Fractional Fornberg Whitham Equation Research Articles

Solving the fractional Fornberg-Whitham equation within Caputo framework using the optimal auxiliary function method

In this work, we solve the fractional-order Fornberg-Whitham (FW) problem in the context of the Caputo operator by using the Optimal Auxiliary Function Method. Tables and figures showing full numerical findings indicate the correctness and efficacy of this strategy. The results provide insights into the solution behavior of the FW equation and demonstrate the applicability of the Optimal Auxiliary Function Method. By giving insight on the behavior of the FW equation in a fractional context, this research advances the use of fractional calculus techniques in the solution of complicated differential equations.

Numerical Investigation of Fractional-Order Fornberg–Whitham Equations in the Framework of Aboodh Transformation

In this investigation, the fractional Fornberg–Whitham equation (FFWE) is solved and analyzed via the variational iteration method (VIM) and Adomian decomposition method (ADM) with the help of the Aboodh transformation (AT). The FFWE is an important model for describing several nonlinear wave propagations in various fields of science and plasma physics. The AT provides a powerful tool for transforming fractional-order differential equations (DEs) into integer-order ones, making them more amenable to analytical solutions. Accordingly, the main objective of this investigation is to demonstrate the effectiveness and accuracy of ADM and VIM in deriving some approximations for the FFWE. Furthermore, we highlight the advantages and potential applications of these methods in solving other fractional-order nonlinear problems in several scientific fields, especially in plasma physics and some engineering problems.

The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique

In the last few decades, it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences. For this reason, fractional partial differential equations (FPDEs) are of more importance to model the different physical processes in nature more accurately. Therefore, the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution. In the current work, the idea of fractional calculus has been used, and fractional Fornberg Whitham equation (FFWE) is represented in its fractional view analysis. A well-known method which is residual power series method (RPSM), is then implemented to solve FFWE. The RPSM results are discussed through graphs and tables which conform to the higher accuracy of the proposed technique. The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution. Because the RPSM procedure is simple and straightforward, it can be extended to solve other FPDEs and their systems.

Numerical and analytical solution to a conformable fractional Fornberg–Whitham equation

Numerical and analytical solution to a conformable fractional Fornberg–Whitham equation

He’s Homotopy Perturbation Method and Fractional Complex Transform for Analysis Time Fractional Fornberg-Whitham Equation

In this article, time fractional Fornberg-Whitham equation of He’s fractional derivative is studied. To transform the fractional model into its equivalent differential equation, the fractional complex transform is used and He’s homotopy perturbation method is implemented to get the approximate analytical solutions of the fractional-order problems. The graphs are plotted to analysis the fractional-order mathematical modeling.

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.

Residual power series method for the time fractional Fornberg-Whitham equation

The purpose of this paper is to solve the time fractional Fornberg-Whitham equation by the residual power series method, where the fractional derivatives are in Caputo sense. According to the definition of generalised fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations. The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional Fornberg-Whitham equation.

Residual power series method for the time fractional Fornberg-Whitham equation

The purpose of this paper is to solve the time fractional Fornberg-Whitham equation by the residual power series method, where the fractional derivatives are in Caputo sense. According to the definition of generalised fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations. The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional Fornberg-Whitham equation.

Comparison Between Two Reliable Methods for Accurate Solution of Fractional Modified Fornberg–Whitham Equation Arising in Water Waves

In this paper, an analytical technique is proposed to determine the exact solution of fractional order modified Fornberg–Whitham equation. Since exact solution of fractional Fornberg–Whitham equation is unknown, first integral method has been applied to determine exact solutions. The solitary wave solution of fractional modified Fornberg–Whitham equation has been attained by using first integral method. The approximate solutions of fractional modified Fornberg–Whitham equation, obtained by optimal homotopy asymptotic method (OHAM), are compared with the exact solutions obtained by the first integral method. The obtained results are presented in tables to demonstrate the efficiency of these proposed methods. The proposed schemes are quite simple, effective, and expedient for obtaining solution of fractional modified Fornberg–Whitham equation.

