Chapter 07: Elliptic Equations with Two Space Variables .fr

u + Pu = 0 at. W. = i . Eigenvalues: s xt. = E 2xt ...... + pU )Pq (T , U ) brings the original equation to the ...... X4 1gr acs '. 2â02 d + 4 2h r acs '. 2â02 df for 2 1 > 0,.

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Chapter 7

Elliptic Equations with Two Space Variables =0 2g The Laplace equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, in heat and mass transfer theory, this equation describes steady-state temperature distribution in the absence of heat sources and sinks in the domain under study. A regular solution of the Laplace equation is called a harmonic function. The first boundary value problem for the Laplace equation is often referred to as the Dirichlet problem, and the second boundary value problem as the Neumann problem. Extremum principle: Given a domain h , a harmonic function i in h that is not identically constant in h cannot attain its maximum or minimum value at any interior point of h . 7.1. Laplace Equation

f

7.1.1. Problems in Cartesian Coordinate System The Laplace equation with two space variables in the rectangular Cartesian system of coordinates is written as j j 2 j ki

2

2 i + j l 2 = 0.

7.1.1-1. Particular solutions and a method for their construction. 1 M . Particular solutions:

k l k l i ( , )=m +n +o , k l k l k l 2 2 i ( , )= m ( − )+n , k l k k l k l l 3 2 i ( , )= m ( −3 ) + n (3 2 − 3 ), k l k l m k + nl i ( , )= +o , 2+ 2 k l k l l i ( , ) = exp(p q )( m cos q + n sin q ), k l k k l i ( , ) = ( m cos q + n sin q ) exp(p q ), k l k k l l i ( , ) = ( m sinh q + n cosh q )( o cos q + h sin q ), k l k k l l i ( , ) = ( m cos q + n sin q )( o sinh q + h cosh q ), k l k k l l i ( , ) = m ln r ( − 0 )2 + ( − 0 )2 s + n , k l where m , n , o , h , 0 , 0 , and q are arbitrary constants.

2 M . Fundamental solution:

t k

l

1 1v ( , )= ln , 2u

v

=w

k

2

+

l

2.

© 2002 by Chapman & Hall/CRC Page 467

468

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES k

l

3 x . If i ( , ) is a solution of the Laplace equation, then the functions k

l

i

1

i

2

= m i (p y + o 1 , p y + o 2 ), k l k l = m i ( cos z + sin z , − sin z + cos z ),

i

3

=m i { k

k

2

l

+

l

2

,k

2

+

l

2

| ,

are also solutions everywhere they are defined; m , o 1 , o signs at y in i 1 are taken independently of each other.

2,

z , and y are arbitrary constants. The

4 x k. A method for constructing particular solutions involves the following. Let } ( ~ ) = l fairly general k l k l ( , ) + ( , ) be any analytic function of the complex variable ~ = + ( and are real k l functions of the real variables and ; 2 = −1). Then the real and imaginary parts of } both satisfy the two-dimensional Laplace equation, 2

= 0,

Recall that the Cauchy–Riemann conditions j

j k

j

= j l ,

j

2

j l

= 0. j j =− k

are necessary and sufficient conditions for the function } to be analytic. Thus, by specifying analytic functions } ( ~ ) and taking their real and imaginary parts, one obtains various solutions of the two-dimensional Laplace equation.

\[

References: M. A. Lavrent’ev and B. V. Shabat (1973), A. G. Sveshnikov and A. N. Tikhonov (1974), A. V. Bitsadze and D. F. Kalinichenko (1985).

7.1.1-2. Specific features of stating boundary value problems for the Laplace equation. 1 x . For outer boundary value problems on the plane, it is (usually) required to set the additional condition that the solution of the Laplace equation must be bounded at infinity. 2 x . The solution of the second boundary value problem is determined up to an arbitrary additive term. 3 x . Let the second boundary value problem in a closed bounded domain h boundary be characterized by the boundary condition*

with piecewise smooth

j

j i

= } (r) for r ,

where is the derivative along the (outward) normal to . The necessary and sufficient condition of solvability of the problem has the form } (r)

= 0.

The same solvability condition occurs for the outer second boundary value problem if the domain is infinite but has a finite boundary.

\[

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964).

* More rigorously, must satisfy the Lyapunov condition [see Babich, Kapilevich, Mikhlin, et al. (1964) and Tikhonov and Samarskii (1990)].

© 2002 by Chapman & Hall/CRC Page 468

7.1. LAPLACE EQUATION

7.1.1-3. Domain: −

469

=0

2

< < , 0 ≤ < . First boundary value problem.

A half-plane is considered. A boundary condition is prescribed:

Solution: \[

( , ) = 1

= } ( ) at } ( )

( − )2 +

−

2

= 0. ¡

=

1

¡ −

2 2

} ( + tan ¢ ) ¢ .

References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), H. S. Carslaw and J. C. Jaeger (1984).

7.1.1-4. Domain: −

< < , 0 ≤ < . Second boundary value problem.

A half-plane is considered. A boundary condition is prescribed: £ ¤ = } ( )

Solution:

( , ) =

1

where ¥ is an arbitrary constant.

\[

( − )2 +

} ( ) ln w

−

= 0.

at

2

+¥ ,

Reference: V. S. Vladimirov (1988).

7.1.1-5. Domain: 0 ≤ < , 0 ≤ < . First boundary value problem. A quadrant of the plane is considered. Boundary conditions are prescribed:

Solution:

\[

( , ) = 4

= } 1 ( ) at

0

[

2

= 0,

= } 2 ( ) at

} 1 (¦ )¦ ¦ 4 + + ( − ¦ )2 ][ 2 + ( + ¦ )2 ]

= 0.

} 2 ( ) . [( − )2 + 2 ][( + )2 + 2 ]

0

Reference: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

7.1.1-6. Domain: −

< < , 0 ≤ ≤ § . First boundary value problem.

An infinite strip is considered. Boundary conditions are prescribed:

Solution:

\[

= } 1 ( ) at

= 0,

( , ) = 1 sin { 2§ § | 1 + sin { 2§ § |

−

−

= } 2 ( ) at

} 1 ( cosh[ ( − ) ¨ } 2 ( cosh[ ( − ) ¨

=§ .

)

§ ] − cos( ¨ § )

) . § ] + cos( ¨ § )

Reference: H. S. Carslaw and J. C. Jaeger (1984).

7.1.1-7. Domain: −

< < , 0 ≤ ≤ § . Second boundary value problem.

An infinite strip is considered. Boundary conditions are prescribed: £ ¤

Solution:

= } 1 ( ) at

( , ) = 1 2 −

= 0,

£ ¤

= } 2 ( ) at

=§ .

} 1 ( ) ln © cosh[ ( − ) ¨ § ] − cos( ¨ § ) ª

1 − } 2 ( ) ln © cosh[ ( − ) ¨ § ] + cos( ¨ § ) ª + ¥ , 2 − where ¥ is an arbitrary constant.

