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