He’s fractional derivative and its application for fractional Fornberg-Whitham equation

Fractional Fornberg-Whitham equation with He?s fractional derivative is studied in a fractal process. The fractional complex transform is adopted to convert the studied fractional equation into a differential equation, and He's homotopy perturbation method is used to solve the equation.

An analytical treatment to fractional Fornberg\u2013Whitham equation

In this paper, an analytical technique, namely the new iterative method (NIM), is applied to obtain an approximate analytical solution of the fractional Fornberg–Whitham equation. The obtained approximate solutions are compared with the exact or existing numerical results in the literature to verify the applicability, efficiency, and accuracy of the method.

Application of new iterative transform method and modified fractional homotopy analysis transform method for fractional Fornberg-Whitham equation

Application of new iterative transform method and modified fractional homotopy analysis transform method for fractional Fornberg-Whitham equation

WITHDRAWN: He’s fractional derivative: A powerful tool to fractional Fornberg–Whitham equation

This article has been withdrawn at the request of the author(s) and/or editor. The Publisher apologizes for any inconvenience this may cause. The full Elsevier Policy on Article Withdrawal can be found at https://www.elsevier.com/about/our-business/policies/article-withdrawal .

On the analytical solution of Fornberg–Whitham equation with the new fractional derivative

Motivated by the simplicity, natural and efficient nature of the new fractional derivative introduced by R Khalil et al in J. Comput. Appl. Math. 264, 65 (2014), analytical solution of space-time fractional Fornberg–Whitham equation is obtained in series form using the relatively new method called q-homotopy analysis method (q-HAM). The new fractional derivative makes it possible to introduce fractional order in space to the Fornberg–Whitham equation and be able to obtain its solution. This work displays the elegant nature of the application of q-HAM to solve strongly nonlinear fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for nonlinear differential equations. Comparisons are made on the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.

Fractional Variational Iteration Method for Fractional Fornberg-Whitham Equation and Comparison with the Undetermined Coefficient Method

DOI: 10.9734/BJMCS/2015/15392 Editor(s): (1) Metin Basarir, Department of Mathematics, Sakarya University, Turkey. Reviewers: (1) Roshdi Khalil, Department of Mathematics, University of Jordan, Jordan. (2) Anonymous, Pakistan. (3) Anonymous, Turkey. (4) Jianfei Huang, College of Mathematics, Qingdao University, China. Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=733&id=6&aid=7607

Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform

In this article, fractional complex transform with optimal homotopy analysis method (OHAM) is used to obtain numerical and analytical solutions for the nonlinear time-space fractional Fornberg-Whitham. Fractional complex transform is proposed to convert time-space fractional Fornberg-Whitham equation to the nonlinear ordinary differential equations and then applied OHAM to the new obtained equations. This optimal approach has general meaning and can be used to get fast convergent series solutions of the different type of nonlinear fractional differential equation. The results reveal that the method is very effective, simple and the OHAM contains a certain auxiliary parameter $h$ which provides us with a simple way to adjust and control the convergence region of convergence of the series solution.

An analytical algorithm for nonlinear fractional Fornberg–Whitham equation arising in wave breaking based on a new iterative method

This work suggests a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving fractional Fornberg–Whitham equation. The fractional homotopy analysis transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. In this paper, it can be observed that auxiliary parameter ℏ, which controls the convergence of the FHATM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is basic and more qualitative difference in analysis between FHATM and other methods. The solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. The proposed method is illustrated by solving numerical example.

New treatment of fractional Fornberg–Whitham equation via Laplace transform

In this paper, a user friendly algorithm based on new homotopy perturbation transform method (HPTM) is proposed to solve nonlinear fractional Fornberg–Whitham equation in wave breaking. The new homotopy perturbation transform method is combined form of Laplace transform, homotopy perturbation method and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive.

Homotopy perturbation method for fractional Fornberg–Whitham equation

This article presents the approximate analytical solutions to solve the nonlinear Fornberg–Whitham equation with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm like homotopy perturbation method. The fractional derivatives are taken in the Caputo sense. Numerical results show that the HPM is easy to implement and accurate when applied to time-fractional PDEs.