© 2002 by Chapman & Hall/CRC Page 469

470

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.1.1-8. Domain: 0 ≤ < , 0 ≤ ≤ § . First boundary value problem. A semiinfinite strip is considered. Boundary conditions are prescribed:

= } 1 ( ) at

= 0,

= } 2 ( ) at

Solution:

( , ) =

2 «X¬ §

exp { −

=1

| sin { §

§

= 0,

= } 3 ( ) at

=§ .

®

|

¦

} 1(¦ ) sin {

0

¦ |

§

1 1 1 − sin { 2( ) | 2§ § cosh[ ( + ) ¨ § ] − cos( ¨ § ) ° ± 0 ¯ cosh[ ( − ) ¨ § ] − cos( ¨ § ) 1 1 1 + sin ² − 3( ) . 2§ § ³ 0 ¯ cosh[ ( − ) ¨ § ] + cos( ¨ § ) cosh[ ( + ) ¨ § ] + cos( ¨ § ) ° ±

+

Example. Consider the first boundary value problem for the Laplace equation in a semiinfinite strip with ´ 1 ( µ ) = 1 and

´ 2 (¶ ) = ´ 3 (¶ ) = 0.

Using the general formula and carrying out transformations, we obtain the solution 2 sin( · µ ¹ º ) . (¶ , µ ) = · arctan ¸ sinh( · ¶ ¹ º )»

¼\[

Reference: H. S. Carslaw and J. C. Jaeger (1984).

7.1.1-9. Domain: 0 ≤ ≤ § , 0 ≤ ≤ ½ . First boundary value problem. A rectangle is considered. Boundary conditions are prescribed:

= 1 ( ) at ± = 3 ( ) at

( , ) =

+

sinh ¿

«X¬

sin ² ¥ ¬

¬

where the coefficients ¬

= Ã ¾ É

¼\[

= Ë

¬

2 2

, ¾ Ä Ä

É

Å 0

0

( § − )À sin ²

½

¾ ¬

=1

=1

Ê

§

,¥

¬

³

sinh ¿

, and

+ ³

1 (Æ

±

3 (Æ

Ã

) sin ² Ç ) sin ² ÌÇ

Æ ½

³

½

³ Ì

= Ã

Æ , È

Á É

Æ

= ½ sinh ² Ç É

Á ¬

=1

«X¬

sin ² Â

=1

Â ±

sinh ²

½

§

sin ² ³

sinh ² ³

½ §

³

³

,

are expressed as

Á

®

=½ .

¬

«X¬

( ½ − )À +

§

¬

½

=§ ,

±

¬

«X¬

= 2 ( ) at ± = 4 ( ) at

= 0,

±

Solution:

= 0,

È

Æ ,

³

,

= Ë

2 2É

Ä Ä

Å 0

® ±

±

Æ

2 (Æ

) sin ² Ç

4 (Æ

) sin ² ÌÇ

0 Â É É ½ Ë = Ì sinh ² Ì Ç . ³ É

½

³

Æ , È

Æ ³

È

Æ ,

References: M. M. Smirnov (1975), H. S. Carslaw and J. C. Jaeger (1984).

© 2002 by Chapman & Hall/CRC Page 470

7.1. LAPLACE EQUATION Í

471

=0

2Î

7.1.1-10. Domain: 0 ≤ Ï ≤ Ì , 0 ≤ Ð ≤ ½ . Second boundary value problem. A rectangle is considered. Boundary conditions are prescribed: £ Ñ Ò

= 1 (Ð ) at ± £ ¤ Ò = 3 (Ï ) at (Ï , Ð ) = −

( Ï − Ì )2 + Ì Ï ¾4 Ì Á4 0

0

Õ

−½ Ô × Ö

=1

ÉÕ

−Ì

−

4½

0

( Ï − ½ )2 +

Â 4½

0

Ð

× Ô

ÕÉ

Ù

É

Ä

Å 0

Ä

2 = Ì

É

Ù Õ

Ì Ë É cos Û Ì Ç Ï Ü cosh Ø Ì Ç ( − Ð )Ú +

2 = Ù ÊÖ

+Ó

1 (Æ

® Þ

Æ

3 ( Æ ) cos Û ÌÇ

Þ

0

Æ

) cos Û Ç Ù

×

= É

Ü Ü Ì

Ç

sinh Û Ç Ù

È

ÄÉ

2Ý = Ù

É Ý

Æ ,

É,Ê

É

Ö

É

Å É Ä

2 = Ì

É

,

0

®

cos Û Ì Ç Ï Ü cosh Û Ì Ç Ð Ü ,

ÂË É

=1

,

Æ , È

Ô

cosh Û Ù Ç Ï Ü cos Û Ù Ç Ð Ü É

Ý

=1

É is an arbitrary constant, and the coefficients

É

2

cosh Ø Ù Ç ( Ì − Ï )Ú cos Û Ù Ç Ð Ü + É

Ð =½ .

±

Ê É

Ô =1

where Ó

Ê

2

Ì Ï = ,

= 2 (Ð ) at ± = 4 (Ï ) at

£ ¤ Ò

Ð = 0,

±

Solution: Ò

£ Ñ Ò

Ï = 0,

Þ

, Â É

×

, and Ë É

2 (Æ

) cos Û Ç Ù

4 (Æ

) cos Û ÇÌ

0 Þ Â É Ù Ë = sinh Û Ì Ç Ü . Ç É

Ü ,

are expressed as

Æ

É

Æ

Ü

Æ , È

Ü È

Æ ,

The solvability condition for the problem in ® question has the form (see Paragraph 7.1.1-2, ® Item 3 ß ) Ä

Å

1 (Ð

Þ

0

) Ð + È

Ä

Å

0

Þ

2 (Ð

) Ð − È

Ä

0

3 (Ï

Þ

Ä

) Ï − È

4 (Ï

Þ

0

) Ï = 0. È

Ù 7.1.1-11. Domain: 0 ≤ Ï ≤ Ì , 0 ≤ Ð ≤ . Third boundary value problem.

A rectangle is considered. Boundary conditions are prescribed: £ Ñ Ò

Ò

− à 1 = 1 (Ð ) at £ ¤ Ò Ò Þ − à 3 = 3 (Ï ) at Þ

£ Ñ Ò

Ï = 0,

Ò

+ à 2 = 2 (Ð ) at £ ¤ Ò Ò Þ + à 4 = 4 (Ï ) at

Ð = 0,

Þ

For the solution, see Paragraph 7.2.2-14 with á ≡ 0.

Ì Ï = , Ù Ð = .

Ù 7.1.1-12. Domain: 0 ≤ Ï ≤ Ì , 0 ≤ Ð ≤ . Mixed boundary value problems.

