{ } , 0 , 1, 2 , . . . k x k  0 1 2 0 . . . a n d l i m k k k x x x x x           0 1 [ 0 , ) , ( 0 , ) a n d { } k R R R R x            1,p   p   p L 1 pp u u d x     , s u p ( ) :p u e s s u x x       |N |= 0 ,x R -N in f s u p ( )u x  p u    ( ) :p pL u u      ( )C     0 ( ) ( ) : ( ) ( ) , 0 , 1, 2 , ..., lim ( ) ( ) , , k x x k k P C y x y x C k y x y x x x           0 ( ) ( )D C       s u p p ( ) : ( ) 0f x x f x     . w D u  u   , ( ) ( ) : ( ) , | | m p p w p W u L D L m         , 2 ( ) m m W H  , 0 ( ) m p W  ( )D  ( ) m H  , 2 0 0 ( ) ( ) m m H W   , ( ) m p W  1 p   1 p   ,k p p k u D u      1 , 1 t h e n p p i np p u u k u u x x          2 2 1 , 2 1 2 1 t h e n n i i u k u x       1 , . n i i u u v u v d x x             n R 2 a n d 1n p    1 , 1 1 , ( ) ( ) , 1 1 1 i f p q q p W L u c u p N p q          1 , 0 2 1 , ( ) ( ) , m a x p p m W C L u c u        1 2 a n d c c ( ., .) :a H H R  2 | ( , ) | | | ,a u u c u u H  u ( , )a u u c L 2 ( )L    2 : 0 , ( ) C L v V v v L       2 :| | , c o n s t ( ) . p S x V x p p L       i i i u v v d x u v n d s u d x x x             i n n R  1 , p  1 , 1 , o n ( ) p p W  1 , 00 , 1 , 00 , , ( ) , ( ) . p p p n u c u u W u u c u H x            H ( , ) :a H H R    :f H R u H ( , ) ( ) ,a u v f v u H   ( , )a   ( , ) 0 ,a u v v H   :J H R 1 ( ) : ( , ) ( ) 2 J v a v v f v  in f ( ) ( ) H u J u J u      2 1 is m e a s u r a b le in f o r a ll ( ) : , c o n tin u o u s in f o r a lm o s t a ll i x L A a x x I          2 : ( , )A a x  1 2 , 2 2 2 1 1 2 1 1 2 ( ) ( , ) ( , ) * . W L v a x a x x x x u c c V x                               0 0 1 0 1 ( , ) ( , m a x ) i n , , 0 , 1, 2 , , , , w i t h m i x e d b o u n d a r y c o n d i t i o n s 0 o n , ( , ) , . i i i k i i k i i i i u a x u g x u x x f x x k x I x u x x x x x u u a x y n h o n x                                           ( )u D  0  (1)   1 1 1 ( , ) , . i i i i u v a x u d x f v d x f g h v d s x x v I x u                  ( )u u x : ( 0 , ] N u x V / V  ,u A u    ( 0 , ] N x / ( ) , ( ) , , ( ) , ( ) , d d v A u v d s A u d s D u P C v V d s d s                  ( , )u u x u ( )J v   (1 ) 0 ( ) ( ) ( ) li m s u p h J v h J v D J V h      c L 0    0 , 0x   0 x    0 ,u x u   0 x 0    0 , 0T T x   0 x    0 0 ,u x u N 0 x x T  (2) ( )T T     / 0 : 0 o nV u P C u       / / V P C  ( )P C  1 , 2 ( )W  ( )P C   :A V V  b V  ( ) , ( , ) , , ( ) i i u v A u v a x u d x u V v D x x               1 2 3 4 , ,b v b b b b x v    1 2 3 , , ,b f v d x b v g d x b f v d x         4 1 . i i i b v I    u V ( ), ,A u u b v v V    1 , i i i b v f v d x f v f v d x v I           (3) (4) (5) 2 2 2 2 2 2 2 2 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) L L L L L L L L f v g v h v c v c v                2 2 2 1( ) ( ) ( ) m a x , , L L L c f g h c         ' :A V V :A V V 0      2 1 2 1 2 1 2 , 0 , ,A u A u u u u u V      ( )A u b b V  ( ) ( )T u u A u b    0  T      1 2 1 2 1 2 , ,T u T u u u u u V                                    22 1 2 1 2 1 2 1 2 1 2 1 2 1 2 22 2 1 2 1 2 1 2 1 2 2 2 2 1 2 , 2 , 1 2 . T u T u u u A u A u u u A u A u u u A u A u u u u u A u A u A u A u u u M                                