Search

A new Caputo-type fractional derivative model with a generalized cardinal sine kernel and its singular kernel extension were presented. In this paper, we used the Laplace transform as an effective tool to study the considered fractional derivative models. Then, we introduced the Riemann–Liouville-type for the studied generalized cardinal sine fractional derivative model. Next, a general framework of the generalized cardinal sine kernel fractional derivative model and its singular kernel extension in relation to functions is presented. Some properties of the studied fractional derivative operators and relationships such as the relation between fractional integral and derivative operators are discussed. The dynamics of some nonlinear fractional order models are simulated, using a numerical algorithm formulated in this paper, to demonstrate the motivations of using the extended operators. The extended version of the considered fractional derivative operators provided useful suggestions regarding the modeling issue.

Some possible definitions of fractional derivative operators with non-singular analytic kernels have been introduced. In this paper, we propose a new generalized class of fractional derivative operators of Caputo-type with non-singular analytic kernels which includes some known operators as special cases. We demonstrate a relationship between the fractional derivative operators of the proposed generalized class and the Riemann-Liouville fractional integral operator. We also, using this relationship, introduce the corresponding fractional integral operators. Then, mainly, we provide extensions to the fractional derivative operators of the proposed generalized class that display integrable singular kernels. The extended fractional derivative operators provide useful insights regarding the modeling issue so that the initialization problem can be overcome. Finally, we discuss some basic properties of the proposed operators that are expected to be widely used in fractional calculus.

A broad framework for time-fractional advection-diffusion equations is considered by incorporating a general class of Caputo-type fractional derivatives that includes many well-known fractional derivatives as particular cases. Within this general context, in this paper, we propose a numerical approach to provide approximate solutions to time-fractional advection-diffusion models that involve fractional derivatives of the generalized class. The developed approach is based on discretizing the studied models with respect to spatio-temporal domains using non-uniform meshes. Moreover, we discuss the stability of the proposed approach, which does not require solving large systems of linear equations. Numerical results and 3D graphics are presented for some time-fractional advection-diffusion models using several fractional derivatives. The feasibility and the reliability of the proposed approach are clearly demonstrated by comparing the numerical solutions of the studied models with the exact solutions in cases with a known solution.

In this paper, we proposed a new fractional derivative operator in which the generalized cardinal sine function is used as a non-singular analytic kernel. In addition, we provided the corresponding fractional integral operator. We expressed the new fractional derivative and integral operators as sums in terms of the Riemann–Liouville fractional integral operator. Next, we introduced an efficient extension of the new fractional operator that includes integrable singular kernel to overcome the initialization problem for related differential equations. We also proposed a numerical approach for the numerical simulation of IVPs incorporating the proposed extended fractional derivatives. The proposed fractional operators, the developed relations and the presented numerical method are expected to be employed in the field of fractional calculus. [ABSTRACT FROM AUTHOR]

In this paper, novel systems of fractional differential equations involving a new generalized Caputo fractional derivative were proposed. The complex dynamic behavior of these systems was studied by numerical simulation. Nonlinear dynamics and chaos in hybrid fractional order systems were investigated using a predictor–corrector algorithm. In particular, the effect of the new generalized fractional derivative parameters on the dynamics of the proposed systems was discussed. The rich variation obtained from the characteristics of the studied systems recommends the implementation of the new generalized derivative in fractional calculus applications.

