New impressive performances for the analytical solutions to the (1

This article includes the extended simple equation method (ESEM) and the Paul-Painleve & PRIME; approach method (PPAM), two different methodologies, to offer a variety of novel analytical performances for the (1 + 1)dimensional Van Der Waals (VDW) gas system. The variational iteration method (VIM), whose initial condition will be obtained from these solutions, will also be used to construct the appropriate numerical solutions for all achieved exact solutions. This article aims to provide abundant new performances of the analytical solutions to the VDW gas system through transforming this coupled system to one equation in terms of the gas volume that is sufficient to get the velocity of gas flow by calculating the wave speed C.The suggested model represents the equation of ideal gas state law that extends to contain the effects of interaction between gas molecules, as well as calculating the finite size of molecules. Furthermore, it describes the one dimensional (1D) longitudinal isothermal motion in elastic bars or fluids. Diverse class of analytical solutions has been extracted by using these inspective methods that are arranged with the same vein, in sequence. Our achieved solutions will contribute not only to understand this important model but also for all related phenomena. A brief comparison is made between our exact and numerical solutions as well as those previously acquired by other researchers who employed other methodologies.