الفهرس 
INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
In the nineteenth century, Sophus Lie introduced a standard method, Lie symmetry transformation, to study the solutions of ordinary and partial differential equations. The mathematical technique of Lie symmetry transformation reduces the order of an ordinary differential equation or dimension of a partial differential equation(s) by one based on the study of the invariance condition under one-parameter Lie group transformation by applying simple assumptions and steps. In this research, a family of invariant solutions of multiples of physical problems, the motion governing equations a natural convective unsteady flow past to a non-isothermal vertical surface and heat transfer equation of lake Tahoe, will be obtained using Lie’s group methods and the results will be evaluated versus other solutions obtained using other symmetries methods including but not limited to translation and scaling method. Lie symmetry group transformation is used to investigate the partial differential equations that model the motion of a natural convective unsteady flow past a non-isothermal vertical surface. The one-parameter Lie group transformation is applied twice consecutively to convert the motion governing equations into a system of ordinary differential equations which is solved numerically using the Lobatto IIIA formula (implicit Runge-Kutta method). The effect of the Prandtl number on the temperature and velocity profiles is illustrated graphically. The one dimension time-dependent heat transform equation in a vertical direction is introduced in terms of the general formula of density and thermal conductivity. One-parameter Lie symmetry group transformation is used to determine the suitable forms of density and thermal conductivity of the water. The general equation is resolved using Lie group analysis after substitution by some special cases of the obtained water density and thermal conductivity forms from the first part, then the obtained partial differential equation is solved numerically or analytically assuming the physical parameters of Lake Tahoe. The thermal stratification phenomenon of lakes is indicated through the temperature distribution across the lake depth from each case.