In this study, an optimal homotopy analysis algorithm is outlined by means of the nonlinear reaction–diffusion systems. This algorithm, the linearization-based algorithm, employs Taylor series approximations of the nonlinear equations to construct an optimal decomposition of the homotopy series solutions. Numerical comparisons between the proposed algorithm and the standard homotopy approach, as tools for analytically solving reaction–diffusion systems, are performed to test the computational efficiency and the pertinent features of the suggested algorithm. The illustrated numerical results demonstrate that the linearization-based algorithm improves the accuracy and the convergence of the homotopy series solutions. The suggested algorithm can be further used to get rapid convergent series solutions for different types of systems of partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the optimized decomposition method, which was developed to solve integer-order differential equations, will be modified and extended to handle nonlinear fractional differential equations. Fractional derivatives will be considered in terms of Caputo sense. The suggested modifications design new optimized decompositions for the series solutions depending on linear approximations of the nonlinear equations. Two optimized decomposition algorithms have been introduced to obtain approximate solutions of broad classes of IVPs consisting of nonlinear fractional ODEs and PDEs. A comparative study was conducted between the proposed algorithms and the Adomian decomposition method by means of some test illustration problems. The implemented numerical simulation results showed that the proposed algorithms give better accuracy and convergence, and reduce the complexity of computational work compared to the Adomian's approach. This confirms the belief that the optimized decomposition method will be used effectively and widely as a powerful tool in solving various fractional differential equations.

This paper is concerned with some features of the modified Caputo-type Mittag-Leffler fractional derivative operator and its associated fractional integral operator. Mainly, new types of solutions for fractional differential equations with Mittag-Leffler kernel are generated based on a numerical algorithm developed in this paper. The suggested algorithm is used to describe the solution behavior of models involving modified Caputo-type Mittag-Leffler fractional derivatives. The results described in this paper are expected to be effectively employed in the area of simulating related fractional models.

•The problem of Q-S synchronization for fractional order systems is investigated.•Synchronization schemes are designed to realize Q-S synchronization behavior.•Numerical simulations are used to demonstrate the effectiveness of the derived results.

Chaotic dynamics and synchronization of fractional-order systems have attracted much attention recently. Based on stability theory of fractional-order systems and stability theory of integer-order systems, this paper deals with the problem of coexistence of various types of synchronization between different dimensional fractional-order chaotic systems. To illustrate the capabilities of the novel schemes proposed herein, numerical and simulation results are given. [ABSTRACT FROM AUTHOR]

In this paper, a new method, called Riccati equation approach, is proposed to seek travelling wave solutions of nonlinear evolution equations. The quasilinear Klein–Gordon equation and the KdV equation are chosen to illustrate the effectiveness and convenience of the suggested method. The obtained solutions which include solitons, kinks and periodic solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The study highlights the power and the effectiveness of the proposed method which can be used in a direct way to handle many other nonlinear evolution equations.

This study expands and modifies the homotopy analysis method to handle differential equations with generalized Caputo-type fractional derivatives. Analytical approximate solutions for such models were successfully provided using the proposed modification. The determination of the valid region of convergence for the proposed method, with respect to the auxiliary control parameter, was discussed when using fractional operators. Then, mainly, the accuracy and effectiveness of the proposed method was verified through illustrative examples and comparisons with the predictor corrector method and RK4 method. Finally, it is expected that the studied generalized operators and the suggested method can be widely applied in the field of fractional calculus.

In this paper, the inverse generalized synchronization problem for different dimensional chaotic dynamical systems in continuous-time is proposed and investigated. New results are derived using new control method and stability theory. Numerical simulations are used to verify the effectiveness of the proposed schemes.

This study provides analytical approximate solutions to classes of nonlinear differential equations with generalized Caputo-type fractional derivatives. The Adomian decomposition method is successfully extended and modified to handle the considered fractional models. Our study displays the useful features of the modified scheme as an effective technique for providing series solutions to differential equations involving the studied fractional derivatives. Analytical solutions to generalized Caputo-type fractional derivative models are discussed and numerical comparisons with a predictor-corrector method are made to verify the applicability, accuracy and efficiency of the method. The influence of the generalized fractional derivative parameters on the dynamics of the studied fractional models is discussed. The used modified method is expected to be effectively employed to handle numerous generalized Caputo-type fractional derivative models.

