Kamel Al-Khaled

Numerical solutions for systems of fractional differential equations by the decomposition method

In this paper we use Adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi-term fractional differential equation by reducing it to a system of fractional equations each of order at most unity. We begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method. Moreover, we show how the method can be applied to a general linear multi-term equation and solve several applied problems.

Numerical approximations for population growth models

This paper aims to introduce a comparison of Adomian decomposition method and Sinc-Galerkin method for the solution of some mathematical population growth models. From the computational viewpoint, the comparison shows that the Adomian decomposition method is efficient and easy to use.

Numerical study of Fisher's reaction-diffusion equation by the Sinc collocation method

Fishers equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by the Sinc collocation method. The derivatives and integrals are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The error in the approximation of the solution is shown to converge at an exponential rate. Numerical examples are given to illustrate the accuracy and the implementation of the method, the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are independent of the initial values.

Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations

We consider solitary-wave solutions of generalized Benjamin-Bona-Mahony-Burgers Equations (shortly BBMB). The decomposition method is proposed for the numerical solution subject to appropriate initial condition. Soliton solutions are constructed to show the nature of the solution. The numerical solutions of our model equation are calculated in the form of convergence power series with easily computable components. The decomposition method performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.

Mathematical methods for a reliable treatment of the (2+1)-dimensional Zoomeron equation

PurposeThis paper investigates an analytical solution to a physical model called (2 + 1)-dimensional Zoomeron equation.MethodsThe solutions of Zoomeron are obtained using direct methods such as the extended tanh, the exponential function and the sechp−tanhpfunction methods.ResultsSeveral soliton solutions are obtained using the proposed methods.ConclusionsThe obtained solutions are new, and each has its own structure.

SINC AND SOLITARY WAVE SOLUTIONS TO THE GENERALIZED BENJAMIN–BONA–MAHONY–BURGERS EQUATIONS

In this paper, we consider the generalized Benjamin–Bona–Mahony–Burgers (BBMB) equations. A variety of exact solutions to the BBMB equations are developed by means of the tanh method. A sinc-Galerkin procedure is also developed to solve the BBMB equations. Sinc approximations to both the derivatives and the indefinite integrals reduce the system to an explicit system of algebraic equations. It is shown that sinc-Galerkin approximations produce an error of exponential order. A comparison of the two methods for solving the BBMB equation was made regarding their solutions. The study outlines the features of the sinc method.

Sinc numerical solution for solitons and solitary waves

Abstract A numerical scheme using Sinc–Galerkin method is developed to approximate the solution for the Korteweg–de Vries model equation. Sinc approximation to both derivatives and indefinite integral reduce the integral equation to an explicit system of algebraic equations, then using various properties of Sinc functions, it is shown that the Sinc solution produce an error of order O ( exp (−c/h)) for some positive constants c , h . The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.

The extended tanh method for solving systems of nonlinear wave equations

Abstract The extended tanh method with a computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. The applied method will be used to solve the generalized coupled Hirota Satsuma KdV equation.

The tanh and sine–cosine methods for higher order equations of Korteweg–de Vries type

The aim of this paper is twofold. Firstly, the tanh method with the aid of Mathematica is used to obtain exact soliton solutions for a new fifth-order nonlinear integrable evolution equation. Secondly, the sine–cosine and the rational sine–cosine methods are proposed for constructing more general exact solutions of the soliton type for two nonlinear evolution equations arising in nonlinear science and theoretical physics, namely the symmetric regularized long wave equation and a new model of the Korteweg–de Vries type, which gives a more realistic version of shallow water waves.

Decomposition method for solving nonlinear integro-differential equations

This paper outlines a reliable strategy for solving nonlinear Volterra-Fredholm integro-differential equations. The modified form of Adomian decomposition method is found to be fast and accurate. Numerical examples are presented to illustrate the accuracy of the method.