Chapter 9
Higher-Order Partial Differential Equations 9.1. Third-Order Partial Differential Equations
1.
3
+
= 0.
3
Linearized Corteveg–de Vries equation. 1 . Particular solutions:
( , ) = (
3
( , ) = (
5
2
− 6 ) +
+ + ,
2
− 60 ) + (
4
− 24 ),
( , ) = sin( + ) + cos( + 3 ) + ,
( , ) = sinh( − 3 ) + cosh( − 3 ) + ,
( , ) = exp − 3 exp + exp − 12 sin 23 + ,
3
where , , , , and are arbitrary constants. 2 . Domain: − < ≤ 0. Boundary value problem. Initial and boundary conditions are prescribed:
= 0 at
= 0,
Solution:
= ( ) at
3 ( , ) = − 2
= 0,
Ai
0
( −
0 as
( ) − #
,
)1 ! 3 "
− .
where Ai ( $ ) is the second derivative of the Airy function. 3 . Domain: 0 ≤ < . The function
( , ) = 3
Ai
0
( ) ( − )1 ! 3 " − #
,
satisfies the equation and the first two conditions specified in Item 2 .
%&
Reference: A. V. Faminskii (1999).
2.
=
6 '
3
3
.
The transformation (
($ , ) =
−2
, $
= 1) ,
=
leads to a constant coefficient equation of the form 9.1.1: (
* *
=
( * 3 −* 3. $
© 2002 by Chapman & Hall/CRC Page 601
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
3
=+ ( )
3.
,
+
3
The transformation
( ) + - ( )
(
+. ( ) .
0
( , ) = ( $ , ) exp /
( ) 21 , #
4
= 3 ( ) + $
=
( ) 3 ( ) , #
( ) 3
3
( ) , #
where 3 ( ) = exp /
( ) 21 , leads to a constant coefficient equation of the form 9.1.1:
#
( *
= *
2
= ('
4.
+5
3
+ 6 )3
3
( * 3 * . $ 3
.
This is a special case of equation 9.6.4.4 with = 1 and 7 = 3. The transformation (
2
( , ) = ( $ , )(
leads to the constant coefficient equation (
*
*
5.
2
2
6
='
3
3
.
The transformation $ = 1 ) ,
6.
2
2
= ('
2
+5
+6 )
(
=
3
3
3
=
−2
+ + ), ( * 3 * $ 3
= $
2
#
+ +
( *
+ (4 − 2 ) *
. $
leads to the constant coefficient equation ( * 2 * 2
( * 3 − * 3 . $
=
.
This is a special case of equation 9.6.4.4 with = 2 and 7 = 3. The transformation
(
2
( , ) = ( $ , )(
leads to the constant coefficient equation ( * 2 * 2
=
+ + ), ( * 3 * $ 3
$
= *
+ (4 − 2 ) *
2
#
+ +
( $
.
9.2. Fourth-Order One-Dimensional Nonstationary Equations 9.2.1. Equations of the Form
8 9 8 :
+ ;
2
4 8
8
≡ 0):
( ) = ?
3
+@
2
+A +B ,
( , ) = ? (
( , ) = ? sin( ) + @ cos( ) + A sinh( ) + B cosh( ) exp(− 4 2 ),
5
− 120 2 ) + @ (
4
− 24 2 ),
where ? , @ , A , B , and are arbitrary constants.
© 2002 by Chapman & Hall/CRC Page 602
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9.2. FOURTH-ORDER ONE-DIMENSIONAL NONSTATIONARY EQUATIONS
9.2.1-2. Domain: 0 ≤ ≤ C . Solution in terms of the Green’s function. 1 . We consider problems on an interval 0 ≤ ≤ C with the general initial condition
= ( ) at
=0
and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as
D
( , ) =
(E ) F ( , E , ) E +
0
#
D
0
(E , ) F ( , E , − ) E >
0
#
. #
2 . Paragraphs 9.2.1-3 through 9.2.1-10 present the Green’s functions for various types of boundary conditions. The Green’s functions can be evaluated from the formula I I
I I
( )
( , E , ) = G F
I
I
HJI
L =1 K
(E ) L
exp(−
2
K
4
2
),
(1)
K ( ) are determined by solving the self-adjoint eigenvalue problem for the where the and fourth-order ordinary differential equation K
4
−
=0
K denotes differentiation with respect to . The subject to appropriate boundary conditions;K the prime norms of eigenfunctions can be calculated by the formula I
L
L 2
I
=
D 0
K
I
2
C
( ) =
2
4
#
K
C
(C ) +
4
I
I 4
( C )
K
2
−
C
2
I
I
( C ) ( C ).
I 4
K
K
(2)
K
Relations (1) and (2) are written under the assumption that = 0 is not an eigenvalue. 9.2.1-3. The function and its first derivative are prescribed at the boundaries:
* M
=
= 0 at
Green’s function: I F
where I I
( , E , ) =
4 HJI C
I
( I) = sinh( C
= 0,
)−sin(
K
= 0 at
I
( )
2
( C )
=1
I C ) cosh(
I
I 4 G
* M
=
K
I
(E ) exp(−
4
I
),
I
) − cosh(
2
K
I
)−cos(
=C.
C
)−cos(
I C ) sinh(
I
)−sin(
) ;
K the are positive roots of the transcendental equation cosh( C ) cos( C ) = 1. The numerical values of the roots can be calculated from the formulas given in Paragraph 9.2.3-2.
9.2.1-4. The function and its second derivative are prescribed at the boundaries:
=
* M M
= 0 at
Green’s function: F
( , E , ) =
2 HI C
I
sin( G =1
= 0,
=
* M M
I
) sin(
= 0 at
I I
E
) exp(−
4
=C.
2
),
= N C
7
.
© 2002 by Chapman & Hall/CRC Page 603
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.2.1-5. The first and third derivatives are prescribed at the boundaries: * M
=
* M M M
= 0 at
Green’s function: 1
( , E , ) = F
2 HJI
+
C
C
* M
= 0, I
* M M M
I
cos( G
=
I
4
) exp(− E
=C.
I
) cos(
= 0 at
2
),
7
= N
.
C
=1
9.2.1-6. The second and third derivatives are prescribed at the boundaries: M M
=
* M M M
= 0 at
M M
= 0,
=
Green’s function:
where I I
1 C
+
3
)−sin( C
=1 K
4
exp(−
I
2
),
I
) − cosh(
)+cos(
)
K
I
C ) cosh(
(E )
2 (C
K
I
I
( ) G
C
=C.
I I
4 HJI
− C )(2E − C ) +
(2 C3
I
( I) = sinh(
= 0 at
I
( , E , ) = F
* M M M
I
)−cos( C
I
C ) sinh(
)+sin(
) ;
K the are positive roots of the transcendental equation cosh( C ) cos( C ) = 1. The numerical values of the roots can be calculated from the formulas given in Paragraph 9.2.3-2.
9.2.1-7. Mixed conditions are prescribed at the boundaries (case 1):
=
* M
= 0 at
Green’s function: ( , E , ) = F
where I I
C
=1
C ) cosh(
= 0 at
K I
4
2
),
I
I
) − cosh(
=C.
I
I
( ) (E ) exp(− | ( C ) ( C )| K K
)−cos(
* M M
I I
I
)−sin( C
4 G
=
I I
K
I
( I) = sinh( K the
2 HJI
= 0,
I
)−cos( C
I
C ) sinh(
)−sin(
) ;
) ;
are positive roots of the transcendental equation tan( C ) − tanh( C ) = 0.
9.2.1-8. Mixed conditions are prescribed at the boundaries (case 2):
=
* M
= 0 at
I
Green’s function: F
where I I
( I) = sinh( K the
)+sin(
4 HJI
( , E , ) =
I C
* M M
= 0,
C
(E )
2 (C
=1 K
K
)
exp(−
I K
)−cos(
= 0 at
=C.
I
I
G
* M M M
I
( )
I C ) cosh(
=
4
2
I
) − cosh(
),
C
I
)+cos(
I C ) sinh(
I
)−sin(
are positive roots of the transcendental equation cosh( C ) cos( C ) = −1.
© 2002 by Chapman & Hall/CRC Page 604
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9.2. FOURTH-ORDER ONE-DIMENSIONAL NONSTATIONARY EQUATIONS
9.2.1-9. Mixed conditions are prescribed at the boundaries (case 3):
= Green’s function:
* M M
( , E , ) = F
= 0 at
2 HI C
I
= 0 at
4
) exp(− E
=C.
I I
) sin(
* M M M
=
I
sin( G
* M
= 0,
2
),
(27 + 1) . 2C
= N
=0
9.2.1-10. Mixed conditions are prescribed at the boundaries (case 4):
= Green’s function:
* M M
= 0 at
I
the
4 HJI
( , E , ) = F
where
= 0,
I
I
C
* M M
I
I
=1 K
I
(E )
2 (C
K
= 0 at
=C.
I
( ) G
* M M M
= I
exp(−
)
4
2
I
K
), I
( ) = sin( C ) sinh( ) + sinh( C ) sin( ); are positive roots of the transcendental equation tan( C ) − tanh( C ) = 0. K
9.2.2. Equations of the Form
2 8
8 :
92
+ ;
2
4 8
8
≡ 0.