A     2 2 1 2 1 A u A u u u            2 1 1 2 2 1 , , i i i i i i u u A u A u a x x a x u x x x                   2 2 1 1 , . i i i i i x i a x u a u a u u u a u x x x x x x x x x                          2 1 1 2 2 1 2 1 1 2 2 2 . i i V u u A u A u L x x u u L u u x x m u u                      2 2 2 1 2 1 2 0m           0 , 2 0 0 2           m i n 1, 2          T  ( )T u u   ( ) 0 T u u A u b u A u b         A ( )A u b ( , )a x u   1 * 1 ( , ) , . n i i i i n i i i u v a u v a x u d x g v d x x x b f v d h v d s v I                  1 1 ( ) ( , ) 2 2 i i u v J v a x u d v g v d x x x              2 1 { , } , , , , , . n n i j i j i j i i j m a x m R x R             u i n f ( ) v V u J v   * ( , ) i i u v a x u g v d x b x x            2 ( )b L  0 : ( )a H H V H    0 ( ) m i n ( ) v H u J v    *1 1 ( ) ( , ) . 2 2 i i i i u v J v a x u g v d x b x x            * , , 0N m  1 :H     1 , s u p , a n d s u p , m a x x u x u V N a x u g x u         2 2 * 1 m i n , . 2 L L N u M v       0 , m in ( ) v V u u x u J v    *1 1 ( ) ( , ) . 2 2 i i i i u v J v a x u d x g v d x b x x            2 2 1 1 * m i n ( ) 1 1 m i n ( , ) 2 2 1 m i n . 2 v J L L u J v u v a x u d x g v d x x x N u M v                        | |x  *   | ( , ) |u x u  ( , )u u x u 1 :H , 1, 2 , 3 i k i  * 1 2 1 3 | | , | | a n d | |f k v h k v I k v   1 :H *1 1 ( , ) 2 2 i i i i u v a x u d x g v d x b x x           1 * * . i i i b f v d x h v d x v I        ( ) 0 , ( ) ( ) a n d ( ) 0 , k k k J v J x x J x D J v v V         { } , 1, 2 , . . . k x k  1 2 &H H * 1 1 ( ) ( , ) 2 i i i i n i i i u v J v a x u g v d x x x f v d x h v d s v I                       1 H 1 * 1 2 2 2 1 2 3 0 . n i i i i f v d x h v d s v I k v d x k v d s k v                2 H ( ) 0J v    0 * 1 ( ) ( ) ( ) l i m s u p 1 2 2 0 2 h J v h J v D J v h g h d x I              * 0 , 0g h  ( , )u u x u   2 : 0 C L v L v   0  0 u  0  *   *      2 2 , m a x ( , ) , , 0 , 1, 2 , . . . 0 o n k p k x k u u c g x u f x u x x k x x u e I x u                i s t h e t h e r m a l h e a t c a p a c i t y o f s y s t e m p c 0  ( , ) , ( ) p x u c x k u u      ( )u Heat source 2 1 1 ( ) 2 2 u u v J v c k d x v g d x v f d x x x x                     1 22 2 2 m a x ( , m a x ) a n d g ( , m a x ) 1 1 m a x ( m a x ) u x g x u x u x x u x u        2 2 ( i ) 0 i n ( ) 0 i n 1 ( ) 0 i n 2 T h e n ( ) 0 i n = a n d t h e t e m p e r a t u r e u i s s u c h t h a t p u u c k x x v i i x i i i g f J v                    22 ( ) 0 ( 1) , 1 2 ( ) 0 1 2 (1 ) i v u x f x A n d x v u x x f            * 0 in .h   J ( v ) 0 i n a n d h v d x = 0     ( ) 0 i n J v   2 1 m a x 0 . 2 2 (1 ) u g f f x       22 m a x ( 1) f o r 1 .u x f x     22 0 m a x , t h e r e f o r e ( 1)u u u x      2 2 1 0 2 2 (1 m a x ( m a x ) ) x g f f x u x u            2 2 2 2 2 m a x 2 ( m a x ) 2 2 2 . x x u x u x u x u f          2 f o r 0 1,y y y y R      2 ( m a x ) m a x i f 0 u m a x u < 1u u   2 - 0 u m a x u < < 1 2 (1 ) x x x f     ( ) 0 J v  , 0 , 0 .f x   +( 1 ) h 0 * ( ) ( ) ( ) l i m 1 ( ) 2 i i J v h J v D J v h g d x f d x h d s I x                 * 0h   , a n d i g f I  ( ) ( ) . k k k J x x J x   k x u 