M A Abdou

Decomposition method for solving a system of coupled fractional-time nonlinear equations

The Adomian decomposition method (ADM) has been successively used to find the explicit and numerical solutions of the time fractional of a coupled nonlinear fractional equations. Four models of special interest with fractional time derivative of order α, 0<α<1 are considered and solved by means of ADM. The behaviour of Adomian solutions and the effects of different values of α are shown graphically. Numerical examples are tested to illustrate the pertinent feature of the proposed algorithm.

New applications of He's homotopy perturbation method for nonlinear differential difference equations

We extend Hes homotopy perturbation method (HPM) with a computerized symbolic computation to find approximate and exact solutions for nonlinear differential difference equations (DDEs) arising in physics. The results reveal that the method is very effective and simple. We find that the extended method for nonlinear DDEs is of good accuracy. To illustrate the effectiveness and the advantage of the proposed method, three models of nonlinear DDEs of special interest in physics are chosen, namely, the hybrid equation, the Toda lattice equation and the relativistic Toda lattice difference equation. Comparisons are made between the results of the proposed method and exact solutions. The results show that the HPM is an attractive method for solving the DDEs.

The Variational-Iteration Method to Solve the Nonlinear Boltzmann Equation

The time-dependent nonlinear Boltzmann equation, which describes the time evolution of a single-particle distribution in a dilute gas of particles interacting only through binary collisions, is considered for spatially homogeneous and inhomogeneous media without external force and energy source. The nonlinear Boltzmann equation is converted to a nonlinear partial differential equation for the generating function of the moments of the distribution function. The variational-iteration method derived by He is used to solve the nonlinear differential equation of the generating function. The moments for both homogeneous and inhomogeneous media are calculated and represented graphically as functions of space and time. The distribution function is calculated from its moments using the cosine Fourier transformation. The distribution functions for the homogeneous and inhomogeneous media are represented graphically as functions of position and time.

The solution of a coupled system of nonlinear physical problems using the homotopy analysis method

In this article, the homotopy analysis method (HAM) has been applied to solve coupled nonlinear evolution equations in physics. The validity of this method has been successfully demonstrated by applying it to two nonlinear evolution equations, namely coupled nonlinear diffusion reaction equations and the (2+1)-dimensional Nizhnik?Novikov Veselov system. The results obtained by this method show good agreement with the ones obtained by other methods.The proposed method is a powerful and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The basic ideas of this approach can be widely employed to solve other strongly nonlinear problems.

New Exact Travelling Wave Solutions of Nonlinear Coagulation Problem with Mass Loss

This paper suggests a generalized F-expansion method for constructing new exact travelling wave solutions of a nonlinear coagulation problem with mass loss. This method can be used as an alternative to obtain analytical and approximate solutions of different types of kernel which are applied in physics. The nonlinear kinetic equation, which is an integro differential equation, is transformed into a differential equation using Laplace’s transformation. The inverse Laplace transformation of the solution gives the size distribution function of the system. As a result, many exact travelling wave solutions are obtained which include new periodic wave solutions, trigonometric function solutions, and rational solutions. The method is straightforward and concise,and it can also be applied to other nonlinear evolution equations arising in mathematical physics.