Solitary wave solutions to some nonlinear fractional evolution equations in mathematical physics (original) (raw)
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Arab Journal of Mathematical Sciences, 2020
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 mig...
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In this paper, the ( G ′ / G , 1 / G ) -expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics. The considered equations comprise the time-fractional, (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation, and the space-time-fractional generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) system in the sense of the conformable fractional derivative. Applying traveling wave transformations to the equations, we obtain the corresponding ordinary differential equations in which each of them provides a system of nonlinear algebraic equations when the method is used. As a result, many analytical exact solutions obtained of these equations are expressed in terms of hyperbolic function solutions, trigonometric function solutions, and rational function solutions. The graphical representations of some obtained solutions are demonstrated to better understan...
International Journal of Sciences: Basic and Applied Research, 2020
We have enucleated new and further exact general wave solutions, along with multiple exact traveling wave solutions of space-time nonlinear fractional Chan-Hillard equation, by applying a relatively renewed technique two variables -expansion method. Also, based on fractional complex transformation and the properties of the modified Riemann-Liouville fractional order operator, the fractional partial differential equations are transforming into the form of ordinary differential equation. This method can be rumination of as the commutation of well-appointed -expansion method introduced by M. Wang et al.. In this paper, it is mentioned that the two variables - expansion method is more legitimate, modest, sturdy and effective in the sense of theoretical and pragmatical point of view. Lastly, by treating computer symbolic program Mathematica, the uniqueness of our attained wave solutions are represented graphically and reveal a comparison in a submissive manner.
Advances in Difference Equations, 2020
We investigate some solitary wave results of time fractional evolution equations. By employing the extended rational exp((−fracpsiprimepsi)(eta))\exp ( (-\frac{{\psi }^{\prime }}{\psi }) ( \eta ) )exp((−fracpsiprimepsi)(eta))exp((−ψ′ψ)(η))-expansion method, a few different results including kink, singular-kink, multiple soliton, and periodic wave solutions are formally generated. It is worth mentioning that the solutions obtained are more general with more parameters. The exact solutions are constructed in the form of exponential, trigonometric, rational, and hyperbolic functions. With the choice of proper values of parameters, graphs to some of the obtained solutions are drawn. On comparing some special cases, our solutions are in good agreement with the results published previously and the remaining are new.
Searching closed form analytic solutions to some nonlinear fractional wave equations
Arab journal of basic and applied sciences, 2021
Numerous tangible incidents in physics, chemistry, applied mathematics and engineering are described successfully by means of models making use of the theory of derivatives of fractional order and research in this area has grown significantly. In this article, we establish exact solutions to some nonlinear fractional differential equations. The recently established rational (G 0 =G)-expansion method with the help of fractional complex transform is used to examine abundant further general and new closed form wave solutions to the nonlinear space-time fractional mBBM equation, the space-time fractional Burger's equation and the space-time fractional ZKBBM in the sense of the Jumarie modified Riemann-Liouville derivative. The fractional complex transform reduces the nonlinear fractional differential equations into nonlinear ordinary differential equations and then the theories of ordinary differential equations are implemented effectively. It is observed that the performance of this method is reliable, useful and gives new and broad-ranging closed form solutions than the existing methods.
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The aim of this study is to consider solving an important mathematical model of fractional order ([Formula: see text])-dimensional breaking soliton (Calogero) equation by Khater method. The derivatives are in the local fractional derivative sense. The fractional transformation equation is utilized to convert the proposed nonlinear fractional order differential equation into nonlinear ordinary differential equation. The Khater method is used to construct the closed-form traveling wave solutions of the said fractional differential equation. In addition, many new exact solutions are constructed. This shows that the Khater method is more convenient, powerful, and easy to solve the nonlinear fractional differential equation arising in mathematical physics.
Journal of Research in Engineering and Applied Sciences, 2021
In this article, we construct a family of closed form traveling wave solutions to the space-time fractional Equal Width equation (EW) and the space-time fractional modified Equal Width equation (mEW) by using newly proposed modified rational fractional (/ 2)-expansion method and the exp-function method. The considered equations are turned into fractional order ordinary differential equations with the help of a complex fractional transformation along with conformable fractional derivative and then the methods are used to investigate their solutions. The achieved solutions are in terms of trigonometric function, hyperbolic function and rational function which might be used to analyze deeply the physical complex phenomena of real world as they are new and bear much more generality. Two more well-established methods, the (′ /)-expansion method and the rational (′ /)-expansion method, are also taken into account to unravel the suggested equations which do not provide any solution. The results reveal that the proposed method is efficient, straightforward and concise which might further be useful tool to examine any other nonlinear evolution equations of fractional order arising in various physical problems.
Pure Traveling Wave Solutions for Three Nonlinear Fractional Models
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Three nonlinear fractional models, videlicet, the space-time fractional Boussinesq equation, - dimensional breaking soliton equations, and SRLW equation, are the important mathematical approaches to elucidate the gravitational water wave mechanics, the fractional quantum mechanics, the theoretical Huygens’ principle, the movement of turbulent flows, the ion osculate waves in plasma physics, the wave of leading fluid flow, etc. This paper is devoted to studying the dynamics of the traveling wave with fractional conformable nonlinear evaluation equations (NLEEs) arising in nonlinear wave mechanics. By utilizing the oncoming - expansion technique, a series of novel exact solutions in terms of rational, periodic, and hyperbolic functions for the fractional cases are derived. These types of long-wave propagation phenomena played a dynamic role to interpret the water waves as well as mathematical physics. Here, the form of the accomplished solutions containing the hyperbolic, rational, an...
Novel Soliton Solutions of the Fractional Riemann Wave Equation via a Mathematical Method
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The Riemann wave equation is an intriguing nonlinear equation in the areas of tsunamis and tidal waves in oceans, electromagnetic waves in transmission lines, magnetic and ionic sound radiations in plasmas, static and uniform media, etc. In this innovative research, the analytical solutions of the fractional Riemann wave equation with a conformable derivative were retrieved as a special case, and broad-spectrum solutions with unknown parameters were established with the improved (G’/G)-expansion method. For the various values of these unknown parameters, the renowned periodic, singular, and anti-singular kink-shaped solitons were retrieved. Using the Maple software, we investigated the solutions by drawing the 3D, 2D, and contour plots created to analyze the dynamic behavior of the waves. The discovered solutions might be crucial in the disciplines of science and ocean engineering.