1 ß . A rectangle is considered. Boundary conditions are prescribed: £ Ñ Ò

= (Ð ) at Ò Þ = ã (Ï ) at

Solution: Ò

Ù Õ

(Ï , Ð ) = − Ç

Ô =1

Þ× É

Ï = 0,

£ Ñ Ò Ò

Ð = 0,

cosh Ø Ç Ù ( Ì − Ï )Ú sin Û Ç Ù

Ð Ü

Ï =Ì , Ù

= â (Ð ) at = ä (Ï ) at Ù Õ

+ Ç

â Ô =1

× É

Ð = .

cosh Û Ç Ù

Ï

Ü sin Û Ç Ù

Ð Ü

É É Éã É Õ Ù Ï ä Ï Ð cos Û Ç Ü sinh Ø Ç ( − Ð ) Ú + Ô cos Û Ç Ë É ® Ì Ë É Ì Ü sinh Û Ç Ì Ü ® Ì =1 =1 Ù Ä Ä É É −Ð É É Ð + Ì Ù ã (Ï ) Ï + Ì Ù ä (Ï ) Ï , È È 0 0 Õ

+ Ô

© 2002 by Chapman & Hall/CRC Page 471

472

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

where Ä

2

= Ù

Þ É É

0

Ä

2 = Ì

ã

Å ®

Æ

(Æ ) sin Û Ç Ù Þ

Æ

ã (Æ ) cos Û Ç Ì

0

×

Ü

â É

Æ ,

Ü È Ì

= sinh Û Ç Ù

É Ë

Ü ,

Å ® â (Æ ) sin Û Ç Ù

0

Ä

2 = Ì

ä

É

å\[

= Ù

Æ , È

Ä

2

ä (Æ ) cos Û Ç Ì Ù

0

= sinh Û Ç Ì É

Æ Ü Æ

Æ , È

Ü

Æ , È

Ü ,

Reference: M. M. Smirnov (1975).

2 ß . A rectangle is considered. Boundary conditions are prescribed: Ò

= (Ð ) Ò

at

Ï = 0,

= ã (Ï ) at

Ð = 0,

Þ

£ Ñ Ò

Ì Ï = , Ù

= â (Ð ) at

£ ¤ Ò

= ä (Ï )

Ð = ,

at

where (0) = ã (0). Þ Solution: Ò

Õ

(Ï , Ð ) = Ô =0

Þ É cosh

ÕÉ

+ Ô =0

where

Þ É

É

2í

Þ

0

æ

0

×

Ü sin Û

Ð É

Ù

Ù Ü cosh Û Ë

(2ê + 1)

(è ) sin Ø é

Ù

î ã (è ) sin Ø é

(2ê í + 1)

−Ð Ù

Ü +

Ô

â ×

É cosh

=0

Ù ÕÉ Ô

É

2

è Ú ë è ,

â ì = Ù æ

è Ú ë è , í

äì =

2í

ï ì = é

É

É

0

Ï É Ï

sin Û Ë

É Ë cosh

É

×

sinh Û ×

ä

Ë

=0

(2ê + 1) Ù , 2

ì = é

Õ

Ì Ì Ü +

É

× å\[

Ì

É

2 = Ù æ ç

ã ì =

Ï

sin Û Ë

coshÉ Ë

É

−Ï

É É

ã

Ì ×

cosh Û ×

Ì Ü sin Û

ç â (è ) sin Ø é

î ä (è ) sin Ø é 0 Ù (2ê í + 1) æ

Ð É Ð

Ù Ü sinh Û Ë

É É

×

(2ê + 1) Ù

Ì Ü

Ù Ü ,

É

è Ú ë è ,

(2ê í + 1)

è Ú ë è ,

.

2

Reference: M. M. Smirnov (1975).

7.1.2. Problems in Polar Coordinate System The two-dimensional Laplace equation in the polar coordinate system is written as £

1ð ð £

Û

ð £ Ò ð £ Ü

7.1.2-1. Particular solutions: ð

Ò

( )= ð

Ò

where õ

= 1, 2, ööö ; Ö

ln + ð ó

ñ

Ö

, ô , and Ý

ð1 2

£

2

£ ñ

Ò ð 2

= 0,

=ò Ï

2

+ Ð 2.

ð

( , )Ö = Û

,

+

, Ý + ð ó Ý

Ü (ô

cos õ

ñ

+

sin õ

ñ

),

Â

are arbitrary constants.

Â

© 2002 by Chapman & Hall/CRC Page 472

7.1. LAPLACE EQUATION ÷

ð

473

=0

2ø

ð

7.1.2-2. Domain: 0 ≤

≤ù

≤

or ù

< ú . First boundary value problem.

The condition Ò

ð ñ

= ( ) at

=ù

Þ

ñ

is set at the boundary of the circle; ð ( ) is a given function. Þ

1 ß . Solution of the inner problem ( ≤ ù ): ð Ò

2û

1 æ 2

ñ

( , )=

Þ

0

é

− 2 ñ cos( − ü ) + ù

ð ù ð

(ü )

− 2ù

2

ð

2

2

ë ü .

This formula is conventionally referred to as the Poisson integral. Solution of the outer problem in series form: í

ð Ò

ñ

0

( , )=

í é Ù

1

ì = é

+ ý

2

æ æ

0

Þ

2û

Þ

0

Û

ì

2û

1

ì =

ð

Õ

ì í

ù

=1

ð ñ

( , )=

(ü ) sin(ê ü ) ë ü ,

ê = 1, 2, 3, ööö

2û

ð

ð

(ü ) Þ

0

é

),

ê = 0, 1, 2, ööö ,

ð

1 æ 2

ñ

Ù

+ ì sin ê

(ü ) cos(ê ü ) ë ü ,

2 ß . Bounded solution of the outer problem ( ≥ ù ): Ò

ñ

( ì cos ê Ü

2

−ù 2 ñ cos( − ü ) + ù

ð

− 2ù

2

2

ë ü .

Bounded solution of the outer problem in series form: Ò í 0,

ñ

( , )= í

where the coefficients

í

ð

Õ

0

+ ý

2

ì

ì í

ùð Û

( ì cos ê Ü

ñ

Ù

+ ì sin ê

ñ

),

=1

Ù ì , and ì are defined by the same relations as in the inner problem.

þ In hydrodynamics and other applications,ð outer problems are sometimes encountered in which one has to consider unbounded solutions for ú . Example. The potential flow of an ideal (inviscid) incompressible fluid about a circular cylinder of radius ÿ with a constant incident velocity at infinity is characterized by the following boundary conditions for the stream function:

ø = 0 at = ÿ , ø

Solution:

ø ( , ) = å\[

sin ÿ

−

2

as

.

sin .

References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. N. Tikhonov and A. A. Samarskii (1990).

ð

7.1.2-3. Domain: 0 ≤ The condition

ð

≤ù

or ù

≤

< ú . Second boundary value problem. £ Ò

ð ñ

= ( ) at Þ

ñ

=ù

is set at the boundary of the circle. The function ( ) must satisfy the solvability condition æ

Þ

2û 0

Þ

ñ

( )ë

ñ

= 0.

