I. GONOSKOV
Copyright © 2013 SciRes. APM
182
0
11
ˆˆ
ˆ
ˆˆ
d,
t
g
t
Gt
tt
Gt
r
00
1
2
,,
d, ,
ˆ
d,
tt
tt
tt
ttV
r
r
rRr
0
12
,d
d, .
n
tt
t
tt
tt
r
r Rr
(31)
Then, we can find the solution, which follows from the
corresponding COD series:
0
0
1
1
ˆ
,1 d
ˆ
nt
t
t
tt
Ar
Sr
(32)
For this solution the initial magnetic field is equal to
r
Rr
and the initial electric field is equal to
.
4. Conclusion
In summary, we propose the theory of Cyclic Operator
Decomposition, which allows one to obtain particular
solutions of linear operator equations for unknown func-
tions. In most cases it is possible to obtain all the possi-
ble solutions, which satisfy the given conditions. We
demonstrate by some reasonings and particular examples
that our approach has the following remarkable proper-
ties: 1) there is a freedom in choosing the COD compo-
nents depending on the certain problem; 2) there is a
rapid uniform convergence for most of the considered
cases; 3) it is possible to find the asymptotic behavior of
the solutions; 4) in many cases when one is analyzing the
approximate solution, it is possible to estimate the accu-
racy; 5) the proposed approach gives good opportunities
for efficient implementation of numerical calculations
due to the recurrent relations that can be used in COD.
5. Acknowledgements
Author would like to thank academician L. D. Faddeev,
M. Yu. Emelin, M. Yu. Ryabikin, and A. A. Gonoskov
for the useful and stimulating discussions.
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