Emanating from the two-field form and using the Greens' formula for the equation (1.3) in a consistent manner yields
First, the weighted time fields 
are chosen to satisfy the following first-order ordinary differential equation associated with the adjoint operator of the original semi-discretized two-field form:
and the initial condition
We first obtain the closed form solution for the above first-order ordinary differential equation, namely, equations (3.95-3.96) to yield the weighted time fields, 
as
where the matrix 
and a matrix 
are defined in series form as
and
Substituting the fundamental solution equation (3.97), into equation (1.22), we have the closed form solution of the equation (3.91) as
A key idea towards the generic theoretical design and development of a new generation of a generalized family of time discretized operators pertaining to Type 2 classification is that the matrix 
is first decomposed into a symmetric and unsymmetric part as
where
and 
are typical control parameters introduced for convenience in the theoretical design and are subsequently discussed for each special case in the sections to follow.
It should be noted that this kind of `sum decomposition' must satisfy a condition that modal superposition can be applied for the linear dynamic system to permit applicability of the Lax equivalence theorem.
Thus, the weighted time fields, which still preserve the original theoretical matrix form may be expressed as
After some algebra we have the closed form explicit representation as
In its present form, the time discretized operator, namely equation (3.103) pertains to Type 1 classification and is still the exact representation of the dynamic equations (3.91-3.92) or is termed as an exact integral operator [EIO].
Any approximation to the load yields an approximate integral operator [AIO] in time.
As is readily seen, the Type 1 classification is the result emanating from a generalized time weighted philosophy of the semi-discretized dynamic equations of motion and is an outcome based on the selection of 
which does not impose any conditions whatsoever upon the dependent field variable(s).
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