Mohamed M. Khader

Numerical solution of two-sided space-fractional wave equation using finite difference method

In this paper, a class of finite difference method for solving two-sided space-fractional wave equation is considered. The stability and consistency of the method are discussed by means of Gerschgorin theorem and using the stability matrix analysis. Numerical solutions of some wave fractional partial differential equation models are presented. The results obtained are compared to exact solutions.

A CHEBYSHEV PSEUDO-SPECTRAL METHOD FOR SOLVING FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method. Finally, some numerical examples are provided to show that this method is computationally efficient, and a comparison is made with existing results. doi:10.1017/S1446181110000830

Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method

The purpose of this study is to introduce a modification of the homotopy perturbation method using Laplace transform and Pade approximation to obtain closed form solutions of nonlinear coupled systems of partial differential equations. Two test examples are given; the coupled nonlinear system of Burger equations and the coupled nonlinear system in one dimensional thermoelasticity. The results obtained ensure that this modification is capable of solving a large number of nonlinear differential equations that have wide application in physics and engineering.

Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays

A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert fractional Logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques.

Approximate solutions to the nonlinear vibrations of multiwalled carbon nanotubes using Adomian decomposition method

Abstract This paper applies the Adomian decomposition method (ADM) to the search for the approximate solutions to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals inter layer forces. The amplitude–frequency curves for large-amplitude vibrations of single-walled, double-walled and triple-walled carbon nanotubes are obtained. The influence of changes in material constants of the surrounding elastic medium and the effect of changes in nanotube geometrical parameters on the vibration characteristics are studied by comparing the results with those from the open literature. This method needs less work in comparison with the traditional methods and decreases considerable volume of calculation, and it’s powerful mathematical tool for solving wide class of nonlinear differential equations. Special attention is given to prove the convergence of the method. Some examples are given to illustrate the determination approximate solutions of the proposed problem.

On the convergence of variational iteration method for nonlinear coupled system of partial differential equations

In this paper, the sufficient conditions that guarantee the convergence of the variational iteration method when applied to solve a coupled system of nonlinear partial differential equations are presented. Especial attention is given to the error bound of the nth term of the resultant sequence. Numerical examples to show the efficiency of the method are presented.

An efficient approximate method for solving linear fractional Klein–Gordon equation based on the generalized Laguerre polynomials

In this paper, a new approximate formula of the fractional derivative is derived. The proposed formula is based on the generalized Laguerre polynomials. Global approximations to functions defined on a semi-infinite interval are constructed. The fractional derivatives are presented in terms of Caputo sense. Special attention is given to study the error and the convergence analysis of the proposed formula. A new spectral Laguerre collocation method is presented for solving linear fractional Klein–Gordon equation (LFKGE). The properties of Laguerre polynomials are utilized to reduce LFKGE to a system of ordinary differential equations, which solved using the finite difference method. Numerical results are provided to confirm the theoretical results and the efficiency of the proposed method.

Numerical simulation of fractional Cable equation of spiny neuronal dendrites

In this article, numerical study for the fractional Cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. A simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method.

Melting phenomenon in magneto hydro-dynamics steady flow and heat transfer over a moving surface in the presence of thermal radiation

The Lie group method is applied to present an analysis of the magneto hydro-dynamics (MHD) steady laminar flow and the heat transfer from a warm laminar liquid flow to a melting moving surface in the presence of thermal radiation. By using the Lie group method, we have presented the transformation groups for the problem apart from the scaling group. The application of this method reduces the partial differential equations (PDEs) with their boundary conditions governing the flow and heat transfer to a system of nonlinear ordinary differential equations (ODEs) with appropriate boundary conditions. The resulting nonlinear system of ODEs is solved numerically using the implicit finite difference method (FDM). The local skin-friction coefficients and the local Nusselt numbers for different physical parameters are presented in a table.

Nonlinear focusing Manakov systems by variational iteration method and adomian decomposition method

In this paper, the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve numerically the focusing Manakov systems of coupled nonlinear Schrodinger equations. The accuracy of the methods are verified by ensuring that the conserved quantities remain almost constant. The results show that VIM is much easier, more convenient, more stable and efficient than ADM.