© 2002 by Chapman & Hall/CRC Page 473

474

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES ð

1 ß . Solution of the inner problem ( ≤ ù ): Ò

ð ñ

ù

( , )=

2

2û

æ

ð

2

(ü ) ln Þ

0

é

ð

ñ

− 2ù

2

cos( − ü ) + ù 2

ù

ë ü +ô ,

where ô is an arbitrary constant; this formula is known as the Dini integral. Series solution of the inner problem: ð Ò

ð

ñ

í

1

ì = é

æ

ì

2û

ê

=1

ù

( , )=−

ù

2

1

ì =

2û

æ

0

ñ

)+ô ,

(ü ) sin(ê ü ) ë ü ,

ð

2 . Solution of the outer problem ( ≥ ù ): ñ

+ 0)

Parabolic coordinates 9 ,:

= 7 9 : , < = 12 7 (: 2 − 9 2) −; = < 9 < = , 0 ≤ : < =

Elliptic coordinates A ,B

= 7 cosh cos B , < = 7 sinh sin B A ; 0 ≤ < = , 0 ≤ B < 2

A

Laplace operator, 8

2A

≤E ≤

1

2.

A

?

1

(cosh D − cos C )2 2

7

2

+ sin2 B ) > ?

7 2(sinh

Bipolar coordinates = , < = C ,D cosh D − cos C cosh D − cos C ; C 0 ≤ < 2 , −= < D < = 7.1.2-7. Domain:

?

1

7 sin C

2 2 1 + 9 2 + : 2) > ? 2 ? : 2@

7 2(9

A

7 sinh D

2

? 2

2

? B ?

2

> ? C

+

2

+

?

@

2

2

? D

2

@

?

Mixed boundary value problem.

An annular domain is considered. Boundary conditions are prescribed: *

= F 1 ( G ) at

E =

1,

= F 2 ( G ) at

E =

2.

?

Solution: (E , G ) =

1 (2) 1 (1) 2 0 + 2 0

ln

1

H

E

+

2

E

H

1

( " cos G + # sin G ) +

=1

=1

( $ cos G + % sin G ). E

Here, the coefficients " , # , $ , and % are expressed as " = $ =

1

+ 2 ( 2 + +1

(2)

2

2

1

1

+1 (1)

2

)

1 −1 (2)

(

2 2

# =

,

− 2 (1) , + 12 )

(2)

+ 2 ( 2 + 2

I

% =

1

1

+1

+1 (1) 2 1

I

)

2

,

1

−1 (2)

(

I 2

2

− 2 I (1) , + 12 )

where the constants ( & ) and I ( & ) (' = 1, 2) are defined by the same formulas as in the first boundary value problem.

J)(

Reference: M. M. Smirnov (1975).

7.1.3. Other Coordinate Systems. Conformal Mappings Method 7.1.3-1. Parabolic, elliptic, and bipolar coordinate systems. In a number of applications, it is convenient to solve the Laplace equation in other orthogonal systems of coordinates. Some of those commonly encountered are displayed in Table 21. In all the coordinate systems presented, the Laplace equation 8 2 = 0 is reduced to the equation considered in Paragraph 7.1.1-1 in detail (particular solutions and solutions to boundary value problems are given there). The orthogonal transformations presented in Table 21 can be written in the language of complex variables as follows: + 'K< = − 21 'L7 (9 + 'K: )2 ; + 'K< = 7 cosh(A + K' B ) ; + 'K< = 'L7 cot M ;

1 C 2(

+ 'KD )N

(parabolic coordinates), (elliptic coordinates), (bipolar coordinates).

© 2002 by Chapman & Hall/CRC Page 476

7.1. LAPLACE EQUATION O

477

=0

2P

The real parts, as well as the imaginary parts, in both sides of these relations must be equated to each other (' 2 = −1). Example. Plane hydrodynamic problems of potential flows of ideal (inviscid) incompressible fluid are reduced to the Laplace equation for the stream function. In particular, the motion of an elliptic cylinder with semiaxes Q and R at a velocity in the direction parallel to the major semiaxis ( Q > R ) in ideal fluid is described by the stream function

Q +R

P (S , T ) = − R U

Q −R V

where S and T are the elliptic coordinates.

J)\

1 W 2 XZY

sin T ,

[

2

2

=Q

− R 2,

References: G. Lamb (1945), J. Happel and H. Brenner (1965), G. Korn and T. Korn (1968).

7.1.3-2. Domain of arbitrary shape. Method of conformal mappings. 1 . Let ] = ] ( ^ ) be an analytic function that defines a conformal mapping from the complex plane ^ = _ + `Ka into a complex plane ] = b + `Kc , where b = b (_ , a ) and c = c (_ , a ) are new independent variables. With reference to the fact that the real and imaginary parts of an analytic function satisfy the Cauchy–Riemann conditions, we have b = c and b = − c , and hence 2f

? _

2

2f

+

? a

? d 2

? e

= gh] i ( ^ ) g 2

2f

> ? b

? e

+

2

2f

? c

2

? d @ .

Therefore, the Laplace equation in the under a conformal mapping into the ? ? _ a -plane transforms ? ? Laplace equation in the b c -plane. 2 j . Any simply connected domain k in the _ a -plane with a piecewise smooth boundary can be mapped, with appropriate conformal mappings, onto the upper half-plane or into a unit circle in the b c -plane. Consequently, a first and a second boundary value problem for the Laplace equation in k can be reduced, respectively, to a first and a second boundary value problem for the upper half-space or a circle; such problems are considered in Subsections 7.1.1 and 7.1.2. Subsection 7.2.4 presents conformal mappings of some domains onto the upper half-plane or a unit circle. Moreover, examples of solving specific boundary value problems for the Poisson equation by the conformal mappings method are given there; the Green’s functions for a semicircle and a quadrant of a circle are obtained. A large number of conformal mappings of various domains can be found, for example, in the references cited below. J)\

References: V. I. Lavrik and V. N. Savenkov (1970), M. A. Lavrent’ev and B. V. Shabat (1973), V. I. Ivanov and M. K. Trubetskov (1994).

7.1.3-3. Reduction of the two-dimensional Neumann problem to the Dirichlet problem. Let the position of any point (_ l , a l ) located on the boundary m of a domain k be specified by a parameter n , so that _ l = _ l ( n ) and a l = a l ( n ). Then a function of two variables, F (_ , a ), is determined on m by the parameter n as well, F (_ , a ) gpo = F (_ l ( n ), a l ( n )) = F l ( n ). The solution of the two-dimensional Neumann problem for the Laplace equation q 2 f = 0 in k with the boundary condition of the second kind f

= F l ( n ) for r s m

?

can be expressed in terms of the solution ? r of the two-dimensional Dirichlet problem for the Laplace equation q 2 b = 0 in k with the boundary condition of the first kind b = t l ( n ) for r s m , where t l ( n ) = u

v l ( n ) w n , as follows: f (_ , a ) = u d 0

b x a

(y , a 0 ) w y − u e 0

b x _

(_ , y ) w y + z .

