Analyses of fixed point and numerical integration are commonly carried out in obtaining asymptotically consistent high Reynolds number solution in case of energy decay equation of homogeneous isotopic turbulence. In this chapter we have solved the \(Von\text{-}K\acute {a}rm\acute {a}n\text{-}Howarth\) equation vis \(\acute {a}\) vis the energy spectrum equation and carried out fixed point analysis for a class of self-preserving solutions of homogeneous isotopic turbulence at high Reynolds number. It is shown that the decay laws obtained by similarity method and fixed point method due to the mathematical applications are compatible.