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Boundary Value Problems for Eaquations of Mixed Type
G.M. de La Penha. L.A. Medeiros (eds.) Contemporary Developments in Continuum Mechanics and Partia\ Differential Equations @North-Holland Publishing Company (1978) BOUNDARY VALUE PROBLEMS FOR EQUATIONS O F M I X E D TYPE
STANLEY O S H E R Mathematics
-
UCLA
Los Angeles, C a l i f .
90024
U. S.A.
We a r e c o n c e r n e d w i t h t h e q u e s t i o n :
which boundary
v a l u e p r o b l e m s a r e w e l l posed f o r t h e d i f f e r e n t i a l e q u a t i o n o f mixed t y p e :
K(y)hxx+u + a u + b u + c u = F YY X Y
(1) where
~ ( y =) s g n y
and
a, by c
=
+1
if
y > O
-1
if
y < O
a r e smooth f u n c t i o n s of
(x,y).
We s h a l l a l s o d e v e l o p a n e f f e c t i v e n u m e r i c a l algorithm (obtained j o i n t l y
w i t h A r t h u r Deacon) b a s e d
on a
r e d u c t i o n o f t h e boundary v a l u e problem i n t o a n e l l i p t i c problem w i t h u n u s u a l non p s e u d o - l o c a l
boundary c o n d i t i o n s f o r
which t h e G a l e r k i n p r o c e d u r e w o r k s w e l l .
T h i s problem i s
s o l v e d b o t h a n a l y t i c a l l y , u s i n g and m o d i f y i n g r e s u l t s
of
K o n d r a t i e n on e l l i p t i c e q u a t i o n s i n c o n i c a l r e g i o n s , and n u m e r i c a l l y u s i n g and m o d i f y i n g s t a n d a r d L a p l a c e i n v e r t e r s . We t h e n u s e t h e i n v e r t e d Cauchy d a t a on t h e p a r a b o l i c l i n e t o o b t a i n t h d s o l u t i o n i n t h e h y p e r b o l i c region. A s a very simple i l l u s t r a t i o n w e c o n s i d e r t h e follow-
i n g problem
3 97
E Q U A T I O N S O F M I X E D TYPE
Yt
in
Qo
we s o l v e
in
fll
we s o l v e
on
C0’
on
C1
which i s s p a c e l i k e , we p r e s c r i b e
on
rl
which i s a c h a r a c t e r i s t i c ,
hxx
+
u
-
xx
uY Y = F o U
YY = F 1
which i s a smooth c u r v e , we p r e s c r i b e
u =
Cp
u = ep 1
nothing i s given.
The aim i s t o r e d u c e all t h e h y p e r b o l i c i n f o r m a t i o n t o the l i n e
y = 0.
T h i s i s done as f o l l o w s :
u = f(x-y)
I n the hyperbolic region write f
and
g
+ g(x+y),
unknown ( h e r e we assume f o r s i m p l i c i t y that F r O ) 1
thus
u
- u
hx
+
X
(2)
u
Y
y=o
Y y=0
= 2f’(x)
= 2g‘ ( x )
Again,
f o r s i m p l i c i t y , we t a k e t h e c u r v e
line,
y = -cx
0 < c < 1
we t h e n h a v e
= cp,(x)
= f(x(l+c))
u(x,-cx)
Cl
+
g(x(1-c))
or
(3)
(-1
cp‘ 1 l + X c 1-c ( K = -) 1+c
= f’ (x)
+
Kg‘
(X
t o be a s t r a i g h t
(1-4) l+C
S T A N L E Y OSHER
398
Using (2) arid ( 3 ) , we have
where
Kx
x
ep,
is always a known function. Thus we have the following n o n pseudo-local boundary
Y
value problem
A
LII = F
B
A F o r the problem analyzed above,
general boundary conditions on
C1,
! ,
= k
X
for
is easily cornpuked.
We now build a parametrix f o r this problem and obtain coerciveness in weighted Sobolev spaces as in Kondratien.
Moreover, we can write an asymptotic expansion u
near
r
A
and
and pj
B
-c
, x j Pj(V)
R e X . + m J
on the boundary with simple formulas for the
moreover it turns out that if
li;le
is
sufficiently small, the hypotheses o f the Lax-Milgran theorem are obeyed f o r this elliptic problem on
HI,
thus a Galerkin
procedure can be shown to converge. We also use as elements i n this procedure the functions having the appropriate singular behavior at A and B. The resulting numerical procedure works very well.
399
E Q U A T I O N S O F M I X E D TYPE
Bibliography
and O s h e r , S.
[l] D e a c o n , A.
121 K o n d r a t i e n , V.A.
-
-
-
T o appear.
B o u n d a r y v a l u e p r o b l e m s for e l l i p t i c
e q u a t i o n s i n d o m a i n s w i t h c o n i c a l o r a n g u l a r points, T r a n s . M o s c o w Math.
SOC.,
T r d y , Vol.
R u s s i a n 2 0 9 - 2 4 2 , T r a n s l a t e d A.M.S.
[3]
O s h e r , S.
-
16 ( 1 9 6 7 )
(1968).
B o u n d a r y v a l u e p r o b l e m s f o r e q u a t i o n s of
m i x e d t y p e I. Cornm. P . D . F .
