Shaher Momani

Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method

Modeling of uncertainty differential equations is very important issue in applied sciences and engineering, while the natural way to model such dynamical systems is to use fuzzy differential equations. In this paper, we present a new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differentiability. The analytic and approximate solutions are given with series form in terms of their parametric form in the space

Approximate analytical solution of the nonlinear fractional KdV-Burgers equation

W_2^2 [a,b]\oplus W_2^2 [a,b].

Control and switching synchronization of fractional order chaotic systems using active control technique

W22[a,b]⊕W22[a,b]. The method used in this paper has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this paper show potentiality, generality, and superiority of our method as compared with other well-known methods.

Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.

New Results on Fractional Power Series: Theories and Applications

In this paper, explicit and approximate solutions of the nonlinear fractional KdV-Burgers equation with time-space-fractional derivatives are presented and discussed. The solutions of our equation are calculated in the form of rabidly convergent series with easily computable components. The utilized method is a numerical technique based on the generalized Taylor series formula which constructs an analytical solution in the form of a convergent series. Five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method. Graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution. The results reveal that the method is very effective and simple in determination of solution of the fractional KdV-Burgers equation.

Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations

In this paper, we generalize the integer-order cable model of the neuron system into the fractional-order domain, where the long memory dependence of the fractional derivative can be a better fit for the neuron response. Furthermore, the chaotic synchronization with a gap junction of two or multi-coupled-neurons of fractional-order are discussed. The circuit model, fractional-order state equations and the numerical technique are introduced in this paper for individual and multiple coupled neuron systems with different fractional-orders. Various examples are introduced with different fractional orders using the non-standard finite difference scheme together with the Grunwald-Letnikov discretization process which is easily implemented and reliably accurate.

A novel expansion iterative method for solving linear partial differential equations of fractional order

This paper discusses the continuous effect of the fractional order parameter of the Lü system where the system response starts stable, passing by chaotic behavior then reaching periodic response as the fractional-order increases. In addition, this paper presents the concept of synchronization of different fractional order chaotic systems using active control technique. Four different synchronization cases are introduced based on the switching parameters. Also, the static and dynamic synchronizations can be obtained when the switching parameters are functions of time. The nonstandard finite difference method is used for the numerical solution of the fractional order master and slave systems. Many numeric simulations are presented to validate the concept for different fractional order parameters.

The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics

In this paper, we investigate the analytic and approximate solutions of second-order, two-point fuzzy boundary value problems based on the reproducing kernel theory under the assumption of strongly generalized differentiability. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation in the space