Evolution Equations In Mathematical Physics Research Articles

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.

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

CONSTRUCTION OF FRACTAL SOLITON SOLUTIONS FOR THE FRACTIONAL EVOLUTION EQUATIONS WITH CONFORMABLE DERIVATIVE

In this paper, the fractional evolutions are described by using the conformable derivative for the first time. We implement fractional functional variable method (FFVM) to obtain some new kinds of fractal soliton wave solutions for these fractional evolution equations. The simplicity and effectiveness of this proposed method are tested on the fractional Drinfeld–Sokolov system and fractional cubic Klein–Gordon equation. The FFVM provides a new perspective to construct exact fractal soliton wave solutions of complex fractional nonlinear evolution equations in mathematical physics.

Non-differentiable fractional odd-soliton solutions of local fractional generalized Broer-Kaup system by extending Darboux transformation

In this paper, a local fractional generalized Broer-Kaup (gBK) system is first de?rived from the linear matrix problem equipped with local space and time fractional partial derivatives, i.e, fractional Lax pair. Based on the derived fractional Lax pair, the second kind of fractional Darboux transformation (DT) mapping the old potentials of the local fractional gBK system into new ones is then established. Finally, non-differentiable frcational odd-soliton solutions of the local fractional gBK system are obtained by using two basic solutions of the derived fractional Lax pair and the established fractional DT. This paper shows that the DT can be extended to construct non-differentiable fractional soliton solutions of some local fractional non-linear evolution equations in mathematical physics.

NEW SOLITARY WAVE SOLUTIONS FOR THE FRACTIONAL JAULENT–MIODEK HIERARCHY MODEL

The main goal of this paper is to study the new solitary wave behaviors of the fractional Jaulent–Miodek hierarchy model (FJMHE) with M-truncated fractional derivative. First, we use the fractional sech-function method (FSFM) to obtain some new solitary wave solutions of the fractional Jaulent–Miodek hierarchy equation. The new method is simple and effective, which provides a more powerful mathematical technique for exploring solitary wave solutions of the fractional evolution equations in mathematical physics. Finally, some 3D and 2D graphs are employed to illustrate the physical properties of the obtained new solitary wave solutions.

Exact solutions of the different dimensional CBS equations in mathematical physics

The focus of this article is to find the exact traveling wave solutions to Calogero–Bogoyavlenskii–Schiff (CBS) equation by employing the exp−Φζ expansion method. The traveling wave solutions are expressed by the hyperbolic function solutions, trigonometric function solutions and rational function solutions. 3D and 2D charts are drawn from the obtained solutions by Mathematica software and discussed the graphical behavior of solutions. Finally, an effective mathematical tools for solving nonlinear evolution equations in mathematical physics are provided by the proposed method. This method is more common and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations.

New and more dual‐mode solitary wave solutions for the Kraenkel–Manna–Merle system incorporating fractal effects

This paper introduces the fractal Kraenkel–Manna–Merle (KMM) system, which explains nonlinear short wave propagation with zero conductivity for saturated ferromagnetic materials in an external field. Fractal models deal with the discontinuous geometry of physical problems. The semi‐inverse technique and the new auxiliary equation method (NAEM) are used to generate a variety of solutions. A collection of exact soliton solutions specifically bright, dark, singular‐shaped, and singular‐periodic are generated using constraint conditions. The fractal parameter impact on these solutions displayed through 2D, 3D, and contour plots taking appropriate parametric values. The arbitrary functions in the solutions are chosen as such unique functions to generate some novel soliton structures. The proposed methods are more straightforward, succinct, and accurate to extract solitons of numerous evolution equations in mathematical physics.

