method to extract the complex, trigonometric, hyperbolic, dark bright soliton solutions of the cubic nonlinear Schrödinger’s equation. Moreover, trigonometric, complex, strain conditions and dark-bright soliton wave distributions are also reported. Furthermore, the modulation instability analysis is also studied in detail. To better understand the dynamic behavior of some of the obtained solutions, several numerical simulations are presented in the paper. According to the obtained results, it is clear that the method has less limitations than other methods in determining the exact solutions of the equations. Despite the simplicity and ease of use of this method, it has a very powerful performance and is able to introduce a wide range of different types of solutions to such equations. The idea used in this paper is readily applicable to solving other partial differential equations in mathematical physics.

Traveling wave solutions of Benny Luke equation via the enhanced (G'/G)-expansion method

In this article, we execute the enhanced (G'/G)-expansion method to search for new and further general closed-form wave solutions to the nonlinear partial differential equation, namely the Benny Luke equation. This method is one of the powerful techniques that come into view in recent time for establishing more exact wave solutions to nonlinear partial differential equations. We have achieved some new exact solutions including soliton and periodic wave solutions with arbitrary parameters and the solutions are expressed in terms of hyperbolic and trigonometric functions. The efficiency of this method for finding exact solutions has been established. It is shown that the enhanced (G'/G)-expansion method is direct, effective and can be used for many other nonlinear partial differential equations in mathematical physics and engineering.

Closed-form solutions to the perturbed NLSE with Kerr law nonlinearity in optical fibers

We implement an efficient solver to achieve solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity. This solver gives closed-form wave structures of the solutions. Namely, various kinds of solutions are achieved. These solutions may be applicable for some physical fields, like optical fibers, plasma fluids and biomolecular dynamical modes. The proposed approach is straightforward, sturdy and fruitful to retrieve some new solutions for nonlinear partial differential equations in mathematical physics. Some figures are plotted for suitable values of the parameters to describe the propagation of traveling wave solutions.