scholarly journals Construction of abundant novel analytical solutions of the space–time fractional nonlinear generalized equal width model via Riemann–Liouville derivative with application of mathematical methods

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 657-668
Author(s):  
Aly R. Seadawy ◽  
Asghar Ali ◽  
Saad Althobaiti ◽  
Khaled El-Rashidy
Keyword(s):  
Differential Equation ◽  
Soliton Solutions ◽  
Magnetic Interaction ◽  
Space Time ◽  
Auxiliary Equation ◽  
Mathematical Methods ◽  
Nonlinear Fractional Differential Equation ◽  
Dispersive Waves ◽  
Composite Function ◽  
Liouville Derivative

Abstract The space–time fractional generalized equal width (GEW) equation is an imperative model which is utilized to represent the nonlinear dispersive waves, namely, waves flowing in the shallow water strait, one-dimensional wave origination escalating in the nonlinear dispersive medium approximation, gelid plasma, hydro magnetic waves, electro magnetic interaction, etc. In this manuscript, we probe advanced and broad-spectrum wave solutions of the formerly betokened model with the Riemann–Liouville fractional derivative via the prosperously implementation of two mathematical methods: modified elongated auxiliary equation mapping and amended simple equation methods. The nonlinear fractional differential equation (NLFDE) is renovated into ordinary differential equation by the composite function derivative and the chain rule putting together along with the wave transformations. We acquire several types of exact soliton solutions by setting specific values of the personified parameters. The proposed schemes are expedient, influential, and computationally viable to scrutinize notches of NLFDEs.

scholarly journals Novel Analytical Approach for the Space-Time Fractional (2+1)-Dimensional Breaking Soliton Equation via Mathematical Methods

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3253
Author(s):  
Abdulmohsen D. Alruwaili ◽  
Aly R. Seadawy ◽  
Asghar Ali ◽  
Sid Ahmed O. Beinane
Keyword(s):  
Differential Equation ◽  
Travelling Wave ◽  
Space Time ◽  
Successful Implementation ◽  
Mathematical Methods ◽  
Wave Solutions ◽  
Complex Transformation ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Solitary Travelling Wave

The aim of this work is to build novel analytical wave solutions of the nonlinear space-time fractional (2+1)-dimensional breaking soliton equations, with regards to the modified Riemann–Liouville derivative, by employing mathematical schemes, namely, the improved simple equation and modified F-expansion methods. We used the fractional complex transformation of the concern fractional differential equation to convert it for the solvable integer order differential equation. After the successful implementation of the presented methods, a comprehensive class of novel and broad-ranging exact and solitary travelling wave solutions were discovered, in terms of trigonometric, rational and hyperbolic functions. Hence, the present methods are reliable and efficient for solving nonlinear fractional problems in mathematics physics.

scholarly journals Oscillation criteria for nonlinear fractional differential equation with damping term

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 119-128 ◽  
Author(s):  
Mustafa Bayram ◽  
Hakan Adiguzel ◽  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Riccati Transformation ◽  
Damping Term ◽  
Nonlinear Fractional Differential Equation ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Oscillation Of Solutions

AbstractIn this paper, we study the oscillation of solutions to a non-linear fractional differential equation with damping term. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, inequalities, and integration average techniquewe establish new oscillation criteria for the fractional differential equation. Several illustrative examples are also given.

scholarly journals Soliton Behaviours for the Conformable Space–Time Fractional Complex Ginzburg–Landau Equation in Optical Fibers

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 219
Author(s):  
Khalil S. Al-Ghafri
Keyword(s):  
Differential Equation ◽  
Optical Fibers ◽  
Power Law ◽  
Elliptic Function ◽  
Landau Equation ◽  
Soliton Solutions ◽  
Space Time ◽  
Optical Soliton ◽  
Ginzburg Landau ◽  
Weierstrass Elliptic Function

In this work, we investigate the conformable space–time fractional complex Ginzburg–Landau (GL) equation dominated by three types of nonlinear effects. These types of nonlinearity include Kerr law, power law, and dual-power law. The symmetry case in the GL equation due to the three types of nonlinearity is presented. The governing model is dealt with by a straightforward mathematical technique, where the fractional differential equation is reduced to a first-order nonlinear ordinary differential equation with solution expressed in the form of the Weierstrass elliptic function. The relation between the Weierstrass elliptic function and hyperbolic functions enables us to derive two types of optical soliton solutions, namely, bright and singular solitons. Restrictions for the validity of the optical soliton solutions are given. To shed light on the behaviour of solitons, the graphical illustrations of obtained solutions are represented for different values of various parameters. The symmetrical structure of some extracted solitons is deduced when the fractional derivative parameters for space and time are symmetric.

scholarly journals Solitary wave solutions in two-Core optical fibers with coupling-coefficient dispersion and Kerr nonlinearity

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 369
Author(s):  
S. Abbagari ◽  
A. Houwe ◽  
H. Rezazadeh ◽  
A. Bekir ◽  
S. Y. Doka
Keyword(s):  
Optical Fibers ◽  
Coupling Coefficient ◽  
Soliton Solutions ◽  
Kerr Nonlinearity ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Mathematical Methods ◽  
Auxiliary Equation Method ◽  
Equivalent Equation ◽  
Intermodal Dispersion

