Volume 65, Issue 12

In this paper, the homotopy perturbation method (HPM) is developed to obtain numerical solutions of linear and nonlinear boundary value problems for fourth-order fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons show that the homotopy perturbation method is very effective and convenient and overcome the difficulty of traditional methods. Numerical examples are presented to illustrate the efficiency, simplicity, and reliability of the method.

In this paper, we develop the analytical solution of the Navier-Stokes equations for a semi-infinite rectangular channel with porous and uniformly expanding or contracting walls by employing the homotopy perturbation method (HPM). The series solution of the governing problem is obtained. Some examples have been included. The results so obtained are compared with the existing literature and a remarkable improvement leads to an excellent agreement with the numerical results.

In this paper, suitable transformations and a so-called exp-function method are used to obtain different types of exact solutions for some nonlinear evolution equations with variable coefficients and nonlinear terms of any orders. The Korteweg-de Vries equation and the Burgers equation with nonlinear terms of any orders are chosen to show how to apply the exp-function method for these kinds of nonlinear equations. These exact solutions are in full agreement with the previous results obtained by Ebaid and by Zhu.

January 21, 2010 In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving delay differential equations which are otherwise too difficult to solve. These equations arise very frequently in signal processing, digital images, physics, and applied sciences. Numerical results reveal the complete reliability and efficiency of the proposed combination.

In this paper, the exp-function method is applied by using symbolic computation to construct a variety of new generalized solitonary and periodic solutions for the Burgers-Kadomtsev-Petviashvili and Vakhnenko equations with distinct physical structures. The results reveal that the exp-function method is very effective and powerful for solving nonlinear evolution equations in mathematical physics.

Although a variational iteration algorithm was proposed by Yildirim (Math. Prob. Eng. 2008 (2008), Article ID 869614) that successfully solves differential-difference equations, the method involves some repeated and unnecessary iterations in each step. An alternative iteration algorithm (variational iteration algorithm-II) is constructed in this paper that overcomes this shortcoming and promises to provide a universal mathematical tool for many differential-difference equations.

In this paper, we have extended the homotopy perturbation method (HPM) to find approximate analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Korteweg-de Vries (mKdV) lattice equation and the discretized nonlinear Schr¨odinger equation are taken as examples to demonstrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons are made between the results of the presented method and exact solutions. The obtained results reveal that the HPM is a very effective and convenient tool for solving such kind of equations.

In this article, a generalized (G´/G)-expansion method is used to find exact travelling wave solutions of the Burgers equation and the Korteweg-de Vries (KdV) equation with variable coefficients. As a result, hyperbolic, trigonometric, and rational function solutions with parameters are obtained. When these parameters are taking special values, the solitary wave solutions are derived from the hyperbolic function solution. It is shown that the proposed method is direct, effective, and can be applied to many other nonlinear evolution equations in mathematical physics.

The present note investigates the influence of slip on the generalized Couette flows of a third-grade fluid. Two flow problems are considered. The resulting equations and the boundary conditions are nonlinear. Analytical solutions of the governing nonlinear problems are found in closed form.

In this study, we use the homotopy perturbation method (HPM) to solve an initial-boundary value problem with a non-classic condition for the one-dimensional wave equation.We will deal with a new type of non-local boundary value problems which are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The method is very reliable and effective and provides the solution in terms of rapid convergent series. Several examples are tested to support our study.

This communication is concerned with the nonlinear flows of an Oldroyd 8-constant fluid when the no-slip condition is not valid. Due to slip effects in terms of shear stress, the arising slip conditions are nonlinear. The resulting mathematical problems involves nonlinear differential equations and nonlinear boundary conditions. To the best of my knowledge, no such analysis for the flows of an Oldroyd 8-constant fluid is available in the literature. Graphs are plotted for the velocity profiles and examined with respect to the sundry emerging parameters.

In this paper, viscous flow with a second-order slip condition over a permeable stretching surface is solved analytically. The current work differs from the previous studies in the application of a new second-order slip velocity model. The closed form solution reported is an exact solution of the full governing Navier-Stokes equations. The effects of slip and mass transfer parameters are discussed.

In this paper, we present a reliable modification of the Adomian decomposition method for solving higher-order singular boundary value problems. He’s polynomials are also used to overcome the complex and difficult calculation of Adomian polynomials occurring in the application of the Adomian decomposition method. Numerical examples are given to illustrate the accuracy and efficiency of the presented method, revealing its reliability and applicability in handling the problems with singular nature.

In this work we present a reliable treatment for two (2+1)-dimensional Korteweg-de Vries-like and Kadomtsev-Petviashvili-like equations. The Hirota bilinear method will be used to show that these two equations are not completely integrable equations. Unlike the completely integrable Korteweg-de Vries and Kadomtsev-Petviashvili equations, where multiple soliton solutions exist, only one-soliton and two-soliton solutions can be derived for each of the Korteweg-de Vries-like and Kadomtsev- Petviashvili-like equations.

