I. GONOSKOV

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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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