Sense Of Conformable Derivative Research Articles

Periodic and solitary wave solutions to a family of new 3D fractional WBBM equations using the two-variable method

For the newly implemented 3D fractional Wazwaz–Benjamin–Bona–Mahony​ (WBBM) equation family, the present study explores exact singular, solitary, and periodic singular wave solutions via the (G′∕G,1∕G)-expansion process. In the sense of conformable derivatives, the equations considered are transformed into ordinary differential equations. In spite of many trigonometric, complex hyperbolic, and rational functions, some fresh exact singular, solitary, and periodic wave solutions to the deliberated equations in fractional systems are attained by the implementation of the (G′∕G,1∕G)-expansion technique through the computational software Mathematica. The unique solutions derived by the process defined are articulated with the arrangement of the functions tanh, sech; tan, sec; coth, csch, and cot, csc. With three-dimensional graphics, some of the latest solutions created have been envisaged, by selecting appropriate arbitrary constraints to illustrate their physical representation. The outcomes which are obtained to show the power of the computational technique for the WBBM equations can be applied to other nonlinear water model equations in ocean and coastal engineering. All the obtained solutions have been verified by the computational software Mathematica.

Comparative study of two techniques on some nonlinear problems based ussing conformable derivative

AbstractIn this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.

On the propagation of alphabetic-shaped solitons to the (2 + 1)-dimensional fractional electrical transmission line model with wave obliqueness

This manuscript deals with the (2 + 1)-dimensional nonlinear electrical transmission line (NETL) equation of fractional-order to explain the dynamics of waves in the nonlinear network lines. The fractional transform with conformable derivative and wave obliqueness is applied to the model equation for converting it into an ordinary differential equation (ODE). To explore alphabetic-shaped solitons, an ansatz and the auxiliary equation methods are utilized to the ODE. The utilized transformation is then put back to the received solutions of the ODE. The solutions of the model equations representing alphabetic solitons are attained in terms of fractionality and wave obliqueness. It is seen that some of the beforehand generated solutions are the particular case of the newly derived solutions. The effects of fractionality, obliqueness, and free parameters on the attained solutions are tested and are demonstrated graphically with physical descriptions. Also, the behavior of the voltage wave propagation of the attained solutions are described. It is found that the alphabetic wave phenomena are changed with the change of obliqueness and fractionality, and the free parameters have substantial effects on the attained solutions. All the attained solitons propagate in the xy-plane as time (t) increases and the soliton propagation is found to be controlled by the free parameters, fractionality, and wave obliqueness. All the reported alphabetic-shaped solitons are found to be new in the sense of conformable derivative, wave obliqueness, and the applied methods.

An Extension of the Double G ′ / G , 1 / G -Expansion Method for Conformable Fractional Differential Equations

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic (CRWP) equation, nonlinear time fractional SharmaTassoOlver (STO) equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the G ′ / G , 1 / G -expansion method is found to be ensuing, effective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the G ′ / G , 1 / G -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs.

Exact and explicit travelling-wave solutions to the family of new 3D fractional WBBM equations in mathematical physics

In this paper, the new 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations newly announced are investigated with the introduction of the spatial and sequential fractional-order derivatives expending in the sense of conformable derivative. Several solutions containing hyperbolic and trigonometric function solutions are created using the modified extended tanh-function (mETF) method. The productivity of the mETF process for building these solutions has been established. This method decreases the large volume of calculations. With the assist of computational software MALAB, the solutions' graphical exemplifications are illustrated in three-dimension (3D) and two-dimension (2D) plot. All plots represent the absolute wave configurations through the obtained solutions to the mentioned equation, preferring suitable parameters. Moreover, it may conclude that the attained solutions and their physical features might help comprehend the shallow water wave propagation in nonlinear dynamics.

On conformable fractional Legendre polynomials and their convergence properties with applications

The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the conformable fractional differentiation. We introduce the CFLPs via different generating functions and provide some of their main properties and convergence results. Subsequently, some pure recurrence and differential recurrence relations, Laplace’s first integral formula, and orthogonal properties are then developed for CFLPs. We append our study by presenting shifted CFLPs and describing an applicable scheme using the collocation method to solve some fractional differential equations (FDEs) in the sense of conformable derivative. Some useful examples of FDEs are treated to support our theoretical results and examine their exact and approximate solutions. To the best of our knowledge, the obtained results are newly presented and could enrich the fractional theory of special functions.

