Sense Of Conformable Fractional Derivative Research Articles

Approximate and Exact Solutions of Some Nonlinear Differential Equations Using the Novel Coupling Approach in the Sense of Conformable Fractional Derivative

Several scientific fields utilize fractional nonlinear partial differential equations to model various phenomena. However, most of these equations lack exact solutions. Consequently, techniques for obtaining approximate solutions, which sometimes yield exact solutions, are essential. In this research, we develop a new approach by combining the homotopy perturbation method (HPM) and the conformable natural transform to solve the gas-dynamic equation (GDE), the Fokker-Planck equation (FPE), and the Swift-Hohenberg equation (SHE) in the context of conformable derivatives. The proposed approach is called the conformable natural homotopy perturbation method (CNHPM). This approach has the advantage of not requiring assumptions about significant or minor physical factors. Consequently, it eliminates some of the constraints associated with conventional perturbation methods and can solve both weak and highly nonlinear problems. We consider the absolute, relative, and residual errors numerically and graphically to assess the correctness of our approach. The results show that our approach serves as a suitable alternative to the approximate methods in the literature for solving fractional differential equations.

Fractional Biswas–Milovic Equation in Random Case Study

We apply two mathematical techniques, specifically, the unified solver approach and the exp(−φ(ξ))-expansion method, for constructing many new solitary waves, such as bright, dark, and singular soliton solutions via the fractional Biswas–Milovic (FBM) model in the sense of conformable fractional derivative. These solutions are so important for the explanation of some practical physical problems. Additionally, we study the stochastic modeling for the fractional Biswas–Milovic, where the parameter and the fraction parameters are random variables. We consider these parameters via beta distribution, so the mathematical methods that were used in this paper may be called random methods, and the exact solutions derived using these methods may be called stochastic process solutions. We also determined some statistical properties of the stochastic solutions such as the first and second moments. The proposed techniques are robust and sturdy for solving wide classes of nonlinear fractional order equations. Finally, some selected solutions are illustrated for some special values of parameters.

New Classifications of All Single Traveling Wave Solutions of the Fractional Approximate Long Water Wave Equations by Using the Complete Discrimination System

First, our work is to transform the space-time fractional approximate long water wave equations into nonlinear ordinary differential equations via the traveling wave transformation in the sense of conformable fractional derivative. Second, we simplify the nonlinear ordinary differential equations into an ordinary differential equation with only one variable by integration and some transformations. Finally, we can further get all single traveling wave solutions of the space-time fractional approximate long water wave equations by the complete discrimination system for the four-order polynomial method; these solutions include the hyperbolic function solutions, rational function solutions, and implicit solutions.

Explicit solutions to the time-fractional generalized dissipative Kawahara equation

In this paper, the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques. The quadratic-case and cubic-case are investigated for the proposed model. Expected solutions are obtained with highlighting to the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation. Some graphical analysis is presented to support the findings of the paper. Finally, we believe that the obtained results in this work will be important and valuable in nonlinear sciences and ocean engineering.

Structures of exact solutions for the modified nonlinear Schrödinger equation in the sense of conformable fractional derivative

Structures of exact solutions for the modified nonlinear Schrödinger equation in the sense of conformable fractional derivative

Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.

Distinct solutions of nonlinear space–time fractional evolution equations appearing in mathematical physics via a new technique

Abstract In this paper, we furnished novel and different traveling wave solutions to the nonlinear space–time fractional Klein–Gordon (KG) equation and the nonlinear space–time fractional symmetric regularized long wave (SRLW) equation. The considered equations are turned into ordinary differential equations with the aid of fractional composite transformation in the sense of conformable fractional derivative and then the proposed improved tanh method along with Riccati equation is employed. Consequently, fresh and further general solutions in the shape of kink, bell, cuspon, compacton, periodic etc. are successfully obtained, which are compared with those in the literature. Further, it is important to point out that some of our solutions is in good harmony for a special case with already published results which in turn validates our other solutions. The comparison ensures the novelty of the proposed technique which might be taken into account for further research in future.

Conformable fractional derivative and its application to partial fractional derivatives

Recently, the study on conformable fractional derivative has been gaining more attention from various researchers. This is due to its importance in numerous practical applications. In this paper, we introduce new definition of conformable fractional derivation to find partial fractional derivatives. The fractional derivatives are taken in the sense of Conformable fractional derivative. This approach is considered as a powerful tool for solving linear or nonlinear equations. The new formulation of fractional calculus is applied to solve many applied problems including fractional partial derivatives. The results obtained through the conformable fractional approach have been evaluated and presented to illustrate the efficiency and good accuracy. It has been observed that the proposed approach is prevailing for the solutions of partial fractional derivatives.

Computing wave solutions and conservation laws of conformable time-fractional Gardner and Benjamin–Ono equations

This paper presents travelling wave solutions for the nonlinear time-fractional Gardner and Benjamin–Ono equations via the exp( $$- \Phi ( \varepsilon ))$$ -expansion approach. Specifically, both the models are studied in the sense of conformable fractional derivative. The obtained travelling wave solutions are structured in rational, trigonometric (periodic solutions) and hyperbolic functions. Further, the investigation of symmetry analysis and nonlinear self-adjointness for the governing equations are discussed. The exact derived solutions could be very significant in elaborating physical aspects of real-world phenomena. We have 2D and 3D illustrations for free choices of the physical parameter to understand the physical explanation of the problems. Moreover, the underlying equations with conformable derivative have been investigated using the new conservation theorem.

