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In this article, the most recent version of an optimal homotopy analysis method (HAM), called linearization‐based approach of HAM or simply LHAM, has been applied to obtain a numerical solution of one of the principal nonlinear fractional‐order hyperbolic problems known as the time‐fractional hyperbolic partial differential equation. Such method is constructed based on employing Taylor series linearization method in order to design an optimal auxiliary linear operator with its corresponding optimal initial guessing. These two optimum contributors will accelerate the convergence of series solutions for the problem at hand. Several numerical comparisons have revealed the efficiency of the proposed method in obtaining a numerical solution of the problem rather than that solution presented by using the standard HAM. All theoretical findings in this work have been verified numerically using MATLAB software package. [ABSTRACT FROM AUTHOR]

Abstract In this study, an optimal homotopy analysis approach will be described to deal with nonlinear fractional differential equations. The proposed approach presents a procedure to find an optimal auxiliary linear operator and the corresponding optimal initial approximation that will accelerate the convergence of series solutions for nonlinear differential equations with fractional derivatives. Then, a reliable modified version of the homotopy analysis method is presented to facilitate the calculations. Numerical comparison will be made to examine the computational efficiency and the pertinent features of the proposed algorithm. This algorithm is expected to be further employed to solve wide classes of nonlinear problems in fractional calculus. [ABSTRACT FROM AUTHOR]

This paper deals with constructing generalized 'fractional' power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non-linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis. [Copyright &y& Elsevier]

Abstract: In this paper, based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter . The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus. [Copyright &y& Elsevier]

The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived analytically to achieve synchronization of the chaotic fractional-order Chen, Rössler and modified Chua systems. Numerical simulations are provided to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: In this article, a novel numerical method is proposed for nonlinear partial differential equations with space- and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor''s formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order. [Copyright &y& Elsevier]

Comparison of homotopy perturbation method (HPM) and variational iteration method (VIM) is made, revealing that the two methods can be used as alternative and equivalent methods for obtaining analytic and approximate solutions for different types of differential equations of fractional order. Furthermore, the former is more general and powerful than the latter. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a new generalization of the one-dimensional differential transform method that will extend the application of the method to differential equations of fractional order. The new generalization is based on generalized Taylor’s formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the obtained results. The new generalization introduces a promising tool for many linear and nonlinear models containing fractional derivatives. [Copyright &y& Elsevier]

Abstract: In this letter we develop a new generalization of the two-dimensional differential transform method that will extend the application of the method to linear partial differential equations with space- and time-fractional derivatives. The new generalization is based on the two-dimensional differential transform method, generalized Taylor’s formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the present method. The results reveal that the technique introduced here is very effective and convenient for solving linear partial differential equations of fractional order. [Copyright &y& Elsevier]

Abstract: In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order. [Copyright &y& Elsevier]

Abstract: This Letter presents numerical solutions for the space- and time-fractional Fokker–Planck equation using the variational iteration method and the Adomian decomposition method. The fractional derivatives are described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. Some examples are given and comparisons are made, the comparisons show that the variational iteration method is very effective and convenient and overcome the difficulty arising in calculating Adomian polynomials. The numerical results show that the two approaches are easy to implement and accurate when applied to space- and time-fractional Fokker–Planck equations. The methods introduce a promising tool for solving many space–time fractional partial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order. [Copyright &y& Elsevier]

Abstract: The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement. [Copyright &y& Elsevier]

Abstract: In this paper, a new generalized Taylor’s formula of the kindwhere a j ∈ R , x > a, 0< α ⩽1, is established. Such an expression is precisely the classical Taylor’s formula in case of α =1. In addition, detailed expressions for and a j (α), that involving Caputo fractional derivatives, are well constructed. Some applications including approximation of functions and solutions of fractional differential equations are also given. [Copyright &y& Elsevier]

Abstract: In this article, we implement relatively a new analytical technique, the Adomian decomposition method, for solving the boundary value problems of time-fractional wave equation. The fractional derivative is described in the Caputo sense. The decomposition method is used to construct analytical approximate solutions of time-fractional wave equation subject to specified boundary conditions. The solutions are calculated in the form of a convergent series with easily computable components. Some examples are given. The results reveal that the Adomian method is very effective and convenient. [Copyright &y& Elsevier]

Abstract: In this paper we propose two algorithms for numerical fractional integration and Caputo fractional differentiation. We present a modification of trapezoidal rule that is used to approximate finite integrals, the new modification extends the application of the rule to approximate integrals of arbitrary order α >0. We then, using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order α >0, where the fractional derivative based on Caputo definition, for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The study is conducted through illustrative examples and error analysis. [Copyright &y& Elsevier]

Abstract: The aim of the present analysis is to apply Adomian decomposition method for the solution of a time-fractional Navier–Stokes equation in a tube. By using an initial value, the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically. The present method performs extremely well in terms of efficiency and simplicity. [Copyright &y& Elsevier]

In this paper, the variational iteration method is implemented to give approximate solutions for nonlinear differential equations of fractional order. In this method the problems are initially approximated by imposing the initial conditions. Then a correction functional for the fractional differential equation is well constructed by a general Lagrange multiplier, which can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. Some examples are given and comparisons are made with the Adomian decomposition method. The comparison shows that the method is very effective and convenient and overcome the difficulty arising in calculating Adomian polynomials. [ABSTRACT FROM AUTHOR]