ABSTRACT
Ö ( Ù , Ú , × ) = è 1( Ù ) è 2(
Ú ) è 3( × ), the equation in question has
more sophisticated solutions in the product form Ö ( Ù , Ú , × ) = é ( Ù , × ) ê ( Ú , × ),
heat Õ
considered in Subsection 1.1.1. 2 ç . Suppose
=
Ö ( Ù , Ú , × ) is a solution of the heat equation. Then the functions Ö
+ Ý 2, â 2 × + Ý 3),
2 = Ü Ö ( Ù cos ì − Ú sin ì + Ý 1,
sin ì + Ú
cos ì + Ý 2, ×
+ Ý 3), Ö
+ Æ
( â 21 + â 22) ×màíÖ ( Ù + 2
+ Ý 1, Ú
+ 2 Æ
+ Ý 2, ×
+ Ý 3), Ö
+ ì × exp ã −
ì ( Ù 2 + Ú 2) 4 Æ
+ ì × ,
, â
− ì
= 1,
where Ü , Ý 1, Ý 2, Ý 3, ì , î
, â , â 1 and â 2 are arbitrary constants, are also solutions of this equation. The signs at â ’s in the formula for
1 are taken arbitrarily, independently of each other. Î"Ï
3 ç . For all two-dimensional boundary value problems discussed in Subsection 2.1.1, the Green’s function can be represented in the product form
ð ( Ù , Ú , ñ , ò , × ) = ð 1( Ù
, ñ ,
, ò ,
× ), where
, ñ ,
× ) and ð 2( Ú
, ò ,
× ) are the Green’s functions of the corresponding one-dimensional boundary value problems (these functions are specified in Subsections 1.1.1 and 1.1.2).