Here, (_ 0 , a 0 ) are the coordinates of danyx point in k , ande z x is an arbitrary constant.

J)\

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964).

© 2002 by Chapman & Hall/CRC Page 477

478

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.2. Poisson Equation {

= – } (x)

2|

7.2.1. Preliminary Remarks. Solution Structure Just as the Laplace equation, the Poisson equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, it describes steady-state temperature distribution in the presence of heat sources or sinks in the domain under study. The Laplace equation is a special case of the Poisson equation with ~ ≡ 0. In what follows, we consider a finite domain with a sufficiently smooth boundary . Let r s and s , where r = {_ , a }, = { , }, |r − |2 = (_ − )2 + (a − )2 . 7.2.1-1. First boundary value problem. The solution of the first boundary value problem for the Poisson equation q

2

f

= − ~ (r)

(1)

in the domain with the nonhomogeneous boundary condition f

= v (r) for r s

can be represented as f (r) = u ~ ( ) (r, ) w − u v ( )

x

w .

(2)

x r

Here, (r, ) is the Green’s function of the first boundary value problem, is the derivative of the Green’s function with respect to , along the outward normal N to the boundary . The integration is performed with respect to , , with w = w w . The Green’s function = (r, ) of the first boundary value problem is determined by the following conditions. 1 j . The function satisfies the Laplace equation in _ , a in the domain everywhere except for the point ( , ), at which has a singularity of the form 21 ln |r−1 | . 2 j . With respect to _ , a , the function satisfies the homogeneous boundary condition of the first kind at the domain boundary, i.e., the condition | = 0. The Green’s function can be represented in the form (r, ) =

1 1 ln +b , 2 |r − |

(3)

where the auxiliary function b = b (r, ) is determined by solving the first boundary value problem for the Laplace equation q 2 b = 0 with the boundary condition b g = − 21 ln |r−1 | ; in this problem, is treated as a two-dimensional free parameter. The Green’s function is symmetric with respect to its arguments: (r, ) = ( , r). / 3

When using the polar coordinate system, one should set

r = { , },

= { , },

|r − |2 =

2

+

2

− 2 cos( − ),

w = w w

in relations (2) and (3).

© 2002 by Chapman & Hall/CRC Page 478

7.2. POISSON EQUATION

2

= − (x)

479

7.2.1-2. Second boundary value problem. The second boundary value problem for the Poisson equation (1) is characterized by the boundary condition f = v (r) for r s . x

The necessary solvability condition for x r this problem is u ~ (r) w + u v (r) w

= 0.

(4)

The solution of the second boundary value problem, provided that condition (4) is satisfied, can be represented as f (r) = u ~ ( ) (r, ) w + u v ( ) (r, ) w + z ,

(5)

where z is an arbitrary constant. The Green’s function = (r, ) of the second boundary value problem is determined by the following conditions: 1 j . The function satisfies the Laplace equation in _ , a in the domain everywhere except for the point ( , ), at which has a singularity of the form 21 ln |r−1 | .

2 j . With respect to _ , a , the function satisfies the homogeneous boundary condition of the second kind at the domain boundary: 1 , = x

0

where 0 is the length of the boundary of x . r The Green’s function is unique up to an additive constant. / 3 The Green’s function cannot be determined by condition 1 j and the homogeneous boundary condition = 0. The point is that the problem is unsolvable for in this case, because, on representing in the form (3), for we obtain a problem with a nonhomogeneous boundary condition of the second kind for which the solvability condition (4) now is not satisfied. 7.2.1-3. Third boundary value problem. The solution of the third boundary value problem for the Poisson equation (1) in the domain with the nonhomogeneous boundary condition f x

+ = v (r) for r s

r is given by formula (5) with z = 0,x where = (r, ) is the Green’s function of the third boundary value problem and is determined by the following conditions:

1 ¡ . The function satisfies the Laplace equation in ¢ , £ in the domain everywhere except for the point ( , ), at which has a singularity of the form 21 ln |r−1 | .

2 ¡ . With respect to ¢ , £ , the function satisfies the homogeneous boundary condition of the third kind at the domain boundary, i.e., the condition ¤ + ¥ = 0. The Green’s function can be represented in the form (3); the auxiliary function is identified by solving the corresponding third boundary value problem for the Laplace equation ¦ 2 = 0. The Green’s function is symmetric with respect to its arguments: (r, ) = ( , r).

§)¨

References for Subsection 7.2.1: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970).

© 2002 by Chapman & Hall/CRC Page 479

480

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.2.2. Problems in Cartesian Coordinate System The two-dimensional Poisson equation in the rectangular Cartesian coordinate system has the form 2

2

x ¢

2

+

x £

x

+ ~ (¢ , £ ) = 0.

2

x

7.2.2-1. Particular solutions of the Poisson equation with a special right-hand side. ª ª 1 ¡ . If ~ (¢ , £ ) = « © ¬ © ®« ¬ exp( ¯ « ¢ + ° ¬ £ ), the equation has solutions of the form =1 =1 ± (¢ , £ ) = − © «

± ¬

=1

«®¬ © =1

¯

2«

+ ° ¬2

exp( ¯ « ¢ + ° ¬ £ ).

ª ª 2 ¡ . If ~ (¢ , £ ) = « © ¬ © ®« ¬ sin( ¯ « ¢ + ² « ) sin( ° ¬ £ + ³ ¬ ), the equation admits solutions of the form =1 =1 ± (¢ , £ ) = © «

± ¬

=1

7.2.2-2. Domain: − ´

=1

< ¢ < ´ , −´

Solution: (¢ , £ ) =

1 2 µ − ¶

«®¬ ©

+ ° ¬2

sin( ¯ « ¢ + ² « ) sin( ° ¬ £ + ³ ¬ ).

≥ 0, ≥ > ≥ 0, 5 5

≥ > ≥ 0, ≥ > ≥ 0.

) sin(7 + * 28

2

(2

for for

( − )] for ( − )] for

8

8

7

(2( + 1) , 2

= 7

sin(7

) sin( * 8 ) cosh( * 8 ) 9

( 5 − )] ( 5 − )]

The Green’s function can be written in form of a double series:

8

sinh(7 sinh(7 0

sin( *

= 7

2

8

)

,

.

7.2.3. Problems in Polar Coordinate System The two-dimensional Poisson equation in the polar coordinate system is written as 1=

=

=

=

+

=

1 2

=

7.2.3-1. Domain: 0 ≤

≤ >

,0≤

2

2

=

=

+ ( , ) = 0,

= 4

2

+ 2.

≤ 2 . First boundary value problem.

A circle is considered. A boundary condition is prescribed: =

= ( ) at

= >

.

© 2002 by Chapman & Hall/CRC Page 485

486

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES Solution: =

=

1

2

2?