9.2.3. Equations of the Form
2 8
8 :
+
92
2 ;
4 8
8
0
#
. #
2 . Paragraphs 9.2.3-2 through 9.2.3-9 present the Green’s functions for various types of boundary I from I the formula conditions. The Green’s functions can be evaluated I I
( , E , ) = F
I
1 HI
I
G
2
=1 K
I
( ) L
L
(E )
K
sin(
2
2
),
(1)
where the and ( ) are determined by solvingK the self-adjoint eigenvalue problem for the fourth-order ordinary differential equation K
4
−
=0
subject to appropriate boundary conditions;K the prime K denotes differentiation with respect to . The norms of eigenfunctions can be calculated by Krylov’s formula [see Krylov (1949)]: I
L
L 2
=
D
I
I
2
C
( ) =
2
4
#
0
(C ) +
I
I C
4
4
( C )
2
−
C
2
I
I
( C ) ( C ).
I 4
(2)
K K K K Relations (1) andK (2) are written under the Kassumption that = 0 is not an eigenvalue.
9.2.3-2. Both ends of the rod are clamped. Boundary conditions are prescribed:
* M
=
= 0 at
= 0, I
Green’s function: ( , E , ) = F I where
I
I
( I) = sinh( C
=
)−sin(
4 HJI C
G
I C ) cosh(
( C ) I K
I
I 2 =1
)−cos(
* M
= 0 at I
( )
2
K
2
I
) − cosh(
=C.
I
(E ) sin( K
),
I C
)−cos(
I C ) sinh(
I
)−sin(
) ;
the are positive K I roots of the transcendental equation cosh( C ) cos( C ) = 1. The numerical values I of the rootsI can be calculated from the formulas %&
= R C
,
where R
1
= 1.875, R
2
= 4.694, R
= N (27 − 1) for 2 7
≥ 3.
Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
© 2002 by Chapman & Hall/CRC Page 606
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9.2. FOURTH-ORDER ONE-DIMENSIONAL NONSTATIONARY EQUATIONS
9.2.3-3. Both ends of the rod are hinged. Boundary conditions are prescribed:
=
* M M
= 0 at
Green’s function: ( , E , ) = F %&
HJI
2C N
2
= 0,
I
1 G =1
7 2
* M M
=
= 0 at
I
sin(
I I
) sin(
=C.
2
) sin( E
),
= N
7
.
C
References: A. N. Krylov (1949), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
9.2.3-4. Both ends of the rod are free. Boundary conditions are prescribed: M M
=
* M M M
= 0 at
M M
= 0,
=
* M M M
Green’s function:
where I
I
( , E , ) = F
I
+ C
3 C3
I
)−sin( C
4 HI
(2 − C )(2E − C ) +
I
( )I = sinh(
= 0 at
C
G
2
=1 K
) − cosh(
K
(E ) sin( 2 (C )
K I
)+cos(
I
I
( )
I
C ) cosh(
I I
2
),
I
I
)−cos( C
=C.
I
C ) sinh(
)+sin(
) ;
K the are positive roots of the transcendental equation cosh( C ) cos( C ) = 1. For the numerical values of the roots, see Paragraph 9.2.3-2. The first two terms in the expression of the Green’s function correspond to the zero eigenvalue L (1) L 2 (1) (2) 0 = 0, to which two orthogonal eigenfunctions 0 = 1 and 0 = 2 − C correspond with =C 0 L L and 0(2) 2 = 13 C 3 .
%&
Reference: A. N. Krylov (1949).
9.2.3-5. One end of the rod is clamped and the other is hinged. Boundary conditions are prescribed:
=
* M
= 0 at
Green’s function: F
where I I
( I) = sinh( K the
( , E , ) =
I C
)−sin(
2 HI C
2 G =1
)−cos(
* M M
= 0 at
I I
K
K
) − cosh(
=C.
I
I
( ) (E ) sin( | ( C ) ( C )| K K
I
=
I I
I C ) cosh(
= 0,
2
I
),
C
I
)−cos(
I C ) sinh(
I
)−sin(
) ;
are positive roots of the transcendental equation tan( C ) − tanh( C ) = 0.
© 2002 by Chapman & Hall/CRC Page 607
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.2.3-6. One end of the rod is clamped and the other is free. Boundary conditions are prescribed:
* M
=
= 0 at
Green’s function:
I where
I
( I) = sinh(
I
I
C ) cosh(
(E ) sin( )
),
I
) − cosh(
2
I
K
)−cos(
2 (C
K
=C.
I
I
2
=1 K
I
)+sin( C
C
= 0 at
I
( ) G
* M M M
=
I
4 HJI
( , E , ) = F
* M M I
= 0,
I
)+cos( C
C ) sinh(
I
)−sin(
) ;
are positive roots of the transcendental equation cosh( C ) cos( C ) = −1.
the K
9.2.3-7. One end of the rod is hinged and the other is free. Boundary conditions are prescribed:
=
* M M
= 0 at
Green’s function: I
where I
the
4 HJI
( , E , ) = F
* M M
= 0,
I
C
( ) = sin(
I I
=1 K
I
) sinh( C
I
2 K
2 (C
(E ) sin( )
2
=C.
),
I
K
) + sinh(
= 0 at I
I
( ) G
* M M M
=
C
I
) sin(
);
are positive roots of K the transcendental equation tan( C ) − tanh( C ) = 0.
9.2.3-8. The first and third derivatives are prescribed at the ends: * M
=
* M M M
= 0 at
Green’s function: F
( , E , ) =
C
2
+
C
HJI
1 =1
2
cos(
* M M M
=
I I
G
* M
= 0,
I
) sin( E
=C. I
I
) cos(
= 0 at 2
),
= N
7 C
.
9.2.3-9. Mixed boundary conditions are prescribed at the ends:
=
* M M
= 0 at
Green’s function: F
( , E , ) =
2 C
HJI
I I
1 G =0
2
* M
= 0,
sin(
9.2.4. Equations of the Form
=
* M M M
I
8
2 8 :
+
92
2
) sin( E
2 ;
8
4 8
≡ 0):
3
( , ) = ( ?
3
( , ) = ? sin( ) + @ cos( ) + A sinh( ) + B cos( ) sin
+@
2
+@
2
P
( , ) = ( ?
+ A + B ) sin T + A + B ) cos T
P
,
,
( , ) = ? sin( ) + @ cos( ) + A sinh( ) + B cos( ) cos where ? , @ , A , B , and are arbitrary constants.
P P
2 4
+ ,
2 4
+ ,
© 2002 by Chapman & Hall/CRC Page 608
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9.2. FOURTH-ORDER ONE-DIMENSIONAL NONSTATIONARY EQUATIONS
9.2.4-2. Domain: 0 ≤ ≤ C . Solution in terms of the Green’s function. 1 . We consider boundary value problems on an interval 0 ≤ ≤ C with the general initial condition
= ( ) at
*
= 0,
4
= ( ) at
=0
and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as
*
( , ) = *
D
0
D 4
(E ) F ( , E , ) E +
#
(E ) F ( , E , ) E + #
0
D
0
(E , ) F ( , E , − ) E >
0
#
. #
2 . Paragraphs 9.2.4-3 through 9.2.4-10 present the Green’s functions for various types of boundary conditions. The Green’s functions can be evaluated I I from the formula I
I
( )
( , E , ) = G F
I
I
HJI
L
(E ) sin TVU 2I
=1 K
L K
2
,
+
2 4 U
+
4
K the self-adjoint eigenvalue problem for the fourthwhere the and ( ) are determined by solving 4 order ordinary differential equation − = 0 subject to appropriate boundary conditions. The K norms of eigenfunctions can be calculated by formula (2) from Paragraph 9.2.3-1. K
K
9.2.4-3. The function and its first derivative are prescribed at the ends:
* M
=
= 0 at
Green’s function: 4 HJI
( , E , ) = F
where I
C
K the
)−sin(
= 0 at
(E ) sin VU 2I
4
2
+
2 4 U
, K
2
2 # K
I
) − cosh(
)= #
(
I
)−cos(
I I
+
I
C ) cosh(
=C.
I
KI I
C
(K C )
K
* M
=
I
( I )
4 G =1
I
( I) = sinh(
I I
= 0,
C
)−cos(
,
I C ) sinh(
I
)−sin(
) ;
are positive roots of the transcendental equation cosh( C ) cos( C ) = 1.
9.2.4-4. The function and its second derivative are prescribed at the ends:
* M M
=
= 0 at
Green’s function: F
( , E , ) =
2 HJI C
I
sin( G
= 0,
* M M
= 0 at
=C.
I I
=
) sin( E
)
sin TVU 2I
I
+
,
+
2 4 U
=1
4
7
= N
.
C
9.2.4-5. The first and third derivatives are prescribed at the ends: * M
* M M M
=
= 0 at
Green’s function: F
( , E , ) =
sin T C
P
P
+
2 HI C
= 0,
* M
I
cos( G =1
=
) cos(
= 0 at
=C.
I I
* M M M
E
)
sin TVU 2I U
2 4
4
+
+
I
,
= N C
7
.
