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REGULARIZATION OF THE CAUCHY PROBLEM FOR THE HELMHOLTZ EQUATION IN A BOUNDED DOMAIN D. A. JURAEV, Z. MALIKOV Samarkand State University We construct a regularized solution of the Cauchy problem for the Helmholtz equation in three-dimensional bounded domain.

3 Let R the real three-dimensional Euclidean space, G R bounded simply connected domain whose boundary consists of a compact T y3 0 and a smooth piece of surface S, lying in the part of the plane y3 0. G G T. ( 1, 2, x3 ) R3, s 2, half ( 1, 2, y3 ) R 3, r y x, 2 ( y1 x1 ) 2 ( y 2 x2 )2.

Cauchy problem. Let U ( y ) P(G ) and U ( y ) 2U ( y ) 0, 0, y G, (1) U U ( y ) f ( y ), ( y ) g ( y ), y S, (2) n where f ( y ) and g ( y ) at the given S continuous function. We need to recover U ( y ) in G.

Theorem. Let U ( y ) P(G ) the part of the plane y3 0 satisfies the inequality U U ( y) ( y) 1 y T, (3) n Then we have x U(x) U(x) B(x)e y3, 1, xG, (4) BIBLIOGRAPHY 1. Tikhonov A.N. On solving ill-posed problems and regularization method / / Dokl. USSR Academy of Sciences. in 1963. - T. 151. 3. P. 501-504.

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G D U ( x) 0, (1) x U ( y ) H (G ), U ( x) N ( y, x )U ( y )ds, x G, (2) y G 1. .. 1- // . , , 1980 . . 147-160.


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. U ( y ) H (G ) y 2 0 U ( y ) 1, y T. x U(x) U (x) C(x)e y2, 1, xG, (3) 1. . // . . 2001 . 3. . 6-8.

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b1 bn (1) [1, . 50], (2) [2, . 81].

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2. . . // . 1997. 1. . 7681.

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http://www.psu.ru/psu2/files/0444/chechulin_teoriya_mnozhestv.pdf 2. . ., . .: - . ., 1953.

3. . ., / . . -. , 2011. 112 .

http://www.psu.ru/psu2/files/0444/chechulin_modeli_ekonomiki.pdf - .. () - .

( ) ( ) . . , - . [2]. , - , [1].

Fortran90 [3]. , .

1. .., .. . .: , 2005. 364.

2. . . .: , 1979.

393.

3. .. . .: , 2000. 450.

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5. . . ANSYS . 2002.224.

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