( , )=

=

( )

0

= >

− 2>

2

2

− 2 cos( − ) + >

where

2

2?

+

=

0

( , ) ( , , , )

0@

=

1 1 1

− ln ln 2 |r − r0= | 2 r = { , }, = cos ,

r0 = { 0 , 0 },

0 |( >

0)

=

− r|

2r

0

,

= sin .

0

=

2

The magnitude of a vector difference is calculated as | r − 5 r 0 | = ( and 5 are any scalars). Thus, we obtain =

=

=

2 2

− 2 5

cos( − ) + 5

2

− 2> = −2

=

4

cos( − ) + > . cos( − ) + 2 ]

2[ 2 >

2 2

=

2 2

1

ln 4

( , , , ) =

,

= sin ,

= cos ,

0

= > =

( , , , ) =

References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. G. Butkovskiy (1979). =

7.2.3-2. Domain: 0 ≤

≤

,0≤ >

≤ 2 . Third boundary value problem.

A circle is considered. A boundary condition is prescribed:

=

A

+

= ( ) at

Solution: =

=

2? >

( ) ( , , > , )

0

where =

( , , , ) =

Here, the

1

( , ) ( , , , )

0@

0

=

8

=0

0

(!

28

(!

=1

>

2 B

= 1,

+

(!

=

7.2.3-3. Domain:

≤ >

(! 8 ) 2 − ( 2 )[ (! 8 8

DEDED

)]2 >

cos[( ( − )],

).

are positive roots of the transcendental equation (! >

) = 0.

,0≤

2>

C =

=2 (( = 1, 2,

B B ( ) are the Bessel functions the ! C and F !

C

C

2?

+

. >

( , )=

=

≤ 2 . First boundary value problem.

The exterior of a circle is considered. A boundary condition is prescribed: =

Solution: 1 ( , )=

2

=

. >

=

=

= ( ) at

2?

0

( )

=

=

2

− 2> =

2

−> 2 cos( − ) + >

2

+

2? 0

=

( , ) ( , , , )

,

where the Green’s function ( , , , ) is defined by the formula@ presented in Paragraph 7.2.3-1.

Reference: A. G. Butkovskiy (1979).

© 2002 by Chapman & Hall/CRC Page 486

7.2. POISSON EQUATION =

7.2.3-4. Domain:

≤

1 >

≤

= − (x)

487

0≤

2, >

2 G

≤ 2 . First boundary value problem.

An annular domain is considered. Boundary conditions are prescribed: = = =

1(

Solution:

2(

( , )=

1 >

1 (

0

2?

+

2

0

@

)$

=

( , , , )

=

=

) at

2. >

2?

&

)

=

2?

−>

2

1

2 (

0

)$

=

( , , , )

)

=

2

( , ) ( , , , )

.

@

@

1

@ =

where

=

=

2

2

=

+I

−2 =

I

(> (>

1 =

ln

>

2 >

2 )

1

1)

2J

2

( ) = for for

=

2

2 J +2

> = 1H

− ln

=0 = H

cos( − ), I

0

=

2

1

2

( , , , ) =

H

(

=

2

+ (I

) − 2

cos( − ), >

= I

H

I

H

= 2 , = 2 + 1, (

,

2 1

.

I

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). =

7.2.3-5. Domain: 0 ≤

≤

,0≤ >

≤ . First boundary value problem.

A semicircle is considered.= Boundary conditions are prescribed: = =

=

= 0,

2 ( ) at

=

= .

3 ( ) at

?

( , ) =−>

1 (

0

)$

−

0@

( , , , )

=

$

=

1

3 ( )

)

=

@ &

'

+

=?

2 (

0@

( , , , )

+

&

)

?

0

1

=

$

( , , , )

=

= =

( , , , ) =

=

>

=0

,

=

− 2 > 2= cos( − ) + > 4 1 − ln 2 [ 2 − 2 cos( − ) + 2 ] 4

'

=

2 2

1

ln 4

&

( , ) ( , , , )

0@

where

=

, >

=

=

1 ( ) at

Solution:

&

Here,

=

1, >

=

=

) at

2 2

=

>

− 2 > 2= cos( + ) + > 4 . 2 [ 2 − 2 cos( + ) + 2 ] =

See also Example 2 in Paragraph 7.2.4-2.

References: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). =

7.2.3-6. Domain: 0 ≤

≤

,0≤ >

≤

2. First boundary value problem.

A quadrant of a circle is considered. Boundary conditions are prescribed:= = = =

Solution:

1(

=

, >

2(

= 0,

) at

? K

( , ) =−>

2

0

1 (

−

=

3(

=

) at

=

=

) at

0@

3 ( )

1

)$ $

( , , , )

=

& )

=

+

&

@ '

=? K

2

+

? K

0

2 (

0@

( , , , )

2.

=

2

)

1

0@

$

=

( , , , ) =

&

'

=0

( , ) ( , , , )

,

© 2002 by Chapman & Hall/CRC Page 487

488

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

where =

( , , , ) =

=

, , , )−

=

=

, , =, 2 − ) − = 1 ( , , , − ) + = 2 2 2 1 = − 2> = cos( − )+> 4 1 ( , , , ) = ln 2 2 . 4 > [ − 2 cos( − ) + 2 ] See also Example 3 in Paragraph 7.2.4-2.

=

1(

1(

, , , + ),

1(

References: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). =

7.2.3-7. Domain: 0 ≤

≤

,0≤ >

≤ L . First boundary value problem.

A circular sector is considered. Boundary conditions are prescribed: = = =

1(

Solution:

( , )=−>

1 (

0M

)$

− 1 1 . For L =

0@

=

$

+

&

)

=

0@

2 (

( , , , )

=

@ &

'

=

0M

$

=

( , , , )

&

'

=0

( , ) ( , , , )

0@

.

−1

=

1( , , , 2

=0 J

=

=

+

=

1( , , , 2

=

2 2

1

ln 4

1( , , , ) =

+ )− L

>

2 1 . For arbitrary L , the Green’s function is given by =

= R SUT

R S Q

,

1

ln 2

( , , , ) =

Q , ¯= '

=

O

R S

'

−

− ), , L

− 2 > 2= cos( − ) + > 4 . 2 [ 2 − 2 cos( − ) + 2 ] =

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

=

1

=L .

) at

=

+

)

3(

( , , , ) =

where

=

=

, where ( is a positive integer, theM Green’s function is expressed as

(

= 0,

) at

( , , , )

2(

=

1

3 ( )

=

, >

=

=

) at

Q

− ¯? Q − ?

? K N

NPO ? K M N N O M

K > N N

K M

, and V 2 = −1.

N N > N N N N M

Q

K

2?

K M

− ( ¯ )? Q − ( O )?

2?