© 2002 by Chapman & Hall/CRC Page 609
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.2.4-6. The second and third derivatives are prescribed at the ends: M M
* M M M
=
= 0 at
M M
= 0,
* M M M
=
Green’s function: F
3
( , E , ) = / 1 +
where I
C2
I
I
( )I = sinh(
+
I
)−sin( C
P
C
I P
sin
(2 − C )(2E − C )1
4 HJI C
(E ) sin VU 2I
2 (C
=1 K
)
K
U
K
) − cosh(
I
( ) G
I
)+cos(
4
2 4
I
+
+
,
I
)−cos( C
=C.
I I
I
C ) cosh(
= 0 at
I
C ) sinh(
)+sin(
) ;
K the are positive roots of the transcendental equation cosh( C ) cos( C ) = 1. For the numerical values of the roots, see Paragraph 9.2.3-2.
9.2.4-7. Mixed boundary conditions are prescribed at the ends (case 1):
=
* M
= 0 at
Green’s function:
where I I
C
G
= 0 at
=C.
I
I ( ) (E ) sin TVU 2I 4 + , | ( C ) ( C )| U 2 4 + K K K I
I
I
) − cosh(
)−cos(
* M M
I
K
C ) cosh(
=
I
4
I
)−sin( C
I I
=1
I
( I) = sinh( K the
2 HJI
( , E , ) = F
= 0,
I
)−cos( C
I
C ) sinh(
)−sin(
) ;
) ;
are positive roots of the transcendental equation tan( C ) − tanh( C ) = 0.
9.2.4-8. Mixed boundary conditions are prescribed at the ends (case 2):
=
* M
= 0 at
Green’s function: I
I
( I) = sinh( K the
C
I C ) cosh(
=1 K
)
K
K
)−cos(
= 0 at
(E ) sin TVU 2I
2 (C
I
)+sin(
* M M M
2 4 U
I
4
+
) − cosh(
,
+
I
=C.
I
( ) G
C
=
I I
4 HJI
( , E , ) = F
where I
* M M
= 0,
C
I
I
)+cos(
C ) sinh(
I
)−sin(
are positive roots of the transcendental equation cosh( C ) cos( C ) = −1.
9.2.4-9. Mixed boundary conditions are prescribed at the ends (case 3):
=
* M M
= 0 at
Green’s function: F
( , E , ) =
2 HJI C
I
sin( G =0
* M
= 0,
) sin(
* M M M
= 0 at
I I
=
E
)
sin TVU U
2I 4 2 4
+
+
=C.
I
,
= N
(27 + 1) . 2C
© 2002 by Chapman & Hall/CRC Page 610
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9.2. FOURTH-ORDER ONE-DIMENSIONAL NONSTATIONARY EQUATIONS
9.2.4-10. Mixed boundary conditions are prescribed at the ends (case 4):
=
* M M
= 0 at
* M M
= 0,
Green’s function: I
=1 K
I
where
( ) = sin( I
K
2 4 U
I
) sinh( C
)
K
I
2I 4
(E ) sin VU
2 (C
=C.
I
( ) G
C
= 0 at
I I
4 HI
( , E , ) = F
* M M M
=
+
+
I
) + sinh(
,
I C
) sin(
);
K the transcendental equation tan( C ) − tanh( C ) = 0. are positive roots of
the
9.2.5. Other Equations 9.2.5-1. Equations containing the first derivative with respect to .
1.
+'
4
2
++
4
= W ( , ). (
The change of variable ( , ) = X − Y ( , ) leads to the equation ( * *
+
( * 4 2* 4
= X Y > ( , ),
which is discussed in Subsection 9.2.1. 2.
8
=' ]
Z [
4
Z
Z \
[ ] 4 Z
.
This is a special case of equation 9.6.4.2 with ^ = 1 and 7 = 4. 3.
4
= + ( )Z
Z [ Z \
\
+ []
Z
[
] 4
,
( ) + - ( )] Z [ \
\
Z
+. ( ) . \ [
]
This is a special case of equation 9.6.4.1 with 7 = 4. The transformation _
4 (
(` , a ) = ( $ , ) exp /cb
where g (a ) = exp icb j
d
(a ) e a2f ,
= ` g (a ) + b $
(a ) g (a ) e a , h
=b ^
(a ) g
4
(a ) e a ,
(a ) e a2f , leads to the constant coefficient equation k l k h
k 4l
= k
, $ 4
which is discussed in Subsection 9.2.1. 4.
Z [
= (m ]
2
+ 5 ] + 6 )4 n
Z \
4o
n p
4
.
This is a special case of equation 9.6.4.4 with q = 1 and 7 = 4.
© 2002 by Chapman & Hall/CRC Page 611
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.2.5-2. Equations containing the second derivative with respect to r . 2o
n
5.
2
n s u (v
+t n
o
2 n
+m
n s
4o
n p
4
= W ( p , s ).
With , r ) ≡ 0 this equation governs transverse vibration of an elastic rod in a resisting medium with velocity-proportional resistance coefficient.l The change of variable w (v , r ) = exp x − 21 q ry (v , r ) leads to the equation k 2l k r 2
k 4l
2
+z
k
which is discussed in Subsection 9.2.4. 2o
n
6.
2
n s
8 n
=m p
4o 4
n p
v 4
− 14 q
2l
1 2 q ry u
= exp x
(v , r ),
.
This is a special case of equation 9.6.4.2 with q = 2 and 7 = 4. 2o
n
7.
2
n s
2
= (m p
+ 5 p + 6 )4 n
4o
.
4
n p
This is a special case of equation 9.6.4.4 with q = 2 and 7 = 4. 8.
n {
n s
2
2
– n
2|
n p
o
= 0.
1 } . General solution (two representations): l l w ( v , r ) = r 1 ( v , r ) + 0 ( v , r ), where
l ~
=
l
l
(v , r ) = v 1 (v , r ) + 0 (v , r ), k l ~ k l ~ (v , r ) is an arbitrary function satisfying the heat equation − = 0; q = 1, 2. w
l ~
2 } . Fundamental solution:
v 2
r
(v , r ) = exp { − | . 2 4r 3 } . Domain: − < v < . Cauchy problem. Initial conditions are prescribed: k w = 0 at r = 0, w = j (v ) at r = 0. Solution: ( v − E )2 r w (v , r ) = b
exp i − f j (E ) e E . 4r 2 − &
Reference: G. E. Shilov (1965).
9.
n
4o
n s
4
– n
4o
n p
4
= 0.
1 } . Fundamental solution:
1 (r + v ) ln |r + v | 2 1 1 1 − (r − v ) ln |r − v | + |r + v | + |r − v | . 2 8 8 2 } . Domain: − < v < . Cauchy problem. Initial conditions are prescribed: k k w = 0 at r = 0, w = 0 at r = 0, w = j (v ) at r = 0. Solution: (v , r ) =
1 2
w
r
&
Reference: G. E. Shilov (1965).
ln
v 2
+ r 2 − v arctan
(v , r ) = b
−
v
r
−
(v − E , r ) j (E ) e E .
© 2002 by Chapman & Hall/CRC Page 612
613
9.3. TWO-DIMENSIONAL NONSTATIONARY FOURTH-ORDER EQUATIONS
n
10.
4o
n s
4o
–2 n
4
2
n s
+ n
2
n p
4o 4
n p
= 0.
General solution (three representations): w
(v , r ) = j 1 (r − v ) + j 2 (r + v ) + r 1 (r − v ) + 2 (r + v ) , w
( v , r ) = j 1 ( r − v ) + j 2 ( r + v ) + v 1 ( r − v ) + 2 ( r + v ) , w
(v , r ) = j 1 (r − v ) + j 2 (r + v ) + (r + v ) 1 (r − v ) + (r − v ) 2 (r + v ),
where j 1 ( ), j 2 ( ), 1 ( ), and 2 ( ) are arbitrary functions.
&
Reference: A. V. Bitsadze and D. F. Kalinichenko (1985).
9.3. Two-Dimensional Nonstationary Fourth-Order Equations 9.3.1. Equations of the Form
+
2
4
+
4
4
=
4
( , , )
9.3.1-1. Domain: 0 ≤ v ≤ 1 , 0 ≤ ≤ 2 . Solution in terms of the Green’s function. We consider boundary value problems in a rectangular domain 0 ≤ v ≤ 1 , 0 ≤ ≤ 2 with the general initial condition w = j (v , ) at r = 0 and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as
1
2
(v , , r ) = b b j ( , ) (v , , , , r ) e e + b w
0
0
0
b
0
1
2
b
u
0
( , , h ) (v , , , , r − h ) e e e h .
Below are the Green’s functions for various types of boundary conditions. 9.3.1-2. The function and its first derivatives are prescribed at the sides of a rectangle: k
= w = 0 at k w = w = 0 at w
Green’s function:
1 2
J
=1 =1 O £ ¤ ¤ ( ¡ 1 ) ¥ ¤ ¤ ( 2 ) 2
¦
(v ) = sinh( £
where the
and ¡
( )
¢
v 2
1 ) − sin(
¢ − cosh( ¡
¥
£
)= ¦
£ ¤ ¤ ( v
Here,
4 4
k
= w = 0 at k w = w = 0 at w
¢
16
(v , , , , r ) =
= 0, = 0, v
2
,
£
¥ ¤ ¤
(v )¥ ( ) £
( )¥ ( ) exp −(
2
¡
( ) = ¦
1 ) cosh(
v
¥ 2
¦
¢ 1 ) − cos( ¡
= 1, = 2. v
v
¢
+ 4 ) z 2 r ,
.
