Marwan Alquran

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.

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.

A New Simplified Bilinear Method for the N-Soliton Solutions for a Generalized FmKdV Equation with Time-Dependent Variable Coefficients

Abstract In this paper a generalized fractional modified Korteweg–de Vries (FmKdV) equation with time-dependent variable coefficients, which is a generalized model in nonlinear lattice, plasma physics and ocean dynamics, is investigated. With the aid of a simplified bilinear method, fractional transforms and symbolic computation, the corresponding N-soliton solutions are given and illustrated. The characteristic line method and graphical analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons and elastic interactions. Finally, the resonance phenomenon for the equation is examined.

Effective approximate methods for strongly nonlinear differential equations with oscillations

PurposeThis paper proposes the use of different analytical methods in obtaining approximate solutions for nonlinear differential equations with oscillations.MethodsThree methods are considered in this paper: Lindstedt-Poincare method, the Krylov-Bogoliubov first approximate method, and the differential transform method.ResultsFigures that are given in this paper give a strong evidence that the proposed methods are effective in handling nonlinear differential equations with oscillations.ConclusionsThis study reveals that the differential transform method provides a remarkable precision compared with other perturbation methods.

Bifurcations of the time-fractional generalized coupled Hirota-Satsuma KdV system

Abstract The Hirota-Satsuma model with fractional derivative is considered to provide some characteristics of memory embedded into the system. The modified system is analyzed analytically using a new technique called residual power series method. We observe thatwhen the value of memory index (time-fractional order) is close to zero, the solutions bifurcate and produce a wave-like pattern.

Solitonic Solutions for Homogeneous KdV Systems by Homotopy Analysis Method

We study two-component evolutionary systems of a homogeneous KdV equations of second and third order. The homotopy analysis method (HAM) is used for analytical treatment of these systems. The auxiliary parameter h of HAM is freely chosen from the stability region of the h-curve obtained for each proposed system.

On the comparison of perturbation-iteration algorithm and residual power series method to solve fractional Zakharov-Kuznetsov equation

Abstract In this paper, we present analytic-approximate solution of time-fractional Zakharov-Kuznetsov equation. This model demonstrates the behavior of weakly nonlinear ion acoustic waves in a plasma bearing cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Basic definitions of fractional derivatives are described in the Caputo sense. Perturbation-iteration algorithm (PIA) and residual power series method (RPSM) are applied to solve this equation with success. The convergence analysis is also presented for both methods. Numerical results are given and then they are compared with the exact solutions. Comparison of the results reveal that both methods are competitive, powerful, reliable, simple to use and ready to apply to wide range of fractional partial differential equations.

A modified algorithm for the Clenshaw-Curtis method

Abstract In this paper, we introduce a modified algorithm for the Clenshaw-Curtis (CC) quadrature formula. The coefficients of the formula are approximated by using a finite linear combination of Legendre polynomials in the Least Squares sense to make the CC method well disposed for numerical solution of the definite integral. We design and analyze the modified CC algorithm. Numerical examples are given to demonstrate the efficiency and the accuracy of the proposed algorithm. The obtained results show that the proposed algorithm perform better and much more efficient than the classical Clenshaw-Curtis method.

Convergence and norm estimates of Hermite interpolation at zeros of Chevyshev polynomials

In this paper, we investigate the simultaneous approximation of a function f(x) and its derivative