Fractal Ion Acoustic Waves of the Space-Time Fractional Three Dimensional KP Equation

Methods known as fractional subequation and sine-Gordon expansion (FSGE) are employed to acquire new exact solutions of some fractional partial differential equations emerging in plasma physics. Fractional operators are employed in the sense of conformable derivatives (CD). New exact solutions are constructed in terms of hyperbolic, rational, and trigonometric functions. Computational results indicate the power of the method.

Computational solutions of conformable space-time derivatives dynamical wave equations: Analytical mathematical techniques

In this article, the instigator sets up the profuse traveling wave solutions four types of fractional nonlinear equations in the sense of conformable derivatives by using the novel form of modified mathematical technique. The constructed traveling wave solutions are articulated in terms of trigonometric, hyperbolic and exponential functions. The derived results are fruitful for the physical demonstrations of problems in mathematical physics and engineering.

Abundant solitary wave solutions for the fractional coupled Jaulent–Miodek equations arising in applied physics

This article explores the abundant solitary wave solutions of the conformable coupled Jaulent–Miodek (JM) equations appearing in applied physics. The aforesaid coupled equations belong to the family of shallow-water wave equations. Two recent modified integration schemes are used for the first time to produce a novel solitary wave, trigonometric and other solutions with some free parameters in the conformable derivative sense. In particular, the modified Kudryashov and [Formula: see text]-expansion schemes are used to illustrate the wave propagations through aforesaid solutions of the JM equations. Furthermore, a comparison is made with some recent results and the dynamics of the obtained solutions are displayed for the reader via soft computation. The outcomes reveal that the methods are effective and provide a direct way of finding novel solutions.

Dynamical analysis of long-wave phenomena for the nonlinear conformable space-time fractional (2+1)-dimensional AKNS equation in water wave mechanics

The main intension of this paper is to extract new and further general analytical wave solutions to the (2 + 1)-dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation in the sense of conformable derivative by implementing the advanced exp(-ϕ(ξ))-expansion method. This method is a particular invention of the generalized exp(-ϕ(ξ))-expansion method. By the virtue of the advanced exp(-ϕ(ξ))-expansion method, a series of kink, singular kink, soliton, combined soliton, and periodic wave solutions are constructed to our preferred space time-fractional (2 + 1)- dimensional AKNS equation. An extensive class of new exact traveling wave solutions are transpired in terms of, hyperbolic, trigonometric, and rational functions. To express the underlying propagated features, some attained solutions are exhibited by making their three-dimensional (3D), two-dimensional (2D) combined, and 2D line plot with the help of computational packages MATLAB. All plots are given to show the proper wave features through the founded solutions to the studied equation with particular preferring of the selected parameters. Moreover, it may conclude that the attained solutions and their physical features might be helpful to comprehend the water wave propagation in water wave mechanics.

New theoretical results and applications on conformable fractional Natural transform

In this paper, we proceed on to develop the classical Natural transform to fractional order in the sense of conformable derivative and set the basic concepts in this new interesting fractional transform version. The conformable fractional Natural transform of certain functions, properties and relationships are derived and discussed. Furthermore, we present the general analytical solution of a generalized conformable Bernoulli’s fractional differential equation based on the new version and Adomain polynomial method. This method is also applied successfully to find the general solutions of some linear and nonlinear conformable fractional differential equations. Finally, the results show that our new approach is simple, accurate and effective.

Solitary and periodic wave solutions to the family of nonlinear conformable fractional Boussinesq‐like equations

The current study investigates the exact solitary, singular, and periodic singular wave solutions to the family of nonlinear fractional Boussinesq‐like (Bq‐like) equations using the (G′/G,1/G)‐expansion method. The considered equations in the sense of conformable derivative are converted into ordinary differential equations with the assist of fractional transformation. Applying the (G′/G,1/G)‐expansion method through the symbolic computation package Maple, some new exact solitary, singular, and periodic wave solutions to the considered equations in fractional forms are acquired in terms of a variety of complex hyperbolic, trigonometric, and rational functions. Moreover, the particular solutions extracted by the mentioned method are expressed with the combination of tanh, sech; coth, csch; tan, sec, and cot, csc functions. Among the produced solutions, some of the new solutions have been visualized by three‐dimensional (3D) and two‐dimensional (2D) graphics under the choice of suitable arbitrary parameters to show their physical interpretation. The acquired results demonstrate the power of the executed method to evaluate the exact solutions of the nonlinear fractional Bq‐like equations, which may be used for applying nonlinear water model to coastal and ocean engineering. All the produced solutions have been verified by substituting back into their corresponding equations with the aid of symbolic computation software Maple 17.

