A Note why Added Mass Concept is Mathematically Needed

One of the resistance to the added mass adaption is the idea that solution of the Navier–Stokes equations provides the solution. Several physical expansions have been provided in the past as why the added mass has be added and why the NS solution is not the total solution of the problem. It could be that the arguments in these explanations which are based on the physics not sufficient to convince the detractors. This paper is an attempt to offer a different view for which emphasizes the mathematical lens and prospective and thus convincing others in the essential
of the added mass. Perhaps this mathematical observation could be used in other situations with similar underline problems.
     The physics is typically described by some mathematical models which contain some kind of differential equation(s).  The cases where integral equation(s) described the situations are somewhat different and a modified approach is needed. Differential equations require boundary and or initial conditions with one degree less or much less. For example, the second order ODE allows only for derivative or the function or combination of both
(no second order derivative on the boundary). The concept of the added mass stems from the mathematical point of view ability or need to impose the same order boundary conditions as the equation(s) (when conditions are linear or elliptic).