K N N

O M

, N

K M N M

=

7.2.3-8. Domain: 0 ≤

0, 0 ( ì í ) if û = ì 4 where í = ä 2 + å 2 , 0 ( ) is the modified Bessel function of the second kind, 0(1) ( ) and 0(2) ( ) are the Hankel functions of the first and second kind of order 0, ä 0 and å 0 are arbitrary constants, and ô 2 = −1. The leading term of the asymptotic expansion of the fundamental solutions, as í î 0, is given by 21 ln 1ð .

(ä , å ) = −

3 Ø . Suppose ü = ü (ä , å ) is a solution of the homogeneous Helmholtz equation. Then the functions ü ü 1 = ( ä + 1 , å + 2 ), ü ü 2 = (− ä + 1 , å + 2 ),

= ü (ä cos + å sin + 1 , −ä sin + å cos + 2 ), where 1 , 2 , and are arbitrary constants, are also solutions of the equation. ö÷ ü

3

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

7.3.2-2. Domain: − ï

< ä

0: ë ü

0: (2) 0 (

1)

+

(2) 0 (

2)

+

(2) 0 (

3)

+

(2) 0 (

4 )

.

© 2002 by Chapman & Hall/CRC Page 496

7.3. HELMHOLTZ EQUATION

7.3.2-7. Domain: − ï

2− 2

=

|1 − % | , 2−%

@ + " 2 A < =?>

2− 2

@ B

for

1

> 0,

@ + " 2D

2− 2

@ B

for

1

< 0,

>

=

< =?>

2 2−% F

|

1|

,

where " 1 and " 2 are arbitrary constants, ; < ( G ) and A < ( G ) are the Bessel functions, and C < ( G ) and D < ( G ) are the modified Bessel functions. The solution of equation (4) is expressed as

1− 2

1− 2

8

3 ( ) = 67 79

: " 1 ; H =I

2− 2

: " 1 CH =?I

2− 2

J = |1 − & | , 2−&

@ + " 2 A H =?I

2− 2

@ B

for

1

< 0,

@ + " 2 D H =?I

2− 2

@ B

for

1

> 0,

I =

2 2−&

| F

1|

,

where " 1 and " 2 are arbitrary constants. The sum of solutions of the form (2) corresponding to different values of the parameter 1 is also a solution of the original equation; the solutions of some boundary value problems may be obtained by separation of variables. 4 . See equation 7.4.3.3, Item 4 , for K = 0.

© 2002 by Chapman & Hall/CRC Page 519

520

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

2. L

L M

N O

M P

+ L

L Q L M

N

=R .

L Q

L

L

This is a two-dimensional equation of the heat and mass transfer theory with constant volume release of heat in an inhomogeneous anisotropic medium. Here, 1 ( ) = and 2 ( ) = are the principal thermal diffusivities. 1 . For % ≠ 2 and &

≠ 2, there are particular solutions of the form #

The function

#

#

=

1+ 2 ( = ) (2 − & )2 2− + (2 − % )2 2− * .

= # (( ),

(1)

(( ) is determined by the ordinary differential equation # ./,-, . +

# ., =! , (

(2)

where

4−% & 4K , ! = . (2 − % )(2 − & ) (2 − % )2 (2 − & )2 The general solution of equation (2) is given by =

8

#

(( ) = 67 "

1

and "

2

1

" 1( 79

where "

" 1 ( 1− 1 + " 2 + 2( 1 S +1) ( 2

ln ( + " 2

+"

2

2

+

+ 1 2!

1 4!

(

for

≠ T 1,

(

2

for

= 1,

2

ln (

for

= −1,

are arbitrary constants.

2 . The substitution #

( , ) = U ( , ) +

K (2 − % )

leads to a homogeneous equation of the form 7.4.3.1: V

V

3. L

L M

N O

M P

(3)

L Q

+ L W

L M

L [

N

V

V

W

N [

L Q L [

V

U

W

X Z

+ V X

V

U V X

N Y

=R

2−

W

= 0.

. Q

This is a two-dimensional equation of the heat and mass transfer theory with a linear source in an inhomogeneous anisotropic medium. 1 \ . For ] ≠ 2 and &

≠ 2, there are particular solutions of the form ^

= ^ (_ ),

X Z c 1d 2 _ = ` (2 − & )2 a 2− b + (2 − ] )2 2− .

Y ordinary differential equation The function ^ = ^ (_ ) is determined by the ^ f/-e e f + g _

^ fe =h ^ ,

(1)

where

4−] & 4K , h = . (2 − ] )(2 − & ) (2 − ] )2 (2 − & )2 The general solution of equation (1) is given by Y g

=

^

(_ ) = _

1− i 2

jlk 1 m n o _ p

^

(_ ) = _

1− i 2

jlk 1 t?n o _ u h

|h | q + k

2r

n o_ p

q + k 2v n o _ u h

| h | q s q s

for h

< 0,

for h

> 0,

where w = 12 |1 − |; k 1 and k 2 are arbitrary constants; m n ( x ) and r n ( x ) are the Bessel functions; and g t?n ( x ) and v n ( x ) are the modified Bessel functions.

© 2002 by Chapman & Hall/CRC Page 520

521

7.4. OTHER EQUATIONS

2 . There are multiplicatively separable particular solutions of the form

( , ) = ( ) ( ),

where ( ) and ( ) are determined by the following second-order linear ordinary differential equations ( 1 is an arbitrary constant): ( ) =

( ) = ( −

,

1

1 )

.

(2)

The solutions of equations (2) are expressed in terms of the Bessel functions (or modified Bessel functions); see equation 7.4.3.1, Item 3 . 3 . There are additively separable particular solutions of the form

( , ) = ( ) + ( ),

where ( ) and ( ) are determined by the following second-order linear ordinary differential equations ( 2 is an arbitrary constant): ( ) − =

( ) − = −

2,

2.

(3)

The solutions of equations (3) are expressed in terms of the Bessel functions (or modified Bessel functions). 4 . The transformation (specified by A. I. Zhurov, 2001) 2− 2

where

2

2 2

= (2 − )2 and

2

= cos ,

2− 2

= (2 − )2 , leads to the equation

4− 1 1 2 2 ( + 2 2 − 2 (2 − )(2 − )

+

= sin ,

− − ) cos 2 + ( − ) (2 − )(2 − ) sin 2

which admits separable solutions of the form ( , ) =

4.

!

) 2 ( ).

!

$ #

( + s)%

( + ) " + #

1 (

= 4 ,

=& .

#

"

The transformation ' = + ( , ) = + * leads to an equation of the form 7.4.3.2:

5.

!

'

+ $

' ,

+ )

+

)

! #

( + s)%

( + ) " + #

'

#

=& "

!

)

,

=.

.

The transformation ' = + ( , ) = + * leads to an equation of the form 7.4.3.3:

6.

+

- . / ! ,

'

+ #

+

'

' ,

+ )

+

)

)

,

= .

+

$ - 0 1 ! # ,

= 0.

