Exact Traveling Wave Solutions for the Space-Time Fractional Modified Third-Order KdV Equation Via an Improved Analytical Technique

In this paper, we present the solution of nonlinear fractional partial differential equations by using the Homotopy Perturbation Aboodh Transform Method (HPATM) and Homotopy Decomposition Method (HDM). The Two methods introduced an efficient tool for solving a wide class of linear and nonlinear fractional differential equations. The results shown that the (HDM) has an advantage over the (HPATM) that it takes less time and using only the inverse operator to solve the nonlinear problems and there is no need to use any other inverse transform as in the case of (HPATM).

In this article, we use fractional sub-equation method to find the exact solutions for some nonlinear partial fractional differential equations, namely space–time fractional coupled Sakharov–Kuznetsov (Z–K) equations, space–time fractional nonlinear coupled Korteweg de Vries (KdV) equations and space–time fractional Hirota–Satsuma KdV system. As a result, three families of exact analytical solutions are obtained. Proposed method is more effective and powerful to construct the exact solutions for nonlinear partial fractional differential equations. Also, we use fractional complex transform and exp function method to find some of the exact solutions for nonlinear partial fractional differential equations.

Elzaki transform and Adomian polynomial is used to obtain the exact solutions of nonlinear fifth order Korteweg-de Vries (KdV) equations. In order to investigate the effectiveness of the method, three fifth order KdV equations were considered. Adomian polynomial is introduced as an essential tool to linearize all the nonlinear terms in any given equation because Elzaki transform cannot handle nonlinear functions on its own. In all the three problems considered, the series solutions obtained converges to the exact solutions. Three dimensional graphs were also plotted to give the shape of the solutions of some KdV equations considered. Hence, Elzaki transform and Adomian polynomial together gives a very powerful and ffective method for solving nonlinear partial differential equations.

In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. An algebraic method is improved to construct uniformly a series of exact solutions for some nonlinear time-space fractional partial differential equations. We construct successfully a series of some exact solutions including the elliptic doubly periodic solutions with the aid of computerized symbolic computation software package such as Maple or Mathematica. This method is efficient and powerful in solving a wide classes of nonlinear partial fractional differential equations. The Jacobi elliptic doubly periodic solutions are generated by the trigonometric exact solutions and the hyperbolic exact solutions when the modulus and , respectively.

In this article, the sub‐equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional‐order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.

Series solution is obtained on solving non-linear fractional partial differential equation using homotopy perturbation transformation method. First of all, we apply homotopy perturbation transformation method to obtain the series solution of non-linear fractional partial differential equation. In this case, the fractional derivative is described in Caputo sense. Then, we present the facts obtained by analyzing the convergence of this series solution. Finally, the established fact is supported by an example.

PurposeThe purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.Design/methodology/approachThe proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.FindingsThe fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.Originality/valueMany engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.

Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations

In this study, the nonlinear fractional partial differential equations have been defined by the modified Riemann–Liouville fractional derivative. By using this fractional derivative and traveling wave transformation, the nonlinear fractional partial differential equations have been converted into nonlinear ordinary differential equations. The modified trial equation method is implemented to obtain exact solutions of the nonlinear fractional Klein–Gordon equation and fractional clannish random walker's parabolic equation. As a result, some exact solutions including single kink solution and periodic and rational function solutions of these equations have been successfully obtained. Copyright © 2015 John Wiley & Sons, Ltd.

In this article, the space-time fractional perturbed nonlinear Schrödinger equation (NLSE) in nanofibers is studied using the improved [Formula: see text] expansion method (ITEM) to explore new exact solutions. The perturbed nonlinear Schrodinger equation is a nonlinear model that occurs in nanofibers. The ITEM is an efficient method to obtain the exact solutions for nonlinear differential equations. With the help of the modified Riemann–Liouville derivative, an equivalent ordinary differential equation has been obtained from the nonlinear fractional differential equation. Several new exact solutions to the fractional perturbed NLSE have been devised using the ITEM, which is the latest proficient method for analyzing nonlinear partial differential models. The proposed method may be applied for searching exact travelling wave solutions of other nonlinear fractional partial differential equations that appear in engineering and physics fields. Furthermore, the obtained soliton solutions are depicted in some 3D graphs to observe the behaviour of these solutions.

Exact solutions of nonlinear fractional order partial differential equations via singular manifold method

In this article, we use the Adomain decomposition method to find the approximate solutions for the linear and nonlinear partial fractional differential equations via the nonlinear Schrodinger partial fractional differential equation and the telegraph partial fractional differential equation. The fractional derivatives are described in the Caputo sense. We compare between the approximate solutions and the exact solutions for the partial fractional differential equations when α,β→1. Also we make the Figures to compare between the approximate solutions and the exact solutions for the partial fractional differential equations when α,β→1. This method is powerfull to find the approximate solutions for nonlinear partial fractional differential equations. Also we will compare between the approximate solutions which obtained by using the variational itearation method and the approximate solutions which obtained by Adomain decomposition methods.

Recent advancement in the field of nonlinear analysis and fractional calculus help to address the rising challenges in the solution of nonlinear fractional partial differential equations. This paper presents a hybrid technique, a combination of Tarig transform and Projected Differential Transform Method (TPDTM) to solve nonlinear fractional partial differential equations. The effectiveness of the method is examined by solving three numerical examples that arise in the field of heat transfer analysis. In this proposed scheme, the solution is obtained as a convergent series and the result is used to analyze the hyper diffusive process with pre local information regarding the heat transfer for different values of fractional order. In order to validate the results, a comparative study has been carried out with the solution obtained from the two methods, the Laplace Adomian Decomposition Method (LADM) and Homotophy Pertubation Method (HPM) and the result thus observed coincided with each other. Inspite of the uniformity between the solutions, the proposed hybrid technique had to overcome the complexity of manupulation of Adomian polynomials and evaluation of integrals in LADM and HPM respectively. The methodology and the results presented in this paper clearly reveals the computational efficiency of the present method. Due to its computational efficiency, the TPDTM has the potential to be used as a novel tool not only for solving nonlinear fractional differential equations but also for analysing the prelocal information of the system.

In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV equations. The solutions are expressed in terms of hyperbolic functions, trigonometric functions and rational functions. The method used is promising method to solve other nonlinear evaluation equations.