BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

46 Petronela Paraschiv
calculation in order to appreciate the influence of diverse factors (parameters)
over a variable, starting from the experimental determined values of that
variable. As “converse problems” can be considered the following types:
i) problems that ask the state determination of a physical process in
anterior moments of time, on the basis of some measurements done over a
variable;
ii) problems that ask for the reconstitution of a physical operator with a
mathematic structure of a known form, but with unknown coefficients, which
can be established on the basis of the information over some functional of the
experimental determined variable.
Therefore, supposing that it was effectuated a set of measurements
(named “functional”) and each functional is associated a function of influence
for the undisturbed problem , i.e. a model in which operator L and its domain of
definition are considered known, there will be solved n problems as such
* *
L φ = p , i = 1, 2, ...., n. (6)
pi
i
*
They are before those n functions of influence φ pi
and it is solved a basic
problem with the operator model “undisturbed L, adjunct of L,
Lφ = q.
(7)
There are built n formulas of the theory of small perturbations as such
*
φ δLφ)= - δJ , i = 1, 2, ...., n, (8)
( pi pi
where δL is the difference between the studied operator L’ and the model one
L, while J pi represents the set of functionals (measurements).
We suppose that the operator L is known, such as
m
L = ∑ [ αuAu + Bu(
βuCu)],
(9)
u−1
where A u , B u and C u are elementary linear operators, for instance differentials or
integral or combination of these ones; αu
( x ) and βu
( x ) are searched
coefficients, usually known with a harsh approximation for the undisturbed
problem (model).
The aim of the mathematic demarche is that of reconstructing the
'
coefficients and β from the expression
α u
'
u
m
'
'
' = [
k k
+
k( k k)
k = 1
L ∑ α A B β C ]. (10)
With the help of the Eqs. (7) and (8) it is obtained
m
δL = ∑ [ δαuAu + Bu( δβuCu)],
(11)
u=
1