Thestudy of free-surface and pressurized water flows in channels has many interesting application, one of the most important being\nthemodeling of the phenomena in the area of natural water systems (rivers, estuaries) aswell as in that of man-made systems (canals,\npipes). For the development ofmajor river engineering projects, such as flood prevention and flood control, there is an essential need\nto have an instrument that be able to model and predict the consequences of any possible phenomenon on the environment and\nin particular the new hydraulic characteristics of the system. The basic equations expressing hydraulic principles were formulated\nin the 19th century by Barre de Saint Venant and Valentin Joseph Boussinesq. The original hydraulic model of the Saint Venant\nequations is written in the form of a system of two partial differential equations and it is derived under the assumption that the\nflow is one-dimensional, the cross-sectional velocity is uniform, the streamline curvature is small and the pressure distribution is\nhydrostatic. The St. Venant equations must be solved with continuity equation at the same time. Until now no analytical solution\nfor Saint Venant equations is presented. In this paper the Saint Venant equations and continuity equation are solved with homotopy\nperturbation method (HPM) and comparison by explicit forward finite difference method (FDM). For decreasing the present error\nbetween HPM and FDM, the st.venant equations and continuity equation are solved by HAM. The homotopy analysis method\n(HAM) contains the auxiliary parameter ? that allows us to adjust and control the convergence region of solution series.The study\nhas highlighted the efficiency and capability of HAMin solving Saint Venant equations andmodeling of unsteady flow through the\nrectangular canal that is the goal of this paper and other kinds of canals.
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