In this paper, we studies chirped solitary waves in two-Core optical fibers with coupling-coefficient dispersion and intermodal dispersion. To construct chirp soliton, the couple of nonlinear Schrödinger equation which describing the pulses propagation along the two-core fiber have been reduced to one equivalent equation. By adopting the traveling-waves hypothesis, the exact analytical solutions of the GNSE were obtained by using three relevant mathematical methods namely the auxiliary equation method, the modified auxiliary equation method and the Sine-Gordon expansion approach. Lastly, the behavior of the chirped like-soliton solutions were discussed and some contours of the plot evolution of the bright and dark solitons are obtained.

scholarly journals Soliton Solutions of Space-Time Fractional-Order Modified Extended Zakharov-Kuznetsov Equation in Plasma Physics

2018 ◽  
Vol 20 ◽  
pp. 1-8
Author(s):  
Muhammad Nasir Ali ◽  
Syed Muhammad Husnine ◽  
Sana Noor ◽  
Turgut Ak
Keyword(s):  
Differential Equation ◽  
Plasma Physics ◽  
Fractional Order ◽  
Physical Phenomenon ◽  
Nonlinear Partial Differential Equation ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Magnetized Plasma ◽  
Ansatz Method

The aim of this article is to calculate the soliton solutions of space-time fractional-order modified extended Zakharov-Kuznetsov equation which is modeled to investigate the waves in magnetized plasma physics. Fractional derivatives in the form of modified Riemann-Liouville derivatives are used. Complex fractional transformation is applied to convert the original nonlinear partial differential equation into another nonlinear ordinary differential equation. Then, soliton solutions are obtained by using (1/G')-expansion method. Bright and dark soliton solutions are also obtain with ansatz method. These solutions may be of significant importance in plasma physics where this equation is modeled for some special physical phenomenon.

New Jacobi elliptic solutions and other solutions with quadratic-cubic nonlinearity using two mathematical methods

2018 ◽  
Vol 13 (02) ◽  
pp. 2050043 ◽  
Author(s):  
Savaissou Nestor ◽  
Mibaile Justin ◽  
Douvagai ◽  
Gambo Betchewe ◽  
Serge Y. Doka ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Soliton Solutions ◽  
Higher Order ◽  
Auxiliary Equation ◽  
Cubic Nonlinearity ◽  
Mathematical Methods ◽  
Auxiliary Equation Method ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Elliptic Solutions

In this paper, we apply two powerful methods, namely, the new extended auxiliary equation method and the generalized Kudryashov method for constructing many exact solutions and other solutions for the higher order dispersive nonlinear Schrödinger’s equation to secure soliton solutions in quadratic-cubic medium. Various solutions of the resulting nonlinear ODE are obtained by using the above two methods.

scholarly journals Exact solutions of the space-time fractional equal width equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2307-2313 ◽  
Author(s):  
Hongcai Ma ◽  
Xiangmin Meng ◽  
Hanfang Wu ◽  
Aiping Deng
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Soliton Solutions ◽  
Space Time ◽  
Bright Soliton ◽  
Liouville Derivative ◽  
Fractional Differential

A class of fractional differential equations is investigated in this paper. By the use of modified Remann-Liouville derivative and the tanh-sech method, the exact bright soliton solutions for the space-time fractional equal width are obtained.

scholarly journals Two Reliable Methods for Solving the (3 + 1)-Dimensional Space-Time Fractional Jimbo-Miwa Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-30 ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert ◽  
Chaowanee Khaopant ◽  
Wanassanun Porka
Keyword(s):  
Exact Solutions ◽  
Dimensional Space ◽  
Soliton Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Space Time ◽  
The Novel ◽  
Wave Solutions ◽  
Liouville Derivative

We investigate methods for obtaining exact solutions of the (3 + 1)-dimensional nonlinear space-time fractional Jimbo-Miwa equation in the sense of the modified Riemann-Liouville derivative. The methods employed to analytically solve the equation are the G′/G,1/G-expansion method and the novel G′/G-expansion method. To the best of our knowledge, there are no researchers who have applied these methods to obtain exact solutions of the equation. The application of the methods is simple, elegant, efficient, and trustworthy. In particular, applying the novel G′/G-expansion method to the equation, we obtain more exact solutions than using other existing methods such as the G′/G-expansion method and the exp-Φ(ξ)-expansion method. The exact solutions of the equation, obtained using the two methods, can be categorized in terms of hyperbolic, trigonometric, and rational functions. Some of the results obtained by the two methods are new and reported here for the first time. In addition, the obtained exact explicit solutions of the equation characterize many physical meanings such as soliton solitary wave solutions, periodic wave solutions, and singular multiple-soliton solutions.

scholarly journals Closed-form wave structures of the space-time fractional Hirota–Satsuma coupled KdV equation with nonlinear physical phenomena

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 555-565 ◽  
Author(s):  
Md Nur Alam ◽  
Aly R. Seadawy ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Kdv Equation ◽  
Expansion Method ◽  
Space Time ◽  
Liouville Derivative ◽  
Coupled Kdv Equation ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
To Receive

AbstractThe present paper applies the variation of (G^{\prime} /G)-expansion method on the space-time fractional Hirota–Satsuma coupled KdV equation with applications in physics. We employ the new approach to receive some closed form wave solutions for any nonlinear fractional ordinary differential equations. First, the fractional derivatives in this research are manifested in terms of Riemann–Liouville derivative. A complex fractional transformation is applied to transform the fractional-order ordinary and partial differential equation into the integer order ordinary differential equation. The reduced equations are then solved by the method. Some novel and more comprehensive solutions of these equations are successfully constructed. Besides, the intended approach is simplistic, conventional, and able to significantly reduce the size of computational work associated with other existing methods.

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