Splitting Decomposition Homotopy Perturbation Method To Solve One -Dimensional Navier -Stokes Equation

Abstract

I S S N 2347-1921 V o l u m e 13 N u m b e r 2 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 7123 | P a g e A p r i l 2 0 1 7 w w w . c i r w o r l d . c o mSplitting decomposition homotopy perturbation method to solve onedimensional Navier-Stokes equation 1-IntroductionNavier-Stokes equations are non-linear partial differential equations that govern the incompressible fluid flow inside specific hollow as it is an equation of parabolic type and the non-linear degree is increased with the increase of Reynolds number . These equations names are taken from the two physicists Claude-Louis Navier and George Gabriel Stokes in the nineteenth century, and resulted from applying Newton second law on the movement of fluids supposing that stress of fluid is totaled of divergence of viscosity (proportional with velocity rate) and pressure. Also, these equations are considered as the most important physical equations which describe a large number of phenomena of different applications in many research fields that may be used in modeling weather, liquid flow in channels and pipes , gas flow round flying bodies, and movement of stars in the galaxy.Navier-Stokes equations are important mathematically, due to their wide applications, where to this day no one has succeeded in proving the existence of general solution or a permanent for Navier-Stokes equations( privately threedimensional). Therefore, this kind of problems is named as the existence and smooth flow of Navier-Stokes. During few last years, many researchers attempted to modify and extend these two methods; for example, Zhang [14 ] modified ADM to solve a class of non-linear boundary value problems. Jafari and Daftardar-Gejji [15] modified ADM to solve a set of non-linear equations led to find approximation solutions are better than those which were obtained by standrad method in measurements of convergent and accuracy. Luo [16] suggested active methods for ADM which is a two-steps Adomian decomposition method (TSADM) to reach the solution. TSADM reduces the repetitive of mathematical processes that are applied to find the solution and also he made comparison for the results. The results showed that TSADM is an active and efficient method which has high accuracy in finding solutions. Also, in many works [17,18,19,20], the authors used ADM to found analytic and approximate solutions for different problems. In the same direction of modification, the HPM is active to find solutions for non-linear equations [10,[21][22][23][24][25][26][27][28][29][30].Depending on the above literature review for attempt of researchers to expand and developed ADM and HPM to solve linear and nonlinear boundary value problems, and depending on our simple information about applications of these methods to solve the problems that under consideration study, we did not find, if any, that is for problems are different from our problem in formulations and conditions. These information encourage us to suggest a new methodology for solving nonlinear problems. The new methodology is con...