Ahmed Elsaid

A new algorithm for the decomposition solution of nonlinear differential equations

In this paper, we introduce a new algorithm for applying the Adomian decomposition method to nonlinear differential and partial differential equations. The proposed algorithm does not require the explicit evaluation of Adomian polynomials to approximate the nonlinearity term. The numerical experiments show that the proposed algorithm is more accurate at larger values for time, though at the cost of more computations.

Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system

Abstract This paper is devoted to introduce a new four-dimensional hyperchaotic system. Existence and uniqueness of the solution of the proposed system are proved. Continuous dependence on initial conditions of the system’s solution and some stability conditions of system’s equilibrium points are studied. The existence of pitchfork bifurcation is demonstrated by using the center manifold theorem and the local bifurcation theory. The Hopf bifurcation is examined in double parameters bifurcation diagrams along with degenerate types of Hopf bifurcations. The rich dynamical behaviors of the system are explored, then circuit implementation of the system is proposed. Numerical simulations are carried out to verify theoretical analysis.

The variational iteration method for solving Riesz fractional partial differential equations

In this paper, the variational iteration method is applied to obtain the solution for space fractional partial differential equations where the space fractional derivative is in the Riesz sense. On the basis of the properties and definition of the fractional derivative, the iterative technique is carried out in a straightforward manner without the need for transforms or numerical approximations. Examples demonstrate that the series solution obtained shows agreement with the exact solutions of the problems solved.

Fractional differential transform method combined with the Adomian polynomials

Abstract A modification of the fractional differential transform method (FDTM) for solving nonlinear fractional differential equations (FDEs) is presented. In this technique, the nonlinear term is replaced by its Adomian polynomial of index k . Then the dependent variable components are replaced in the recurrence relation by their corresponding differential transform components of the same index. Thus nonlinear FDEs can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the Adomian polynomials. Numerical examples with different types of nonlinearities are solved and good results are obtained.

A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains

Abstract In this paper, a homotopy perturbation technique is proposed to solve a class of initial-boundary value problems of partial differential equations of arbitrary (fractional) orders over finite domains. The basic idea of this technique is to utilize both the initial and boundary conditions in the recursive relation of the solution scheme so that we can obtain a good approximate solution. Numerical examples are presented to illustrate the validity of the proposed technique.

A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders

The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.

Solving the 2-D heat equations using wavelet-Galerkin method with variable time step

A wavelet-Galerkin method to solve nonhomogenous 2-D heat equations in finite rectangular domains is presented. Integrals involving nonhomogenity terms with scaling function are evaluated based on available connection coefficients. Variable time step technique is used to delay the error blow up hence obtaining better results for solutions that reach steady state.

Hyperchaotic Fractional-Order Systems and Their Applications

Research about fractional-order hyperchaotic systems gains a lot of interest from both theoretical and applied point of view. Some fractional-order hyperchaotic systems have been investigated, such as the fractional-order hyperchaotic Rossler system and the fractional-order hyperchaotic Chen system. Recent publications in this area include nonlinear circuits, secure communication, laser applications, spread spectrum communication, communication in star coupled network, video encryption communication, color image encryption algorithm, and applications of different types of synchronization. We are pleased to announce the publication of this special issue focusing on novel topics in hyperchaotic fractional-order systems and their applications. The main objective of this special issue is to provide an opportunity to study the new developments related to novel chaotic systems, synchronization schemes, bifurcations, and control in hyperchaotic fractional-order systems along with their applications. Among the articles that were submitted for review, our editorial team has selected seven articles for publication. These articles cover the topics of adaptive fuzzy synchronization, image encryption algorithm, dynamical analysis of a novel hyperchaotic system, eigenvalue problems, BAM neural networks with distributed delays and impulses, complex synchronization scheme between integer-order and fractional-order chaotic systems with different dimensions, and fractional-order FPGA implementation. We are confident that this special issue advances the understanding and research of hyperchaotic fractional-order systems and their applications.