) − cos(
¢ 1 ) sinh( ¡
4
v
)
¢ ) − sin( ¡
v
) ,
= sinh( ¡ ¢ 2 ) − sin( ¡ ¢ 2 ) cosh( ¡ ¢ ) − cos( ¡ ¢ ) − cosh( 2 ) − cos( 2 ) sinh( ) − sin(
) ,
are positive roots of the transcendental equations ¢ ¢ ¢
cosh( 1 ) cos( 1 ) = 1, ¡
¡
cosh( 2 ) cos( 2 ) = 1
( = 1 § 2 ). ¡
© 2002 by Chapman & Hall/CRC Page 613
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.3.1-3. The function and its second derivatives are prescribed at the sides of a rectangle:
= ¨ w = 0 at V w =¨ w = 0 at w
Green’s function:
4
(v , , , , r ) =
= ¨ w = 0 at V w =¨ w = 0 at w
sin( =1 =1 ¡
1 2
= 0, = 0, v
¢ ¢
= © , 1
¡
+ 4 ) z 2 r ,
¡
.
2
¢
4
) sin( ) exp −(
= ª
¢
) sin( ) sin( v
= 1, = 2. v
¡
9.3.1-4. The first and third derivatives are prescribed at the sides of a rectangle: w
=¨ w = 0 at VV w =¨ w = 0 at
Green’s function:
J
1
(v , , , , « ) =
1 2
v
= 0, = 0,
=0 =0 ¬
¢
= © , 1
) sin( ) cos(
¡
= ª
=¨ w = 0 at VV w =¨ w = 0 at
¢
cos(
¬
w
=®
,
2
¬
¡
v
= 1, = 2.
¢
) cos( ) exp −(
¡
¢
4
+ 4 ) z 2 « ,
¡
1 for = 0, 2 for © ≠ 0. ©
9.3.1-5. The second and third derivatives are prescribed at the sides of a rectangle: ¯ ¯ V
¯ =¨ = 0 at VV ¯ =¨ = 0 at
Green’s function:
2 (
, ,« ) = , ,« ) =
Here, £
1 1
1 2
+ +
3 3
− 2 )(2 − 2 ) +
(2 23
( ) = sinh(
( )
¢
= 1, = 2.
, , « )
2 (
, , « ),
4
£
( ) £ ( ) exp(− £ 2 ( 1)
¥
( )¥ ( ) exp(− 4 z 2 « ). ¥ 2 ( 2 )
1
1 ) − sin(
=1
4
2 =1
¢ − cosh( ¡
¥
1(
− 1 )(2 − 1 ) +
(2 13
¯ =¨ = 0 at VV ¯ =¨ = 0 at
( , , , , « ) =
1(
¯ ¯ V
= 0, = 0,
¢ 1 ) − cos( ¡
¢ 1 ) sinh( ¡
4
z 2«
),
¡ ¢
1 ) cosh(
) + cos( ¢ ) + sin( ¡
)
) ,
= sinh( ¡ ¢ 2 ) − sin( ¡ ¢ 2 ) cosh( ¡ ¢ ) + cos( ¡ ¢ ) − cosh( 2 ) − cos( 2 ) sinh( ) + sin(
) ,
and are positive roots of the transcendental equations
where the
¢
¡
cosh( 1 ) cos( 1 ) = 1, ¡
¢
cosh( 2 ) cos( 2 ) = 1.
¡
© 2002 by Chapman & Hall/CRC Page 614
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9.3. TWO-DIMENSIONAL NONSTATIONARY FOURTH-ORDER EQUATIONS
9.3.1-6. Mixed boundary conditions are prescribed at the sides of a rectangle:
= ¨ ¯ = 0 at V ¯ = ¨ ¯ = 0 at ¯
Green’s function:
( , , , ,« ) =
4 1 2
¯ =¨ = 0 at ¯ VV ¯ ¨ =¨ = 0 at
= 0, = 0,
¯
¨
¢ ¢
(2 + 1) = © , 2 1
=
¡
(2
¢
) sin( ) sin(
sin( =0 =0 ¡
= 1, = 2.
ª
) sin( ) exp −( + 1)
2 2
4
¢
+ 4 ) z 2 « ,
¡
.
¡
9.3.2. Two-Dimensional Equations of the Form
2
2
+
2° °
=0
This equation governs two-dimensional free transverse vibration of a thin elastic plate; the unknown ¯ is the deflection (transverse displacement) of the plate’s midplane points relative to the original plane position. Here, ± ± = ± 2 and ± is the Laplace operator that is defined as 2
±
= ²
³
³ ´ ³
2
+
2 2
+
2
³
1³
in the Cartesian coordinate system,
2
+ ³
µ ³ µ
³ µ
2
1
2
µ
³ ¶
Particular solutions: 9.3.2-1.
¯
( , · , « ) = ¸¹
1
sin( º 1 ) + »
¯
( , · , « ) = ¸¹
1
sin( º
¯
(
, · , « ) = ¸¹
1
sinh( º
1
¯
sinh( º
1
( , · , « ) = ¸¹
1
¯
(½ , ¾ , « ) = ¸¹ Ç
1¿ À
(½ , ¾ , Æ
) = ¸¹
)¼ ¸¹
)+»
1
cosh( º
1
)+»
1
cosh( º
1
(º ½ ) + ¹ (º ½ ) + ¹
1¿ À
cos( º 1 )¼ ¸¹
1
) + » 1 cos( º 1
1
in the polar coordinate system.
2
³
(º ½ ) + ¹
2Á À
(º ½ ) + ¹
2Á À
2
sin( º 2 · ) + »
2
sin( º 2 · ) + »
2 1 +º 2 2 cos( º 2 · )¼ cos ¸ ( º 1 + º 2
)¼ ¸¹
2
sinh( º 2 · ) + »
2
cosh( º 2 ·
)¼ ¸¹
2
sinh( º 2 · ) + »
2
cosh( º 2 ·
(º ½ ) + ¹
3 ÂVÀ
(º ½ ) + ¹
3 ÂVÀ
À
À
( º ½ )¼ cos(Ä ¾ ) sin( º
4Ã 4Ã
2 2 ) z «¼ , 2 2 ) z «¼ , )¼ sin ¸ ( º 12 + º 22 ) z « ¼ )¼ cos ¸ ( º 12 + º 22 ) z « ¼ 2Å Æ
cos( º 2 · )¼ sin ¸ ( º
( º ½ )¼ sin(Ä ¾ ) cos( º
, ,
),
2Å Æ
),
where ¹ 1 , ¹ 2 , ¹ 3 , ¹ 4 , » 1 , » 2 , º , º 1 , º 2 are arbitrary constants, the ¿ À (È ) and Á À (È ) are the Bessel functions of the first and second kind, the  À (È ) and à À (È ) are the modified Bessel functions of the first and second kind, ½ = É Ê 2 + · 2 , and Ä = 0, 1, 2, ËËË 9.3.2-2. Domain: − Ì
< Ê < Ì , −Ì
< · < Ì . Cauchy problem.
Initial conditions are prescribed: Ç
Æ
= Í (Ê , · ) at
= 0,
Î Ï
Ç
= Ð (Ê , · ) at
Æ
= 0.
Poisson solution: Ç
1 (Ê , · , Æ ) = Ñ Ò Ó Ò Ó −
Ï
+ 2È Ö Å Æ , · + 2× Ö Å Æ Ø sin ÔTÈ
Í ÔÕÊ
−
2
+ × 2Ø Ù È Ù ×
1 Ó Ó + Ñ Ò Ù Ú Ò Ó Ò Ó Ð ÔTÊ + 2È Ö Å Ú , · + 2× Ö Å Ú Ø sin ÔTÈ 0
Green’s function: Û
Þß
−
Ó
−
2
+ × 2Ø Ù È Ù × .
Ó
1 (Ê , · , È , × , Æ ) = Ñ Å Ò 4
Ï 0
sin Ü
( Ê − È )2 + ( · − × ) 2 Ù Ú Ý Ú . 4Å Ú
References: A. N. Krylov (1949), I. Sneddon (1951), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
© 2002 by Chapman & Hall/CRC Page 615
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.3.2-3. Domain: 0 ≤ Ê ≤ à 1 , 0 ≤ · ≤ à 2 . Solution in terms of the Green’s function. We consider boundary value problems in a rectangular domain 0 ≤ Ê ≤ à 1 , 0 ≤ · ≤ à 2 with the general initial conditions Ç Ç = Í (Ê , · ) at Æ = 0, Î Ï = Ð (Ê , · ) at Æ = 0 and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as Û Û Ç
1 2 1 2 Î (Ê , · , Æ ) = Æ Ò á Ò á Í (È , × ) (Ê , · , È , × , Æ ) Ù × Ù È + Ò á Ò á Ð (È , × ) (Ê , · , È , × , Æ ) Ù × Ù È . Î
0
0
0
0
Paragraphs 9.3.2-4 through 9.3.2-6 present the Green’s functions for three types of boundary conditions. 9.3.2-4. Domain: 0 ≤ Ê ≤ à 1 , 0 ≤ · ≤ à 2 . All sides of the plate are hinged. Boundary conditions are prescribed: Ç
= Î Ç = 0 at V Ç = Î ´ ´ Ç = 0 at
Ç
= Î Ç = 0 at V Ç = Î ´ ´ Ç = 0 at
= 0, · = 0, Ê
Ê
= à 1, · = à 2.
