الفهرس 
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Abstract
In this thesis the author reviewed some of the most known methods for the numerical solution of partial diffe-rential equations he present work deals with the numerical solution of problems that involve free surfaces, that is boundaries which denends on the solutions to be obtained . A description of’ a numerical method introduce by G.H.Meyer [15]and its apolication to specific enGineering problems is illustrated.The mentioned method combines the method of lines, successive over rlaxation and invariant imbedding. Numerical results of the illustrated examples reveal the efficiency of the approach. The analysis and convergence of the method is also considered .