T h e L a u r e n t i e n - B i t s a d z e model.
2, ( 1 9 7 7 ) , '199-547.
STANLEY O S H E R Mathematics
-
UCLA
Los Angeles, C a l i f .
90024
U. S.A.
We a r e c o n c e r n e d w i t h t h e q u e s t i o n :
which boundary
v a l u e p r o b l e m s a r e w e l l posed f o r t h e d i f f e r e n t i a l e q u a t i o n o f mixed t y p e :
K(y)hxx+u + a u + b u + c u = F YY X Y
(1) where
~ ( y =) s g n y
and
a, by c
=
+1
if
y > O
-1
if
y < O
a r e smooth f u n c t i o n s of
(x,y).
We s h a l l a l s o d e v e l o p a n e f f e c t i v e n u m e r i c a l algorithm (obtained j o i n t l y
w i t h A r t h u r Deacon) b a s e d
on a
r e d u c t i o n o f t h e boundary v a l u e problem i n t o a n e l l i p t i c problem w i t h u n u s u a l non p s e u d o - l o c a l
boundary c o n d i t i o n s f o r
which t h e G a l e r k i n p r o c e d u r e w o r k s w e l l .
T h i s problem i s
s o l v e d b o t h a n a l y t i c a l l y , u s i n g and m o d i f y i n g r e s u l t s
of
K o n d r a t i e n on e l l i p t i c e q u a t i o n s i n c o n i c a l r e g i o n s , and n u m e r i c a l l y u s i n g and m o d i f y i n g s t a n d a r d L a p l a c e i n v e r t e r s . We t h e n u s e t h e i n v e r t e d Cauchy d a t a on t h e p a r a b o l i c l i n e t o o b t a i n t h d s o l u t i o n i n t h e h y p e r b o l i c region. A s a very simple i l l u s t r a t i o n w e c o n s i d e r t h e follow-
i n g problem
3 97
E Q U A T I O N S O F M I X E D TYPE
Yt
in
Qo
we s o l v e
in
fll
we s o l v e
on
C0’
on
C1
which i s s p a c e l i k e , we p r e s c r i b e
on
rl
which i s a c h a r a c t e r i s t i c ,
hxx
+
u
-
xx
uY Y = F o U
YY = F 1
which i s a smooth c u r v e , we p r e s c r i b e
u =
Cp
u = ep 1
nothing i s given.
The aim i s t o r e d u c e all t h e h y p e r b o l i c i n f o r m a t i o n t o the l i n e
y = 0.
T h i s i s done as f o l l o w s :
u = f(x-y)
I n the hyperbolic region write f
and
g
+ g(x+y),
unknown ( h e r e we assume f o r s i m p l i c i t y that F r O ) 1
thus
u
- u
hx
+
X
(2)
u
Y
y=o
Y y=0
= 2f’(x)
= 2g‘ ( x )
Again,
f o r s i m p l i c i t y , we t a k e t h e c u r v e
line,
y = -cx
0 < c < 1
we t h e n h a v e
= cp,(x)
= f(x(l+c))
u(x,-cx)
Cl
+
g(x(1-c))
or
(3)
(-1
cp‘ 1 l + X c 1-c ( K = -) 1+c
= f’ (x)
+
Kg‘
(X
t o be a s t r a i g h t
(1-4) l+C
S T A N L E Y OSHER
398
Using (2) arid ( 3 ) , we have
where
Kx
x
ep,
is always a known function. Thus we have the following n o n pseudo-local boundary
Y
value problem
A
LII = F
B
A F o r the problem analyzed above,
general boundary conditions on
C1,
! ,
= k
X
for
is easily cornpuked.
We now build a parametrix f o r this problem and obtain coerciveness in weighted Sobolev spaces as in Kondratien.
Moreover, we can write an asymptotic expansion u
near
r
A
and
and pj
B
-c
, x j Pj(V)
R e X . + m J
on the boundary with simple formulas for the
moreover it turns out that if
li;le
is
sufficiently small, the hypotheses o f the Lax-Milgran theorem are obeyed f o r this elliptic problem on
HI,
thus a Galerkin
procedure can be shown to converge. We also use as elements i n this procedure the functions having the appropriate singular behavior at A and B. The resulting numerical procedure works very well.
399
E Q U A T I O N S O F M I X E D TYPE
Bibliography
and O s h e r , S.
[l] D e a c o n , A.
121 K o n d r a t i e n , V.A.
-
-
-
T o appear.
B o u n d a r y v a l u e p r o b l e m s for e l l i p t i c
e q u a t i o n s i n d o m a i n s w i t h c o n i c a l o r a n g u l a r points, T r a n s . M o s c o w Math.
SOC.,
T r d y , Vol.
R u s s i a n 2 0 9 - 2 4 2 , T r a n s l a t e d A.M.S.
[3]
O s h e r , S.
-
16 ( 1 9 6 7 )
(1968).
B o u n d a r y v a l u e p r o b l e m s f o r e q u a t i o n s of
m i x e d t y p e I. Cornm. P . D . F .
T h e L a u r e n t i e n - B i t s a d z e model.
2, ( 1 9 7 7 ) , '199-547.