Green’s Û function: â
â
4 (Ê , · , È , × , Æ ) = Å Ó à 1à 2
sin(ä À Ê ) sin( å ã · ) sin(ä À È ) sin( å ã × )
Ó
À =1 ã =1 Ñ Ä
=
ä À
à1
å ã
,
=
Ñ ç à2
æ À ã
,
= ä 2À + å 2ã .
æ À ã
,
sin( æ À ã Å Æ )
9.3.2-5. Domain: 0 ≤ Ê ≤ à 1 , 0 ≤ · ≤ à 2 . The 1st and 3rd derivatives are prescribed at the sides: Ç
Ç =Î = 0 at Ç Ç Î ´è = Î ´ èV´ èV´è = 0 at Î
= 0, · = 0, Ê
Ç
Ç =Î = 0 at Ç Ç Î ´è = Î ´ èV´ èV´è = 0 at Î
Ê
= à 1, · = à 2.
Green’s function: Û â
â
1 (Ê , · , È , × , Æ ) = Å Ó à 1à 2
If Ä = ç
Ñ
=
ä À
Ä
À
À =0 ã =0 é é Ñ ç å ã = , à2
,
à1
Ó
ã
cos(ä À Ê ) cos( å ã · ) cos(ä À È ) cos( å ã × ) æ À ã
= ä 2À + å 2ã ,
=ê
é
À
sin( æ À ã Å Æ ) æ À ã
,
1 for Ä = 0, 2 for Ä ≠ 0.
= 0, the ratio sin( æ À ã Å Æ ) ë æ À ã must be replaced by Å Æ .
9.3.2-6. Domain: 0 ≤ Ê ≤ à 1 , 0 ≤ · ≤ à 2 . Mixed boundary conditions are set at the sides: Ç
Ç
=Î Î è
Ç
=0
=´ ´Î èVèVè
Ç
= 0, · = 0,
at = 0 at Ê
Ç
=Î Î è
Ç
Ç
=0
=´ ´Î èVèVè Ç
at = 0 at
= à 1, · = à 2. Ê
Green’s function: Û â
â
2 (Ê , · , È , × , Æ ) = Å Ó à 1à 2 ä À
=
Ñ
Ä à1
,
å ã
ã
Ó
À =1 ã =0 é Ñ ç
=
à2
,
sin(ä À Ê ) cos( å ã · ) sin(ä À È ) cos( å ã × ) æ À ã
= ä 2À + å 2ã , é
ã
=ê
1 for ç 2 for ç
sin( æ À ã Å Æ ) æ À ã
,
= 0, ≠ 0.
© 2002 by Chapman & Hall/CRC Page 616
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9.3. TWO-DIMENSIONAL NONSTATIONARY FOURTH-ORDER EQUATIONS
9.3.2-7. Domain: 0 ≤ ½ < Ì , 0 ≤ ¾ ≤ 2Ñ . Cauchy problem. Initial conditions for the symmetric case in the polar coordinate system: Ç
Solution:
Æ
= Í (½ ) at
= 0,
1 (½ , Æ ) = Å Æ Ò Ó È Í (È ) ¿ 2 0 Ç
Ç
Î Ï
0ì
Æ
= 0 at
È ½ 2Å Æ í
sin ì
= 0.
È 2
+½ 4Å Æ
2
í
Ù È
,
where ¿ 0 ( î ) is the zeroth Bessel function.
Þß
References: I. Sneddon (1951), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
9.3.2-8. Domain: 0 ≤ ½ ≤ ï , 0 ≤ ¾ ≤ 2Ñ . Transverse vibration of a circular plate. Initial and boundary conditions for symmetric transverse vibrations of a circular plate of radius ï with clamped contour in the polar coordinate system: Ç
Æ
= Í (½ ) at Ç =0 at
Ç
= Ð (Æ ) at Ç Î =0 at
= 0, ½ =ï ,
Î Ï
Æ
= 0; ½ =ï .
µ
Solution: â
Ç
(½ , Æ ) = Ó À =1 ð À
cos( Å º 2À Æ ) + » À sin( Å º 2À Æ )¼ ð À (½ ),
¸¹ À
(½ ) = Â 0 ( º À ï ) ¿ 0 ( º À ½ ) − ¿ 0 ( º À ï ) Â 0 ( º À ½ ),
where the º À are positive roots of the transcendental equation (the prime denotes the derivative) ) Â 0ñ ( º ï ) − Â 0 ( º ï ) ¿ 0ñ ( º ï ) = 0,
¿ 0(º ï
and the coefficients ¹ À and » À are given by
Þß
1
= ò
¹ À
ð À
ò
2 Ò
ò
0
(½ ) ð À (½ )½ Ù ½ , Í
ó
ò 2
ð À
= 14 ï
6
¸ ð Àñ ñ ( ï
1 ò Ò ó Ð (½ ) ð À (½ )½ Ù ½ , = Å 2ò º À ð À 2 0 = ï 2 ¿ 02 ( º À ï ) Â 02 ( º À ï ).
» À
)¼
2
Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
9.3.3. Threeand õ 2ö
õ ÷
2
+
ø
2ù
-Dimensional Equations of the Form ô
ù
=0 ö
9.3.3-1. Three-dimensional case. Cauchy problem. Domain: − Ì
< Ê < Ì , −Ì Ç
< ú < Ì , −Ì
< î < Ì . Initial conditions are prescribed: Æ
= Í (Ê , ú , î ) at
= 0,
Î Ï
Ç
= 0 at
Æ
= 0.
Solution: Ç Þß
(Ê , ú , î , Æ ) =
1
Ñ Å Æ Ø 3 Ò −Ó Ô 2Ö
Ò
−
Ó
Ò
−
Ó
Í
(Ê + È , ú + × , î + û ) cos ì
È 2
+× 2+û 4Å Æ
2
−
3Ñ Ù È Ù × Ù û . 4 í
Ó Ó Ó Reference: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).
© 2002 by Chapman & Hall/CRC Page 617
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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.3.3-2. Three-dimensional case. Boundary value problem. Domain: 0 ≤ Ê ≤ à 1 , 0 ≤ ú ≤ à 2 , 0 ≤ î ≤ à 3 (rectangular parallelepiped). Initial conditions: Ç
Æ
= Í (Ê , ú , î ) at
= 0,
Ç
Î Ï
Æ
= Ð (Ê , ú , î ) at
= 0.
Boundary conditions: = Î ü ü Ç = 0 at = Î èVè Ç = 0 at Ç Ç Ç
= Î ý ý Ç = 0
at
Ç Ç
= 0, î
= Î ü ü Ç = 0 at = Î èVè Ç = 0 at Ç
= 0, ú = 0, Ê
= Î ý ý Ç = 0
Solution:
= à 1, = à 2, Ê
at
ú
= à 3. î
Û Ç
1
2
3
2
3
Î (Ê , ú , î , Æ ) = Æ Ò á Ò á Ò á Í (È , × , û ) (Ê , ú , î , È , × , û , Æ ) Ù û Ù × Ù È Û Î 0 0 0 1
+ Ò á Ò á Ò á Ð (È , × , û ) (Ê , ú , î , È , × , û , Æ ) Ù û Ù × Ù È , 0
0
0
where Û âJþ
8
â
(Ê , ú , î , È , × , û , Æ ) = Å Ó à 1à 2à 3 þ
=
Ñ
ä
Ä
=
å ã
,
à1
þ
âÿ
Ó Ó æ =1 ã =1 =1
,
à2
sin(ä
ã
=
Ñ
à3
ÿ Ê
þ
× sin(ä ÿ
Ñ ç
þ ÿ
1
) sin( å ã ú ) sin( î ) ÿ
È )þ sin( ÿ å ã × þ ) sin(
, æ
ã
=ä
2
) sin( æ ÿ û
þ ã
ÿ Å Æ
),
+ å 2ã + 2 .
þ
9.3.3-3. -dimensional ÿcase. Cauchy problem. Domain:
= {− Ì
< Ì ; = 1, ËËË , }. Initial conditions are prescribed:
=
= 0,
≡ =
=0
2
+
2
2 2
.
Particular solutions:
( , )=
.
( , ), B
=1
.
are solutions of the Helmholtz equations
where the
− 5
characteristic equation
= 0.
M
')(
B
L
=0
= 0 and the L
are roots of the L
Reference: A. V. Bitsadze and D. F. Kalinichenko (1985). ? ?
9.
N
@? >
[ ] = 0.
=0
Here, is any constant coefficient linear differential operator with arbitrarily many independent variables 1 , , . Particular solutions: O
!"!"!
C
( 1,
!"!"!
,
)=
C
=1
C
( 1,
are solution of the equations [ ] −
M
B
=0
L
= 0, and the
,
),
C
C
O
!"!"!