Soliton Solutions for (3+1)-Dimensional Nonlinear Evolution Equations by Tanh Method

In this paper, the Tanh method is employed to compute the exact solutions for nonlinear PDEs. These equations are (3+1)-dimensional Jimbo-Miwa , potential-YTSF, and generalized KP equations. The solutions obtained by this method are compared with the solutions obtained through other methods. It is shown that the Tanh method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Algebraic computational methods for solving three nonlinear vital models fractional in mathematical physics

Abstract This research paper uses a direct algebraic computational scheme to construct the Jacobi elliptic solutions based on the conformal fractional derivatives for nonlinear partial fractional differential equations (NPFDEs). Three vital models in mathematical physics [the space-time fractional coupled Hirota Satsuma KdV equations, the space-time fractional symmetric regularized long wave (SRLW equation), and the space-time fractional coupled Sakharov–Kuznetsov (S–K) equations] are investigated through the direct algebraic method for more explanation of their novel characterizes. This approach is an easy and powerful way to find elliptical Jacobi solutions to NPFDEs. The hyperbolic function solutions and trigonometric functions where the modulus and, respectively, are degenerated by Jacobi elliptic solutions. In this style, we get many different kinds of traveling wave solutions such as rational wave traveling solutions, periodic, soliton solutions, and Jacobi elliptic solutions to nonlinear evolution equations in mathematical physics. With the suggested method, we were fit to find much explicit wave solutions of nonlinear integral differential equations next converting them into a differential equation. We do the 3D and 2D figures to define the kinds of outcome solutions. This style is moving, reliable, powerful, and easy for solving more difficult nonlinear physics mathematically.

Lie symmetry reductions, abound exact solutions and localized wave structures of solitons for a (2 + 1)-dimensional Bogoyavlenskii equation

By applying the two efficient mathematical methods particularly with regard to the classical Lie symmetry approach and generalized exponential rational function method, numerous exact solutions are constructed for a (2 + 1)-dimensional Bogoyavlenskii equation, which describes the interaction of Riemann wave propagation along the spatial axes. Moreover, we obtain the infinitesimals, all the possible vector fields, optimal system, and Lie symmetry reductions. The governing Bogoyavlenskii equation is converted into various nonlinear ordinary differential equations through two stages of Lie symmetry reductions. Accordingly, abundant exact closed-form solutions are obtained explicitly in terms of independent arbitrary functions, rational functions, trigonometric functions, and hyperbolic functions with arbitrary free parameters. The dynamical behavior of the resulting soliton solutions is presented through 3D-plots via numerical simulation. Eventually, single solitons, multi-solitons with oscillations, kink wave with breather-type solitons, and single lump-type solitons are obtained. The proposed mathematical techniques are effective, trustworthy, and reliable mathematical tools to work out new exact closed-form solutions of various types of nonlinear evolution equations in mathematical physics and engineering sciences.

Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics

PurposeFractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.Design/methodology/approachThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.FindingsAchieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.Originality/valueThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

On the exact solutions to Biswas–Arshed equation involving truncated M-fractional space-time derivative terms

In this work, two different schemes, the extended hyperbolic auxiliary and the simplest equation method, are employed to construct the exact solutions involving parameters of the Biswas–Arshed model (BAM) with truncated M-fractional derivative. Moreover, semi-inverse variational principle are applied to underlying equation to acquire analytical solution. These methods have a broad applicability to many other nonlinear evolution equations in mathematical physics. Different traveling wave solutions have been investigated by invoking these methods. 3D graphic representations are given to explain the principal effect of parameter μ on dynamical properties of the soliton solutions. The stability property of the obtained solutions is tested to show the ability of our obtained solutions through the physical experiments. Moreover, the general solution of nonlinear ordinary differential equation corresponding to underlying equation is found using traveling wave reduction.

New exact solutions of space and time fractional modified Kawahara equation

One of the important nonlinear evolution equations in mathematical physics is the modified Kawahara equation. In this work, by utilizing an analytic method based on the Jacobi elliptic functions, a group of exact solutions of the fractional modified Kawahara equation had been found. The method can be applied to all of the time, space and space–time fractional equations. Four problems had also been given to demonstrate the application of the method and some solutions had been illustrated by the graphics.