In this study, we investigate the magnetohydrodynamic (MHD) boundary layer flow by employing the homotopy perturbation method (HPM) and Pad´e approximation. The series solution of the governing nonlinear problem is developed. Generally, the truncated series solution is adequate only in a small region when the exact solution is not reached. We overcame this limitation by using the Pad´e techniques, which have the advantage in turning the polynomials approximation into a rational function, and applied it to the series solution to improve the accuracy and enlarge the convergence domain. A comparison of the present solution with the existing solution is made and excellent agreement is noted.

This paper investigates the magnetohydrodynamic (MHD) boundary layer flow of a Jeffery fluid induced by a nonlinearly stretching sheet with mass transfer. The relevant system of partial differential equations has been reduced into ordinary differential equations by employing the similarity transformation. Series solutions of velocity and concentration fields are developed by using the homotopy analysis method (HAM). Effects of the various parameters such as Hartman number, Schmidt number, and chemical reaction parameter on velocity and concentration fields are discussed by presenting graphs. Numerical values of the mass transfer coefficient are also tabulated and analyzed.

An analysis has been carried out for peristaltic flow and heat transfer of a Carreau fluid in an asymmetric channel with slip effect. The governing problem is solved under long wavelength approximation. The variations of pertinent dimensionless parameters on temperature are discussed. Pumping and trapping phenomena are studied.

The influence of slip on the magnetohydrodynamic (MHD) peristaltic flow in a planar channel with compliant walls is examined. An incompressible Maxwell fluid saturates the porous medium. An established solution is valid for small wave number. The mathematical expression of the stream function is presented. Several interesting flow parameters are sketched and examined.

In this paper, we apply He’s variational iteration method (VIM) coupled with an auxiliary parameter which proves very effective to control the convergence region of an approximate solution. Moreover, a convenient way is considered for choosing a suitable auxiliary parameter via a residual function. The proposed algorithm is tested on some nonlinear age-structured population models, and numerical results explicitly reveal the complete reliability, efficiency, and accuracy of the suggested technique. It is observed that the approach may be implemented on other nonlinear models of physical nature.

The quantum screening effects on the entanglement fidelity for the elastic collision are investigated in electron-ion quantum plasmas. The partial wave analysis and modified Debye-Hückel interaction potential are employed to obtain the entanglement fidelity function in electron-ion quantum plasmas as a function of the collision energy, charge of the ion, and quantum wave number. It is found that the quantum screening effects significantly enhance the entanglement fidelity in electron-ion quantum plasmas. It is also found that the entanglement fidelity increases with an increase of the ion charge. The quantum screening effects on the entanglement fidelity is also found to be increased with increasing plasma density. In addition, it is found that the quantum screening effects decreases with an increase of the collision energy.

The finite size effects of the charged dust grain on the electron-dust grain collisions are investigated in complex dusty plasmas. The stationary phase analysis and the effective potential due to the renormalized dust charge are employed to obtain the phase shift for the scattering of the electron and the spherically charged dust grain as a function of the impact parameter, collision energy, Debye length, and dust radius. It is found that the size effect of the dust grain enhances the electron-dust grain scattering cross section in dusty plasmas. It is also found that the size effect on the scattering cross section increases with increasing plasma density. In addition, it is found that the size effect on the electron-dust scattering cross section decreases with an increase of the plasma temperature.

In this paper, a new projective equation is used to obtain the variable separation solutions with two arbitrary functions of the (2+1)-dimensional Broek-Kaup system (BKK). Based on the derived solitary wave solutions and by selecting appropriate functions, some novel localized excitations such as fusion and annihilation of solitary waves are investigated.

Self assembly systems formed by malic acid and alkyloxy benzoic acids are characterized. The ferroelectric ingredient malic acid formed double hydrogen bond with p-n-alkyloxy benzoic acids. Various hydrogen bonded complexes have been synthesized with malic acid and pentyl to dodecyloxy benzoic acid, respectively. Fourier transformation infrared (FTIR) studies confirm the hydrogen bond formation. Polarizing optical microscopic (POM) studies revealed the textural information while the transition and enthalpy values are calculated from differential scanning calorimetry (DSC) studies. A phase diagram has been constructed from the POMand DSC studies. A new smectic ordering, smectic X*, has been identified which exhibits a finger print type texture. This phase has been characterized by POM, DSC, helix, and tilt angle studies. The transition from traditional cholesteric to smectic X* phase is observed to be first order. The tilt angle data in this phase has been fitted to a power law and the temperature variation of the tilt angle follows mean field theory predictions. The results of FTIR, POM, DSC, tilt angle, and helicoidal studies are discussed.

A numerical simulation is performed to study a completely open cavity based on left-handed materials. This cavity can restrict electromagnetic waves with no surrounding reflective wall. A closed path with zero optical paths is formed. Due to the effect of resonance, this cavity can be used as a filter.