C
P
where the equation
B C
C
DC
L
= 0 the L
are roots of the characteristic
are arbitrary constants. P
© 2002 by Chapman & Hall/CRC Page 643
644
HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.6. Higher-Order Linear Equations with Variable Coefficients 9.6.1. Equations Containing the First Time Derivative 9.6.1-1. Statement of the problem for an equation with two independent variables.
Consider the linear nonhomogeneous partial differential equation
−
where O
,
O
,
[ ] = ( , ),
is a general linear differential operator of order
with respect to the space variable ,
;
O
,
(1)
[ ]≡
( , )
=0
,
(2)
whose coefficients = ( , ) are sufficiently smooth functions of both arguments for ≥ 0 and ≤ 2 . The subscripts and indicate that the operator , is dependent on the variables 1 ≤ and . We set the initial condition = ( ) at =0 (3)
O
and the general nonhomogeneous boundary conditions
(1)
[ ]≡
J
B
B
(2)
()
H
=
( ) at
= 1,
(
1
A
, ),
!"!"!
Q
(4) B
=
B
,
=0
(1)
=
−1
J
()
H
=0
[ ]≡
(1)
(2)
−1
(2) ,
( ) at
=
(
2
= + 1, A
Q
, ),
!"!"!
B
B
where ≥ 1 and ≥ + 1. We assume that both sets of the boundary forms (1) [ ] ( = 1, (2) [ ] ( = + 1, , ) are linearly independent, which means that for any nonzero the following relations hold: Q
Q
, ) and = ()
A
!"!"!
J
A
Q
!"!"!
B
C
J
R
B
(1)
() R
[ ]
0, S
C
J
=1 B
[ ]
B
B
0. S
J
= +1 B
(2)
() R
Q
R
B B
In what follows, we deal with the nonstationary boundary value problem (1), (3), (4). B
B
9.6.1-2. The case of general homogeneous boundary conditions. The Green’s function.
The solution of equation (1) with the initial condition (3) and the homogeneous boundary conditions
J
(1)
[ ] = 0 at
(2)
[ ] = 0 at
B
=
=
1
(
2
(
= 1, A
, ),
!"!"!
Q
= + 1, A
Q
!"!"!
(5)
, )
J
can be written as B
( , )=
#
2
( ) ( , , , 0) .
Here,
9
.
&
+ .
#
0
1
2
( , ) ( , , , )
#
.
T
( , , , ) is the Green’s function that satisfies, for > ;
9
.
T
− 9
O
,
[ ]=0 9
9
.
T
&
.
&
T
.
(6)
1 T
≥ 0, the homogeneous equation (7)
© 2002 by Chapman & Hall/CRC Page 644
645
9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
with the special nonhomogeneous initial condition = ( − ) at 9
.
=
(8) T
and the homogeneous boundary conditions J
(1)
[ ] = 0 at
(2)
[ ] = 0 at
B
=
9
9
=
1
(
2
(
= 1, A
, ),
!"!"!
Q
= + 1, A
Q
(9)
, ).
!"!"!
J
The quantities and appear in problem (7)–(9) as free parameters ( 1 ≤ ≤ 2 ), and ( ) is the Dirac delta function. It should be emphasized that the Green’s function is independent of the functions ( , ), ( ), (1) ( ), and (2) ( ) that characterize various nonhomogeneities of the boundary value problem. If the coefficients , (1) , and (2) determining the differential operator (2) and boundary conditions (4) are independent of time , then the Green’s function depends only on three arguments, ( , , , ) = ( , , − ). B
.
T
.
9
,
,
;
B
B
H
H
9
B
9
')(
.
.
T
B
T
Reference: Mathematical Encyclopedia (1977, Vol. 1).
9.6.1-3. The case of nonhomogeneous boundary conditions. Preliminary transformations. To solve the problem with nonhomogeneous boundary conditions (1), (3), (4), we choose a sufficiently smooth “test function” = ( , ) that satisfies the same boundary conditions as the unknown function; thus, (1) [ ] = (1) ( ) at = 1 ( = 1, , ), (10) (2) (2) [ ]= ( ) at = 2 ( = + 1, , ). U
U
U
,
U
,
A
A
!"!"!
Q
J
B
B
Q
!"!"!
J
Otherwise the choice of the “test function” is arbitrary and is not linked to the solution of the equation in question; there are infinitely many such functions. B
Let us pass from
= V
B
U
( , ) to the new unknown V
W
X
= ( , ) by the relation Y
Y
W
X
( , ) = ( , ) + ( , ). V
W
X
Y
W
X
Z
W
(11)
X
Substituting (11) into (1), (3), and (4), we arrive at the problem for an equation with a modified right-hand side, [
[
− Y
[
\
[ ]=
,^ ]
Y
( , ), _
W
X
( , )= _
W
X
( , )− _
W
+ Z
X
[
X
\
,^ ]
[ ],
(12)
Z
X
subject to the nonhomogeneous initial condition = ( ) − ( , 0) Y
`
W
Z
at
W
=0 X
(13)
and the homogeneous boundary conditions a
a
(1) b
[ ] = 0 at
(2) b
[ ] = 0 at
=
Y
W
Y
= W
W
1
(
W
2
(
= 1, c
, ),
deded
f
= + 1, c
f
dedgd
(14)
, ). h
The solution of problem (12)–(14) can be found using the Green’s function by formula (6) in which one should replace by , ( , ) by ( , ), and ( ) by ( ) = ( ) − ( , 0). Taking into account relation (11), for we obtain the representation V
Y
_
W
X
_
W
X
`
W
2 ]
( , )= V
W
X
i
j
2
i
k
W
j
X
Z
+
lmj
j
k
W
j
X
W
2 ]
^ i
( , , , ) k
W
j
X
n
\
o
2 ]
i
Z
i
n
k
W
j
X
n
lmj
+ ( , )
lmn
Z
W
X
[ Z
( , ) ( , , , ) j
[
0 j
2 ]
^ i
j
1 ]
]
[ ( , )]
,p
( , ) ( , , , ) _
0
−
lmj
V
^ i
1 ]
0
W
( , 0) ( , , , 0) Z
i
`
1 ]
+
W
( ) ( , , , 0) `
]
−
`
n
1
n
k
W
j
X
n
lmj
lmn
lmj
n
lmn
.
(15)
1 ]
Changing the order of integration and integrating by parts with respect to , we find, with reference to the initial condition (8) for the Green’s function, n
^ i
[ Z k [
0 n
lmn
= ( , ) ( − ) − ( , 0) ( , , , 0) − Z
j
X
q
W
j
Z
j
k
W
j
X
[ ^
i
Z
0
( , ) j
n
k
( , , , ) W
[
j
X
n
lmn
.
(16)
n
© 2002 by Chapman & Hall/CRC Page 645
646
HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
We transform the inner integral of the last term in (15) using the Lagrange–Green formula [see Kamke (1977)] to obtain 2 ]
k
1 ]
\
w
[ ]
,p o
w
Z
k
r
,
Z
y
\
[ ]
,p
or
1
w
z8{
(−1) u
( , ) j
k
n
k
, |
}
−1
vx~
k
t o
y ~
v
y
(−1)
+ =
}
}
j
~
z7{
}
, ) n
k
|
,
j
and
+1 ( j
j
n
(17) y
y
= ( , ); and
[ ] of (2);
,
k
u
=0
[ ] is the differential form adjoint with
Z
=] 2 , =] 1
o
[ , ]
s
t
[ , ]≡
s
+
lmj
j
=0
where
i
y
vxw
[ ]≡
,p
or
=
lmj
] w
\
2 ]
i
are nonnegative integers.
Using relations (16) and (17), we rewrite solution (15) in the form
2
( , )=
( ) ( , , , 0)
1
+
2
0
( , ) ( , , , )
1
+
[ , ]
s
0
= =
8
2 1
. (18)
This formula was derived taking into account the fact that the Green’s function with respect to and satisfies the adjoint equation*
+
,
[ ] = 0.
For subsequent analysis, it is convenient to represent the bilinear differential form
C
s
−1
[ , ]=
[ ]=
[ ],
C
C
¡
[ , ] as
(−1)
=0
C
− −1
s
C
+ +1 (
, )
. ¢
(19)
=0
Note that in the special case where operator (2) is binomial,
[ ]=
,
+
0(
= const,
, ) ,
the differential forms in (19) are written as
£
[ , ]=
(−1) =0
− −1
− −1
[ ]=
,
− −1
− −1
−1
− −1
(−1)
− −1
.
9.6.1-4. The case of special nonhomogeneous boundary conditions. Consider the following nonhomogeneous boundary conditions of special form that are often encountered in applications: ¥¤
¥¤
¥¤
(1)
=
¥¤
¦
=
( ) at
1
(
2
(
= 1, A
!"!"!
, ), §
(20)
¥¤
¥¤
=
(2) ¦
( ) at
=
= + 1, A
§
!"!"!