Solitary wave solutions to some nonlinear fractional evolution equations in mathematical physics

The objective of this article is to construct new and further general analytical wave solutions to some nonlinear evolution equations of fractional order in the sense of the modified Riemann-Liouville derivative relating to mathematical physics, namely, the space-time fractional Fokas equation, the time fractional nonlinear model equation and the space-time fractional (2 + 1)-dimensional breaking soliton equation by exerting a rather new mechanism (G'/G,1/G) -expansion method. We use the fractional complex transformation and associate the fractional differential equations to the solvable integer order differential equations. A comprehensive class of new and broad-ranging exact traveling and solitary wave solutions are revealed in terms of trigonometric, rational and hyperbolic functions. The attained wave solutions are sketched graphically by using Mathematica and make a comparison to the results attained by the presented technique with other techniques in a comprehensive manner. It is notable that the method can be considered as a reduction of the reputed (G'/G) -expansion method commenced by Wang et al. It is noticeable that, the two variable (G'/G,1/G) -expansion method appears to be more reliable, straightforward, computerized and user-friendly.

General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics

This paper is devoted to improving the general Kudryashov method by a new and general auxiliary equation. So, a new method is introduced, which we call ‘the general improved Kudryashov method’, to produce exact solutions for nonlinear evolution equations arising in mathematical physics. As application examples, exact traveling wave solutions for the combined Korteweg–de Vries and modified Korteweg–de Vries (KdV–mKdV) equation and the (2+1)-dimensional Zakharov–Kuznetsov equation are obtained. These solutions can be classified as solitary and periodic wave solutions. Some of the obtained solutions are graphically sketched.

Abundant closed form wave solutions to some nonlinear evolution equations in mathematical physics

The propagation of waves in dispersive media, liquid flow containing gas bubbles, fluid flow in elastic tubes, oceans and gravity waves in a smaller domain, spatio-temporal rescaling of the nonlinear wave motion are delineated by the compound Korteweg-de Vries (KdV)-Burgers equation, the (2+1)-dimensional Maccari system and the generalized shallow water wave equation. In this work, we effectively derive abundant closed form wave solutions of these equations by using the double (G′/G, 1/G)-expansion method. The obtained solutions include singular kink shaped soliton solutions, periodic solution, singular periodic solution, single soliton and other solutions as well. We show that the double (G′/G, 1/G)-expansion method is an efficient and powerful method to examine nonlinear evolution equations (NLEEs) in mathematical physics and scientific application.

Exact Solutions of Rosenzweig-Macarthur (RM) Model Equations by Using Exp Function Method

He’s exp function method promising for finding exact solutions of nonlinear evolution equations in mathematical physics. The exact solutions of the Rosenzweig-MacArthur (RM) model equations is obtain by using the exp-function method. The method is by transformation used to construct solitary and soliton solutions of nonlinear evolution equations. Thefree parameters in the obtained generalized solutions might imply some meaningful results in physical process.

New exact solutions for Chan-Hilliard equation by new sub-equation method

In this present study, new sub-equation method is used to construct exact travelling wave solutions of the Cahn-Hilliard equation (CH). As a consequence, many new general exact travelling wave solutions are obtained including generalized hyperbolic (GH) function, generalized trigonometric (GT) function, exponential function and rational function solutions. Compared with the most existing similar new sub-equation method, the proposed method gives new and more general exact travelling wave solutions. More importantly, the method with the help of symbolic computation prepares a powerful mathematical tool for solving a great many nonlinear evolution equations in mathematical physics.

Analytical Wave Solutions for Foam and KdV-Burgers Equations Using Extended Homogeneous Balance Method

In this paper, extended homogeneous balance method is presented with the aid of computer algebraic system Mathematica for deriving new exact traveling wave solutions for the foam drainage equation and the Kowerteg-de Vries–Burgers equation which have many applications in industrial applications and plasma physics. The method is effective to construct a series of analytical solutions including many types like periodical, rational, singular, shock, and soliton wave solutions for a wide class of nonlinear evolution equations in mathematical physics and engineering sciences.

Further Results about Traveling Wave Exact Solutions of the (2+1)-Dimensional Modified KdV Equation

We used the complex method and the exp(-ϕ(z))-expansion method to find exact solutions of the (2+1)-dimensional mKdV equation. Through the maple software, we acquire some exact solutions. We have faith in that this method exhibited in this paper can be used to solve some nonlinear evolution equations in mathematical physics. Finally, we show some simulated pictures plotted by the maple software to illustrate our results.