The new types of wave solutions of the Burger’s equation and the Benjamin-Bona-Mahony equation

In this article,we suggest the two variable(G/G,1/G)-expansion method for extracting further general closed form wave solutions of two important nonlinear evolution equations(NLEEs)that model one-dimensional internal waves in deep water and the long surface gravity waves of small amplitude propagating uni-directionally.The method can be regarded as an extension of the(G/G)-expansion method.The ansatz of this extension method to obtain the solution is based on homogeneous balance between the highest order dispersion terms and nonlinearity which is similar to the(G/G)method whereas the auxiliary linear ordinary differential equation(LODE)and polynomial solution differs.We applied this method to find explicit form solutions to the Burger’s and Benjamin-Bona-Mahony(BBM)equations to examine the effectiveness of the method and tested through mathematical computational software Maple.Some new exact travelling wave solutions in more general form of these two nonlinear equations are derived by this extended method.The method introduced here appears to be easier and faster comparatively by means of symbolic computation system.

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

In this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method.Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation,nonlinear fractional discrete Schrodinger equation,fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation.The obtained solution is verified by comparison with exact solution whenα=1.

The double auxiliary equations method and its application to space-time fractional nonlinear equations

This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differential equation.The proposed scheme has been successfully applied on two very important evolution equations,the space-time fractional differential equation governing wave propagation in low-pass electrical transmission lines equation and the time fractional Burger’s equation.The obtained results show that the proposed method is more powerful,promising and convenient for solving nonlinear fractional differential equations(NFPDEs).To our knowledge,the solutions obtained by the proposed method have not been reported in former literature.

Wave solutions and numerical validation for the coupled reaction-advection-diffusion dynamical model in a porous medium

The current study examines the special class of a generalized reaction-advection-diffusion dynamical model that is called the system of coupled Burger’s equations.This system plays a vital role in the essential areas of physics,including fluid dynamics and acoustics.Moreover,two promising analytical integration schemes are employed for the study;in addition to the deployment of an efficient variant of the eminent Adomian decomposition method.Three sets of analytical wave solutions are revealed,including exponential,periodic,and dark-singular wave solutions;while an amazed rapidly convergent approximate solution is acquired on the other hand.At the end,certain graphical illustrations and tables are provided to support the reported analytical and numerical results.No doubt,the present study is set to bridge the existing gap between the analytical and numerical approaches with regard to the solution validity of various models of mathematical physics.