, ). ¨
Without loss of generality, we assume that the following inequalities hold: C
¨
−1≥ ©
1
>
2 ©
>
ª"ª"ª
>
C
, ©
¨
−1≥ ©
C
>
+1
©
+2
>
ª"ª"ª
>
. ©
The Green’s function satisfies the corresponding homogeneous boundary conditions that can be obtained from (20) by replacing by and setting (1) ( ) = (2) ( ) = 0. ¥¤
¥¤
¦
¦
* This equation can be derived by considering the case of homogeneous initial and boundary conditions and using arbitrariness in the choice of the test function = ( , ); it should be taken into account that the solution itself must be independent of the specific form of , because does not occur in the original statement of the problem. By appropriately selecting the test function, one can also derive the boundary conditions (21). }
}
}
«
¬
}
© 2002 by Chapman & Hall/CRC Page 646
647
9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
The adjoint homogeneous boundary conditions, with respect to (20), which must be met by the Green’s function with respect to and have the form
®
[ ] = 0 at
®
=
[ ] = 0 at
=
1
(
2
(
≠
©
¯
©
≠ ¯
= + 1,
, ©
©
°
, B ²
§
= 1,
°
, ;
!"!"!
¨
, ;
!"!"!
§
= 1, ±
= + 1, ±
§
, ),
!"!"!
§
(21)
, ).
!"!"!
¨
These conditions involve the linear differential forms [ ] defined in (19). For each endpoint of the interval in question, the set { } of the indices in the boundary operators (21) together with the set { } of the orders of derivatives in the boundary conditions (20) make up a complete set of nonnegative integers from 0 to − 1. Taking into account the fact that the test function must satisfy the boundary conditions (20) and the Green’s function to conditions (21), we rewrite solution (18) to obtain
©
©
¯
²
¨
2
( , )=
( ) ( , , , 0)
1
−
2
+
0
¥¤
( )
¦
[ ]
=
1
¥¤
( )
¦
[ ]
0
= +1
(2)
¥¤
²
+
1
( , ) ( , , , )
0
=1
(1)
³
²
¥¤
¥¤
=
2
,
(22)
where the [ ] are differential operators with respect to , which are defined in (19). If the Green’s function is known, formula (22) can be used to immediately obtain the solution of the nonhomogeneous boundary value problem (1), (3), (20) for arbitrary ( , ), ( ), (1) ( ) ( = 1, , ), and (2) ( ) ( = + 1, , ). ³
¥¤
¥¤
±
!"!"!
§
¦
±
§
!"!"!
¦
¨
9.6.1-5. The case of general nonhomogeneous boundary conditions. On solving (4) for the highest derivatives, we reduce the boundary conditions (4) to the canonical form ¥¤
¥¤
´
¥¤
−1
µ´
+
¥¤
´
(1)
¥¤
+
¶
´
= 1,
(
1
±
, ),
!"!"!
§
(23)
¥¤ ´
(2)
=
( ) at
·
´
(2)
=
()
²
(1)
=
−1
µ´
()
²
¥¤
=0
¥¤
´
( ) at
·
=
(
2
= + 1, ±
§
, ),
!"!"!
¨
=0 ¶
where the leading terms in different boundary conditions are different, −1≥ ¨
>
1 ©
2 ©
>
ª"ª"ª
>
, ©
−1≥ ¨
>
+1 ©
The sums in (23) do not contain the derivatives of orders 1 , (for = 2 ); thus, (1) ( ) = 0 at = ( = 1, ³
³
>
+2 ©
,
(for
³
©
!"!"!
©
>
ª"ª"ª
. ©
=
1)
´
(2) ¶
¸
( ) = 0 at
²
©
= ¸
¹
!"!"!
( = + 1, E
©
¹
+1 , ©
³
E
´
²
and
§
!"!"!
, ©
³
, ), §
, ).
!"!"!
¨
It can be shown that the solution of problem (1), (3), (23) is given by ¶
2
( , )=
1
¥¤
=1
¥¤
(1)
³
²
−
( ) ( , , , 0)
0
·
2
+
0
¥¤
1
[ ]
=
1
+
¥¤
(2)
²
( )
( , ) ( , , , )
= +1
·
0
( )
¥¤
[ ]
=
2
,
(24)
where the [ ] are differential operators with respect to , which are defined in (19). Relation (24) is similar to (22) but contains the Green’s function satisfying the more complicated boundary conditions that can be obtained from (23) by substituting for and setting (1) ( ) = (2) ( ) = 0. ³
¥¤
¥¤
·
·
© 2002 by Chapman & Hall/CRC Page 647
648
HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.6.2. Equations Containing the Second Time Derivative 9.6.2-1. The case of homogeneous initial and boundary conditions. Consider the linear nonhomogeneous differential equation
2
+ ( , )
2
º
−
= ( , ).
( , )
(1)
=0
We set the homogeneous initial conditions = 0 at = 0 at
= 0, =0
(2)
and the homogeneous boundary conditions (1) »
[ ] = 0 at
»
=
²
(2)
[ ] = 0 at
²
=
1
(
2
(
= 1, ±
!"!"!
, ), §
= + 1, ±
§
(3)
, ),
!"!"!
¨
where the boundary operators (1) [ ] and (2) [ ] are defined in Paragraph 9.6.1-1. The solution of problem (1)–(3) can be represented in the form* »
»
²
=
2
( , )=
Here,
²
( , ) ( , , , )
0
1
( , , , ) is the Green’s function; for >
2
2
+ ( , ) º
.
(4)
≥ 0, it satisfies the homogeneous equation
−
( , )
=0
(5)
=0
with the special semihomogeneous initial conditions =0 at = ( − ) at
¼
= , =
(6)
and the corresponding homogeneous boundary conditions (1) »
² »
(2) ²
[ ] = 0 at
=
[ ] = 0 at
=
1
(
2
(
= 1, ±
!"!"!
= + 1, ±
§
, ), §
!"!"!
(7)
, ). ¨
The quantities and appear in problem (5)–(7) as free parameters ( 1 ≤ ≤ 2 ), and ( ) is the Dirac delta function. One can verify by direct substitution into the equation and the initial and boundary conditions (1)–(3) that formula (4) is correct, taking into account the properties (5)–(7) of the Green’s function.
¼
9.6.2-2. The case of nonhomogeneous initial and boundary conditions. Consider the linear nonhomogeneous differential equation (1) with the general nonhomogeneous initial conditions = 0 ( ) at = 0, (8) = 1 ( ) at =0
* Problem (1)–(3) is assumed to be well posed.
© 2002 by Chapman & Hall/CRC Page 648
649
9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
and the nonhomogeneous boundary conditions, reduced to the canonical form (see Paragraph 9.6.1-5): ¥¤
¥¤
´
¥¤
+
¥¤
´
−1
´
¥¤
+
·
=
( ) at
= 1,
(
1
±
, ),
!"!"!
§
´
(9)
¥¤ ´
(2)
(1)
= ´
(2)
=
()
²
()
¶
−1
´
²
¥¤
=0
¥¤ ´
(1)
·
( ) at
=
(
2
= + 1, ±
§
, ).
!"!"!
¨
=0 ¶
Introducing a test function = ( , ) that satisfies the nonhomogeneous initial and boundary conditions (8), (9) and using the same line of reasoning as in Paragraph 9.6.1-3 for a simpler equation, we arrive at the solution of problem (1), (8), (9) in the form
( , )=
2
−
−
0(
)
·
¾
2
( )
[ ]
=
1
+
=0
0
¥¤
(1)
½
¥¤
( , , , )
=1
³
¥¤ ²
1
1 2
( , ) ( , , , )
0
) ( , 0)
º
[ ]
0
( , , , 0)
¢¥
¥¤
( )
·
0(
(2)
= +1
)+
¥¤
²
1(
1
+
=
2
,
(10)
where the [ ] are differential operators with respect to , which are defined in relations (19), Paragraph 9.6.1-3. If the coefficients of equation (1) and those of the boundary conditions (9) are time independent, i.e., ³
¿
À¥Á
Â
îÄ
Å
= ( ), º
º
=
( ),
´
´
(1)
(2)
= const,
²
then in solution (10) one should set
= const,
²
¶
¶
( , , , )=
( , , − ),
Æ
( , , , )
=0
=−
( , , ).
Æ
9.6.3. Nonstationary Problems with Many Space Variables 9.6.3-1. Equations with the first-order partial derivative with respect to .
Consider the following linear differential operator with respect to variables ¥Ë
[ ]≡
Ç
È
x,
¥Ë
1,
ÉÊÉÊÉ
( 1,
,
!"!"!
,
, )
¥Ë
1
1 + ÌÊÌÊÌ 1
+
!"!"!
1,
!"!"!
,
:
¥Ë
.
(1)
È
The coefficients of the operator are assumed to be sufficiently smooth functions of 1, , , and (and also bounded if necessary). The coefficients of the highest derivatives 1, are assumed to be everywhere nonzero.
ÉÊÉÊÉ
!"!"!
1 . Cauchy problem ( ≥ 0, x ). The solution of the Cauchy problem for the linear nonhomogeneous parabolic differential equation with variable coefficients Í
Î
Ï
−
[ ] = (x, )
Ç
x,
(2)
under the initial conditions
= (x) at
=0
(3)
© 2002 by Chapman & Hall/CRC Page 649
650
HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
is given by Ë
(x, ) =
Ë
(y, ) (x, y, , ) y Ð
0
Ñ
+
(y) (x, y, , 0) y, Ð
Ñ
y=
1
!"!"!
