I- . 1! "#" $ %%&'&() #" *&(+"#,#"#' -# "#'( (+ &. , */0' $# * 1 (.# , , , . (,# #"#(: , , , ; ! ; ; " ) # . “ " ” # . x1 , x2 ,..., xn , n ≥ 2 Ω En $ n− x ( 1– #2' %. & α = (α 1 , α 2 , ..., α n ) , !% " ) . n , "3* " n - α = α1 + α 2 + ... + α n ($&4 " , . u ( x ) = u ( x1 , x2 ,..., xn ) α = α1 + α 2 + ... + α n x ∈Ω # ! ∂ ( )u D u = D1 D2 ...Dn u = α1 α 2 , D 0u = u ( x ) αn ∂ x1 ∂ x2 ...∂ xn x = αi . ' " α ∂ iu ∂u α ∂ 2u α D u = α = D i i u , Di u = = u xi , D i2 u = = u x x ,... . 2 i i i ∂ x ∂ x ∂ xi i i α α1 α α2 αn 2– #2' %. ( F ( x, ..., % " Dα u, ...) = 0 u ( x) = u ( x1, x2 , ..., xn ) # 5 # ! " (1.1) m − *#'* 5" . " (1.1) 3– #2' %. . ) Pα = Dα u, ( α = 0,1, ..., m ) # ! # 4– #2' %. ( ) u xu . 1. + 2 xy , F Pα . 6 7" , (1.1) . F, α = m , (1.1) 4-#6 . 6 7" !". * " ) 1– u " ) # Pα " . . + 2 xuu yy − 3 xyu y − u = 0. , . xy 2 ) u y + 3 x u ⋅ u xy + 2 u x − f ( x , y ) u = 0. 3x 2 ⋅ u – u xy . 1. & # ! . # % - , u ) sin x ⋅ u xx + xcos y ⋅ u xy − 2 xy ⋅ u x − u = 0. . 1. , u, u x , u xx , u xy # !. , , . 5– #2' %. Ω u ( x) # ! (1.1) , " ! (1.1) '&+ "8' (4"# 4) &. , " , u ( x) " [4]. 2– !". a uxxy + b uxxx + c u yyy = 0 . uxxy ∈C ( D) , uxxx ∈C ( D) , uyyy ∈C ( D) , " . 6– #2' %. )" ( – ! " (9"&,&(*#') &. , " ( ! , u ( x) ) " (1.1) [4]. 1 1 ln , r = !". E ( x, y ) = 2π r 3– ) % ($#,&(*#" 2 ( x − x0 ) + ( y − y0 ) u xx + u yy = 0 2 # ! . 7– #2' %. ( Lu ≡ α ≤m aα ( x ) D α u = f ( x ) m − *#'* 5" " # " (1.2) . 6 ) x∈Ω (1.2) , (1.2) 5 ' : ( " , f ( x) : ( " 5/",#+#( " . 4– !". 1) 2 x2U xx + 3x yU xy + 5U yy = 0 f ( x) , 5 ' # ! 2) 5 x yU xy + 6U xx = 5 x 2 + y 2 x ≠ 0, y ≠ 0 8 – #2' %. ' ; " , . " *#'* 5 # " - . 1) a u x − b u y = f ( x ) − 2) U xx −U yy = f ( x, y) − !". 5– " 3) U xyy − U xxx = 0 − ; ; . 9– #2' %. ( ∂ 2u Aij ( x ) + ∂ xi ∂ x j i , j =1 n n i =1 Bi ( x ) ∂u + C ( x )u = f ∂ xi n − /6+#' -. " # " " – ", " . , (1.1) ! " . " , !% . + (1.3) " . (1.3) Aij u x i x j , A ji u x j x i (1.3) i≠ j # ! # x∈Ω Aij = A ji Aij , Bi , C , f - , Aij ( i, j = 1,..., n ) (1.3) (1.3) " Ω x = ( x1, x2 ,..., xn ) (x) ( Aij + A ji ) u x x i j " . m = 2, n = 2 , " 44 (. *#'* 5" . 6 7" $ %%&'&() #" *&(+"#,# F ( x, y , u , u x , u y , u xx , u xy , u yy ) = 0 (1.4) a ( x, y )u xx + 2b ( x, y )u xy + c ( x, y )u yy + a1 ( x, y )u x + b1 ( x, y )u y + c1 ( x, y )u = 0 (1.5) a( x, y), b( x, y), c( x, y), a1 ( x, y), b1 ( x , y ), c1 ( x , y ) − , # ! . (1.4) " # " 5#'#' " 1 1#'*: ∂F ∂F ∂F + + ≠ 0. ∂ u xx ∂ u xy ∂ u yy 7 (1.6) !". 1) sin2 ( u xx + u xy ) + cos 2 ( u xx + u yy ) − u = 0 6– # 2) 2 u xx + u 2yy ; " = f ( x, y ), - " # ∂F ∂F = 2U xx , = 2u yy , 2 ( u xx + u yy ) ≠ 0 . ∂u xx ∂u yy , (1.3) ) A ij ( x ) = 0 (1.3) .) " . 7– !". y ( u xx + u yy ) + a u x + bu y + cu = f ( x, y ) !% , " ! . "#" $ %%&'&() #" *&(+"#,# . " y=0 ., , , 1− u = 0 - y=0 ( (+ &. , +# ! $ !""#': 1) ux = 0 2) uxy = 0 3) aux + cu = 0 4) u xxx = 0 u ( x, y ) = ϕ ( y ) ; u ( x, y ) = ϕ ( y ) + φ ( y ) ; " " u xx = ϕ ( y ) u = f ( y )e " ux = ϕ 1( y) + ϕ2 ( y) −c x a ; u ( x , y ) = x 2ϕ 1 ( y ) + x ϕ 2 ( y ) + ϕ 3 ( y ) . #-6 ( , *#;4#,"#1 . ( #-!""#': 1. m " 2. . 3. 4. + " 5. / 6. / " . # " # . " . # . # " " " 7. 0 " 8. 0 " # # # . ! ( ) . # ( . 8 ) .