In this paper, we present an optimal homotopy analysis transform method to deal with nonlinear PDEs. The suggested method constructs an optimal decomposition of the homotopy analysis transform series solutions based on a linear approximation of the nonlinear equation. Numerical comparisons between the suggested approach and the homotopy analysis transform approach, as analytical tools for solving nonlinear differential equations, are performed to examine the features and the computational efficiency of the suggested approach. The illustrative numerical results demonstrate that the series solutions of the proposed approach have better accuracy and convergence than those obtained using the homotopy analysis transform method. The proposed approach can be further implemented to provide rapid convergent series solutions for numerous types of nonlinear differential equations.

Abstract: In this paper, the homotopy perturbation method is implemented to construct solitary solutions for variants of the KdV equations with fractional time derivatives. In this scheme, the solution takes the form of a convergent series with easily computable components. The chosen initial solution or trial function plays a major role in changing the physical structure of the solution. Two models are investigated and the obtained results reveal that the method is very effective and convenient for constructing solitary solutions for nonlinear problems with fractional time derivatives. [Copyright &y& Elsevier]

Abstract: In this paper, we propose algorithms to compute the fractional derivatives of a function by a weighted sum of function values at specified points. The fractional derivatives are considered in the Caputo sense. The error analysis of the algorithms and some illustrative examples are presented. The numerical results confirm that the new algorithms are accurate, efficient and readily implemented. [Copyright &y& Elsevier]

Fractional advection‐dispersion equations are used in groundwater hydrologhy to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper we present two reliable algorithms, the Adomian decomposition method and variational iteration method, to construct numerical solutions of the space‐time fractional advection‐dispersion equation in the form of a rabidly convergent series with easily computable components. The fractional derivatives are described in the Caputo sense. Some examples are given. Numerical results show that the two approaches are easy to implement and accurate when applied to space‐time fractional advection‐dispersion equations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Abstract: In this paper, a modification of He’s homotopy perturbation method is presented. The new modification extends the application of the method to solve nonlinear differential equations of fractional order. In this method, which does not require a small parameter in an equation, a homotopy with an imbedding parameter p ∈[0,1] is constructed. The proposed algorithm is applied to the quadratic Riccati differential equation of fractional order. The results reveal that the method is very effective and convenient for solving nonlinear differential equations of fractional order. [Copyright &y& Elsevier]

This paper presents numerical solutions for the space‐ and time‐fractional Korteweg–de Vries equation (KdV for short) using the variational iteration method. The space‐ and time‐fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals 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. The numerical results show that the approach is easy to implement and accurate when applied to space‐ and time‐fractional KdV equations. The method introduces a promising tool for solving many space–time fractional partial differential equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007

Abstract: In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear 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 fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented. [Copyright &y& Elsevier]

In this paper, we present new ideas for the implementation of homotopy asymptotic method (HAM) to solve systems of nonlinear fractional differential equations (FDEs). An effective computational algorithm, which is based on Taylor series approximations of the nonlinear equations, is introduced to accelerate the convergence of series solutions. The proposed algorithm suggests a new optimal construction of the homotopy that reduces the computational complexity and improves the performance of the method. Some numerical examples are tested to validate and illustrate the efficiency of the proposed algorithm. The obtained results demonstrate the improvement of the accuracy by the new algorithm.

• A fractional Caputo-difference form of Ikeda map is presented. • Chaotic dynamics of fractional Ikeda map is studied. • Control schemes to stabilize and synchronize fractional Ikeda map are discussed. This paper is concerned with a fractional Caputo-difference form of Ikeda map. The dynamics of the proposed map is investigated numerically through phase plots and bifurcation diagrams considered from different perspectives. In addition, a stabilization controller is proposed and the asymptotic convergence of the states is established using the stability theory of linear fractional-order discrete systems. Furthermore, a new synchronization scheme is introduced whereby a new 2D fractional-order chaotic map is considered as the master system and the fractional-order Ikeda map is considered as the response system. Experimental investigations and numerical simulations are also provided to confirm the main findings of the study. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper [Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order, Appl Math Comput. submitted for publication] the authors presented a new generalization of the differential transform method that would extended the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form with combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization. [Copyright &y& Elsevier]