.
(4)
Here, = (x, y, , ) is the fundamental solution of the Cauchy problem, which satisfies for > ≥ 0 the equation Ñ
Ñ
−
Ñ
[ ]=0
Ç
x,
(5)
Ñ
and the special initial condition
Ñ
=
= (x − y).
(6)
¼
The quantities y and appear in problem (5), (6) as free parameters (y ), and (x) is the -dimensional Dirac delta function. of operator (1) are independent of time , then the fundamental If the coefficients 1, , solution depends on only three arguments, (x, y, , ) = (x, y, − ). If the coefficients of operator (1) are constants, then (x, y, , ) = (x − y, − ).
Î
Ï
¼
¥Ë
¨
È
ÉÊÉÊÉ
Ñ
Ñ
Ñ
Ñ
2 . Boundary value problems ( ≥ 0, x ). The solutions of linear boundary value problems in a spatial domain for equation (2) with initial condition (3) and homogeneous boundary conditions for x (these conditions are not written out here) are given by formula (4) in which the domain should be replaced by . Here, by we mean the Green’s function that must of integration satisfy, apart from equation (5) and the boundary condition (6), the same homogeneous boundary conditions for x as the original equation (2). For boundary value problems, the parameter y belongs to the same domain as x, i.e., y . Í
Î
Ò
Ò
Î
Ò
Ï
Ò
Ñ
Î
Ò
Î
Ó)Ô
Ò
Reference: Mathematical Encyclopedia (1977, Vol. 1).
9.6.3-2. Equations with the second-order partial derivative with respect to . Õ
1 . Cauchy problem ( ≥ 0, x ). The solution of the Cauchy problem for the linear nonhomogeneous differential equation with variable coefficients Í
Õ
Î
Ï
2 Ö
−
2 Õ
[ ] = (x, )
Ç
Õ
under the initial conditions
= (x) at = (x) at Ö
¦
is given by (x, ) = Õ
×
Ø
×
Ù
Ú
(y, ) (x, y, , ) y Û
Ü
0
− ×
Ù
Ú
(y) ß
à
á
Ý
Ý
Õ
Ü
Þ
(x, y, , ) â
Ü
ã
ä
á
=
Þ
(8)
Ü
Þ
=0
Ü
Ý
Õ
Ö
Here,
= 0, =0 Õ
Ö
(7)
Ö
x,
y+ ×
Ù
Ú
å
(y) (x, y, , 0) y. Ý
â
Þ
(x, y, , ) is the fundamental solution of the Cauchy problem
Ý
â
Ü
2 á â
Ý
2
−
[ ] = 0,
x, æ
Ý
Ø á
ä Ý
= ç ç
Ø
= 0,
Ý ç
á
ç
ä
= ç
â ç á
= (x − y), è
where y and play the role of parameters. If the coefficients of operator (1) are independent of time , then the fundamen1, , tal solution depends on only three arguments, (x, y, , ) = (x, y, − ), and the relation (x, y, , ) =0 = − (x, y, ) holds. If the coefficients of operator (1) are constants, then (x, y, , ) = (x − y, − ). Ø
Ü
Ú
é
ì
ì
ä
ê
â
ì
Ü
â
Ý
ì
ä
Ý
ëÊëÊë
ê
ç
Ý
â
Ü
Ý
â
Ü
â
ç
Ý
â
Ü
Ý
â
Ø
Ü
© 2002 by Chapman & Hall/CRC Page 650
651
9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
2 . The solution of the Cauchy problem for the more complicated linear nonhomogeneous differential equation with variable coefficients í
2 î
â
î
+ (x, )
2
á
ï
â
−
á
[ ] = (x, ) î
x, æ
Û
â
â Ø
á
á
with initial conditions (8) is expressed as (x, ) = î
â
Ø ×
Ù ×
Ú
(y, ) (x, y, , ) y Û
Ü
0
−
Ù ×
Ú
(y) ß
à
Ý
á
â
Ü
Þ
(x, y, , )
Ý
â
Ü
ã
Ü
y+ Þ
ä
=0
Ü á
Here,
Þ
Ù ×
Ú
(y) + (y, 0) (y)
ð7å
ï
ß
ñ
(x, y, , 0) y.
Ý
â
(9)
Þ
(x, y, , ) is the corresponding fundamental solution of the Cauchy problem,
Ý
â
Ü
2
+ (x, )
Ý
2
á â
ï
−
Ý
â
á
á
= 0, ä
= ç ç
Ý
Ø
á
Ý
[ ] = 0,
x, æ
â
Ý ç
ä
ç á
è
= ç
â
Ø
= (x − y).
ç
á
3 . Boundary value problems ( ≥ 0, x ). The solutions of linear boundary value problems in a spatial domain for equation (7) with initial condition (8) and homogeneous boundary conditions for x (these conditions are not written out here) are given by formula (9) in which the domain of integration should be replaced by . Here, by we mean the Green’s function that must satisfy, apart from equation (7) and the initial conditions (8), the same homogeneous boundary conditions as the original equation (7). Ø
í
â
ò
ó
ó
ò
ó
á
ô
õ
ó
Ý
9.6.4. Some Special-Type Equations 1. ö
= ( ) ÷
ù
ö
ú
+ ÷
( )+ ( ) ý
ð
ø
û
The transformation ö
ø
ö
û
ÿ
â
â
à
ö ñ
×
() ß
â
Þ
à
×
â
+ ( ) . ÷
þ
ø
ø ö
Ü
where ( ) = exp
ø
ú
( , ) = ( , ) exp î
ü
() â
Þ
â
, ã
=
ÿ
( )+ â
=
á
=
÷
2
ö
ú
( ) ( ) â
â
Þ
â
, Ü
= ×
( ) â
õ
() â
Þ
â
,
õ
á
. ÷
ú û
The transformation = 1 ö
å
.
õ á
Ü
ö
×
, leads to the simpler constant coefficient equation ã
á
2.
÷
û
ø
ö
û
ú
ÿ
=
,
î ÿ
1−
leads to the constant coefficient equation õ
ê
á ê
= (−1)
3. ö
ö
÷
ø
=
ú
=0
á
ö
.
õ õ á
â
õ
á
. ÷
û ö
û
The change of variable = ln | | leads to a constant coefficient equation. ÿ
© 2002 by Chapman & Hall/CRC Page 651
652
HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
4. ö
÷
2
= (
+
+ )
û ö
ú ö
. ÷
ú û
ø
ö
The transformation
û ú
( , ) = ( , )| î
ÿ
â
â
2 ÿ
+
¥ÿ
+ |
−1 2 , õ
=
Þ ×
ÿ
2
ÿ
+
¥ÿ
+
leads to a constant coefficient equation. 5.
–
ö
= 0, ú
= 1, 2,
÷ ö
ø
Here, is a linear differential operator of any order with respect to the space variable coefficients can depend on .
whose ÿ
ÿ
1 . General solution: í
( , )= î
ÿ
−1 õ
â
ê
ÿ
â
=0 ê
where the = ( − ) = 0.
( , ),
ê
â
= 1:
( , ) are arbitrary functions that satisfy the original equation with
ê
ê
ÿ
â
ê
á
2 . Fundamental solution: Ø
í
( , )=
Ý
ÿ
â
õ
where
ÿ
Â
2
6.
ö
ö
( − 1)!
1(
Ý
, ), ÿ
â
â
¥Á
−1
, ) is the fundamental solution of the equation with = 1.
1(
Ý
âeõ
The linear differential operator
–
2
= 0, ú
= 1, 2,
can involve arbitrarily many space variables.
÷
ø
Here, is a linear differential operator of any order with respect to the space variable coefficients can depend on .
whose ÿ
ÿ
1 . General solution: í
( , )= î
ÿ
−1 õ
â
= = 0. ê
ê
ê
ê
ÿ
â
=0 ê
where the ( − )
( , ),
ê â
( , ) are arbitrary functions that satisfy the original equation with ÿ
â
= 1:
á
2 . Suppose that the Cauchy problem for the special case of the equation with = 1 is well posed if only one initial condition is set at = 0; this means that the constant coefficient differential operator is such that the equation with = 1 is regular with regularity index = 1. Then the fundamental solution of the original equation can be found by the formula Ø
Ø
í
â
( , )=
Ý
ÿ
â
õ
where
Ý
¥Á
1( Â
ÿ
âeõ
−1
( − 1)!
Ý
1( ÿ
, ), â
, ) is the fundamental solution for = 1.
â
The linear differential operator
can involve arbitrarily many space variables.
© 2002 by Chapman & Hall/CRC Page 652
653
9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
7.
[ ] = 0. ÷
=0
Here, is any linear differential operator with arbitrarily many independent variables Particular solutions: ( 1, î
ÿ
where the equation ê
!"!"!
, ÿ
õ
)=
!
( 1,
ÿ
are solutions of the equations [ ] − # $
=0
ê
# ê
= 0, and the
!"!"!
,
= 0, the
1,
!"!"!
, ÿ
õ
.
),
ÿ
=1 "
ÿ
õ
#
are roots of the characteristic
are arbitrary constants. "
© 2002 by Chapman & Hall/CRC Page 653