New Exact Traveling Wave Solutions of the Time Fractional Complex Ginzburg-Landau Equation via the Conformable Fractional Derivative

In this study, the exact traveling wave solutions of the time fractional complex Ginzburg-Landau equation with the Kerr law and dual-power law nonlinearity are studied. The nonlinear fractional partial differential equations are converted to a nonlinear ordinary differential equation via a traveling wave transformation in the sense of conformable fractional derivatives. A range of solutions, which include hyperbolic function solutions, trigonometric function solutions, and rational function solutions, is derived by utilizing the new extendedG′/G-expansion method. By selecting appropriate parameters of the solutions, numerical simulations are presented to explain further the propagation of optical pulses in optic fibers.

Conformable fractional derivative and its application to partial fractional derivatives

Recently, the study on conformable fractional derivative has been gaining more attention from various researchers. This is due to its importance in numerous practical applications. In this paper, we introduce new definition of conformable fractional derivation to find partial fractional derivatives. The fractional derivatives are taken in the sense of Conformable fractional derivative. This approach is considered as a powerful tool for solving linear or nonlinear equations. The new formulation of fractional calculus is applied to solve many applied problems including fractional partial derivatives. The results obtained through the conformable fractional approach have been evaluated and presented to illustrate the efficiency and good accuracy. It has been observed that the proposed approach is prevailing for the solutions of partial fractional derivatives.

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.

Series solutions for nonlinear time-fractional Schrödinger equations: Comparisons between conformable and Caputo derivatives

In the present paper, we present the exact analytical solution of the time-fractional Schrödinger equation (TFSE) in the sense of conformable fractional derivative (Co-FD) based on the residual power series method (RPSM). Moreover, we make graphical comparisons between the solution of this problem in the sense of Co-FD and the obtained solutions in previous studied in the sense of Caputo fractional derivative (Ca-FD). Comparisons indicate that Co-FD is a suitable alternative to Ca-FD in the modeling of TFSE. The main advantage for employing the RPSM is the simplicity of computing the coefficients of the series solution by applying differential operators without having to use integrated operators. Finally, our proposed technique can be applied easily in the solution of higher dimension without any restriction on the type of equation and the Co-FD can be used to formulate other types of partial differential equation.

New results on complex conformable integral

A new theory of analytic functions has been recently introduced in the sense of conformable fractional derivative. In addition, the concept of fractional contour integral has also been developed. In this paper, we propose and prove some new results on complex fractional integration. First, we establish necessary and sufficient conditions for a continuous function to have antiderivative in the conformable sense. Finally, some of the well-known Cauchy′s integral theorems will also be the subject of the extension that we do in this paper.

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.

Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative.

In this paper, the time-fractional nonlinear dispersive (TFND) partial differential equations (PDEs) in the sense of conformable fractional derivative (CFD) are proposed and analyzed. Three types of TFND partial differential equations are considered in the sense of CFD, which are the TFND Boussinesq, TFND Klein-Gordon, and TFND B(2, 1, 1) PDEs. Solitary pattern solutions for this class of TFND partial differential equations based on the residual fractional power series method is constructed and discussed. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions. The results suggest that the algorithm presented here offers solutions to problems in a rapidly convergent series leading to ideal solutions. Furthermore, the results obtained are like those in previous studies that used other types of fractional derivatives. In addition, the calculations used were much easier and shorter compared with other types of fractional derivatives.

Application of the extended exp(-φ(ξ))-expansion method to the nonlinear conformable time-fractional partial differential equations

This paper investigates the new exact solutions of the three nonlinear time fractional partial differential equations namely the nonlinear time fractional Clannish Random Walker’s Parabolic (CRWP) equation, the nonlinear time fractional modified Kawahara equation, and the nonlinear time fractional BBM-Burger equation by utilizing an extended form of exp(-φ(ξ))-expansion method in the sense of conformable fractional derivative. As outcomes, some new exact solutions are obtained and signified by hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Some solutions have been plotted by MATLAB software to show the physical significance of our studied equations. In the point of view of our executed method and generated results, we may conclude that extended exp (-φ(ξ))-expansion method is more efficient than exp(-φ(ξ))-expansion method to extract the new exact solutions for solving any types of integer and fractional differential equations arising in mathematical physics.  Â

New soliton solutions of conformable time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation in modeling wave phenomena

In this paper, new traveling wave solutions of time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation have been revealed by the help of extended Kudryashov’s method. In this case, the time fractional derivative has been considered in the sense of conformable fractional derivative. The salient feature of the proposed extended Kudryashov’s method is to take full advantage of solutions for the Riccati and the Bernoulli equations involving parameters. Using some particular wave transformation, the given equation is converted to the nonlinear ODE by the help of chain rule and the derivative of a composite function; and then, a novel analytical method viz. extended Kudryashov’s method has been used to solve the resulting equation. Consequently, some new exact solutions have been successfully obtained. The exact solutions devised by the proposed method manifest that the proposed method is effective and easy to implement.

Closed-form travelling wave solutions to the nonlinear space-time fractional coupled Burgers’ equation

Fractional order nonlinear evolution equations play important roles to give a depiction of the complex physical phenomena of real world. The main aim of this article is to extract exact analytic solutions to the space-time fractional coupled Burgers’ equation in the sense of conformable fractional derivative. A suitable composite transformation is implemented to reduce the considered equation into an ordinary differential equation of fractional order. Then a new approach, called the rational fractional -expansion method, the Exp-function method and the extended tanh method, is employed to construct the closed-form solutions. This effort makes available the travelling wave solutions in terms of hyperbolic function, trigonometric function and rational function, which are found to be newer and more general than the existing results in the literature. The proposed method is highly efficient and may further be used to invstigate entirely new solutions to many other fractional evolution equations.

Closed form solutions of two time fractional nonlinear wave equations

Abstract In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized ( G ′ / G ) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.