Reliable methods to study some nonlinear conformable systems in shallow water

In this paper, consider the eminent coupled Boussinesq–Burger (BB) equations and the coupled Whitham–Broer–Kaup (WBK) equations with time fractional derivative arising in the investigation of shallow water waves. The derivative is described in the sense of conformable derivative. We introduce the fundamental {(G'} / {G} )-expansion method and its extension, namely the two-variable {(G'} / {G}, {1} / {G} )-expansion method, to establish general solutions, some typical wave solutions existing in the literature, and some new and compatible soliton solutions comprised with certain parameters. For the definite values of these parameters, we derive and show in figures the well-known kink, singular kink, bell-shape soliton, periodic soliton, cuspon, and so on. The obtained solutions affirm that the introduced methods are reliable and efficient techniques to examine a wide variety of nonlinear fractional systems in the sense conformable derivative.

New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

The major purpose of this article is to seek for exact traveling wave solutions of the nonlinear space-time Sharma–Tasso–Olver equation in the sense of conformable derivatives. The novel ( G ′ G ) -expansion method and the generalized Kudryashov method, which are analytical, powerful, and reliable methods, are used to solve the equation via a fractional complex transformation. The exact solutions of the equation, obtained using the novel ( G ′ G ) -expansion method, can be classified in terms of hyperbolic, trigonometric, and rational function solutions. Applying the generalized Kudryashov method to the equation, we obtain explicit exact solutions expressed as fractional solutions of the exponential functions. The exact solutions obtained using the two methods represent some physical behaviors such as a singularly periodic traveling wave solution and a singular multiple-soliton solution. Some selected solutions of the equation are graphically portrayed including 3-D, 2-D, and contour plots. As a result, some innovative exact solutions of the equation are produced via the methods, and they are not the same as the ones obtained using other techniques utilized previously.

Series solutions for the Laguerre and Lane-Emden fractional differential equations in the sense of conformable fractional derivative

In the present paper, we use efficient and simple algorithms of the fractional power series and Adomain polynomial methods that provide effective tools for solving such linear and nonlinear fractional differential equations in the sense of conformable derivative. The first method succeeded in finding the fractional power series solution for the Laguerre linear fractional differential equation with some important special cases. The second method is applied to predict and construct the fractional Adomain approximation solution of the general conformable Lane-Emden fractional differential equation with some interesting linear and nonlinear special cases with their graphs. Finally, the results show that the Adomain approximation solutions of those problems are stable at different values of α and assure that this way can be applied successfully for solving many linear and nonlinear fractional differential equations in conformable sense.

New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative

The exact wave solutions to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) and space time fractional Zakharov-Kuznetsov Benjamin-Bona-Mahony (ZKBBM) equations are studied in the sense of conformable derivative. The existence of chain rule and the derivative of composite functions permit the nonlinear fractional differential equations (NLFDEs) to convert into the ordinary differential equation using wave transformation. The wave solutions of these equations are examined by means of the expanding and effective two variable ( G' / G ,1/ G )-expansion method. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The method is efficient, convenient, accessible and is the generalization of the original ( G' / G )-expansion method.

New analytical solutions of (2 + 1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques

In this paper, the new exact solutions for the (2 + 1) dimensional time fractional Zoomeron equation have been derived via two efficient analytical techniques, which are the extended exp(−Φ(ξ))-expansion technique and the novel exponential rational function technique. The fractional derivative is designated based on the conformable derivative sense. Consequently, many new closed form solutions of this equation are obtained including hyperbolic function solutions, trigonometric function solutions and exponential function solutions by using these techniques. The obtained results show that the applied methods are very effective, reliable and simple for solving other nonlinear fractional differential equations in mathematical physics and nonlinear optics.

Rational exponential solutions of conformable space-time fractional equal-width equations

AbstractIn this paper, the rational exponential solutions of two space-time fractional equal-width (FEW) equations are explored in the conformable derivative sense. The way to reach explicit exact solutions is to transform the fractional order PDEs into a nonlinear ODEs of discrete order through some properties of conformable derivatives and a fractional complex transforms. The subsequent equations have been elucidated by employing theexpafunction approach. Some new exact solutions of the said equations are effectively formulated and graphically conveyed with the aid of symbolic computation in Mathematica and MATLAB respectively.