الفهرس 
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Abstract
Hidden symmetries of non linear differential equations are here investigated. These
symmetries were defined in the early nineties by Barbara Shrauner [1-5] for problems
having no Lie vectors or no closed form solution. The subject was then left dormant for
twenty years.
We in this thesis track the hidden syrnmetries, type I and type II for nonlinear
ordinary and partial differential equations using Lie extended differential method. Then
for highly non linear partial differential equations construct of an optimal set of Lie vectors
[6] and explore its hidden syrnmetries. Using these vectors we reduce the partial
differential equations to ordinary ones, sometimes to a fourth reduction level, tracking
hidden symmetries. For each application analytical solutions were obtained and algebraic
rules were deducted. These rules are new and will help future workers in the field. For
each analytical solution a comparison was done whenever available with the most of recent
researches. Many of the obtained new solutions appear III real life.