This is a two-dimensional equation of the heat and mass transfer theory in an inhomogeneous

anisotropic medium. Here, 1 ( ) = 243 and 2 ( ) = 5 2 6 are the principal thermal diffusivities.

© 2002 by Chapman & Hall/CRC Page 521

522

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

1 . Particular solutions ( , , 7 are arbitrary constants):

( , ) = 2 −3

( , ) =

( , ) =

+ 2 −6

+7 ,

(8 + 1) 2 −3 8 2 − − 6 + . 2 3

(9 9 2

−

+ 1) 2 − 6

+ ,

2 . There are multiplicatively separable particular solutions of the form

( , ) = ( ) ( ),

(1)

where ( ) and ( ) are determined by the following second-order linear ordinary differential equations ( 1 is an arbitrary constant):

( 2 3 ) = − 1 , ( 52 6 ) = 1 .

(2) (3)

The solution of equation (2) is given by ( ) = :

; 2
( 2 −3

; 2
( 2 −3

+7

2@ 1 > ( 2

+7

2C

− ; 2 ?5A

3

1> ( 2

− ; 2 ?5A

3

for

1

> 0,

for

1

< 0,

where ( = −(2 D 8 ) E | 1 | D ; 7 1 and 7 2 are arbitrary constants; = 1 ( F ) and @ 1 ( F ) are the Bessel functions; and B 1 ( F ) and C 1 ( F ) are the modified Bessel functions. The solution of equation (3) is given by

( ) = :

; 2< ; 2? 2 −6 7 1 = 1 > *G2 − 6 ; 2< ; 2? 2 −6 7 1 B 1 > *G2 − 6

+7

2 @ 1 > *G2

+7

2C

− ; 2 ?5A

6

1 > *G2

− ; 2 ?5A

6

for

1

< 0,

for

1

> 0,

where * = −(2 D 9 ) E | 1 | D ; 7 1 and 7 2 are arbitrary constants. The sum of solutions of the form (1) corresponding to different values of the parameter also a solution of the original equation.

1

is

3 . See equation 7.4.3.8, Item 3 , for = 0.

7.

- . / !

+

$ - 0 1 !

+ #

,

=& .

# +

,

This is a two-dimensional equation of the heat and mass transfer theory with constant volume release

of heat in an inhomogeneous anisotropic medium. Here, 1 ( ) = 2 3 and 2 ( ) = 52 6 are the principal thermal diffusivities. The substitution

( , ) = H ( , ) − (8 + 1) 2 −3 2 8

leads to a homogeneous equation of the form 7.4.3.6:

8.

+

- . / !

,

+ #

+

2 3

H

,

+

+

$ - 0 1 ! # ,

=&

!

+

52 6

H

,

= 0.

.

This is a two-dimensional equation of the heat and mass transfer theory with a linear source in an inhomogeneous anisotropic medium.

© 2002 by Chapman & Hall/CRC Page 522

523

7.4. OTHER EQUATIONS

1 . For 8 9 ≠ 0, there are particular solutions of the form

= (I ),

;

= > 9 2 2 −3 + 8 2 2 − 6 )1 2 . The function = (I ) is determined by the ordinary differential equation 1 K 4 KLK J − = , = . 2 2 I

I

8

9

For the solution of this equation, see 7.4.3.3 (Item 1 for = −1). 2 . The original equation admits multiplicatively (and additively) separable solutions. See equation

7.4.3.12 with ( ) = 2 3 and ( ) = 52 6 . 3 . The transformation (specified by A. I. Zhurov, 2001) − ; 2

2

where

2

= 8

2

and

= cos ,

3

2

− ; 2

6

2

= sin ,

= 9 2 , leads to the equation 2 2

1

2 2 − cot 2 2 2 2 which admits separable solutions of the form ( , ) = 1 ( ) 2 ( ! $ - . 1 ! ! 9. + # =& . + , + ,

−

+

1

= 4 , ).

1 . For ≠ 2 and 8 ≠ 0, there are particular solutions of the form

=

2

2 −3

2−

+ . (2 − )2 8 2 The function = ( ) is determined by the ordinary differential equation ( ),

2 2

+

=

1

= 4 .

2− For the solution of this equation, see 7.4.3.3 (Item 1 ).

2 . The original equation admits multiplicatively (and additively) separable solutions. See equation 7.4.3.12 with ( ) = and ( ) = 52M3 . 3 . The transformation (specified by A. I. Zhurov, 2001) 1 1− 2

2

where

2

2

= (2 − ) and 2 2

1 2 −23

= cos ,

= sin ,

2

= 8 , leads to the equation

2 (1 − ) cos 2 + 1 + + − 2 2− (2 − ) sin 2 which admits separable solutions of the form ( , ) = 1 ( ) 2 ( ).

10.

ON

!

where V

=7

1

=7

1

= [7

1R

( ) + 7

2 ]

= [7

1R

( ) + 7

2 ]

7 1 + 7 3 ( ) Q

2

+7

2

−2P

3

+7

2

− 6 P

1, V 2, V 3, V 4,

and V

5

= 4 ,

= 0.

2

1 . Particular solutions:

2 2 2

1

2!

( ) " + #

1

+7 2

+7

+7

4,

7 1 + 7 3 ( ) Q

+7

( ) + 7

R

( ) = P

−2P S

3R 3R

( ) + 7

4,

4

4,

Q

( )

1 P [7 ( )

, 1R

( ) + 7

2]

Q

T

Q U

,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 523

524

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

2 W . Separable particular solution: X = (V

+V

1Y Z [

2Y

−

Z [ )\

( ), U

where V 1 , V 2 , and ] are arbitrary constants, and the function \ differential equation [ ^ ( ) \ ` _ ] _` + ] 2 \ = 0.

= \ ( ) is determined by the ordinary U

U

X 3 W . Separable particular solution:

= [V

sin( ] a ) + V

1

2

cos( ] a )] b ( ), U

where V 1 , V 2 , and ] are arbitrary constants, and the function b = b ( ) is determined by the ordinary U differential equation [ ^ ( ) b ` _ ] _` − ] 2 b = 0. U

4 W . Particular solutions with even powers of a : X

e

e

where h

e g

U

( )+j e

i

− 2 m (2 m − 1) k

U

5 W . Particular solutions with odd powers of a : X

e

where h 11.

r

sOt

U

=0

( )=h U

−1 (

( )q u q

q r

2 +1

( )a )

q r

l U

T

l U

,

,

( )+j U

e

U

c i

)=h

e

, ( ) U 1 k ^ ( )n

U

i c

( )+j

^

− 2 m (2 m + 1) k

U

i

( )=k

,

l U

+ q v

( )q u

q w

w

syx

q w

e )

( ) U

U

are arbitrary constants ( m = o , ppp , 1).

and j

U

U

c

)

( ) g

U

= ) ( ) are defined by the recurrence relations

) e

de

e

e e

= c e

where the functions )

e

l U

^

are arbitrary constants ( m = o , ppp , 1).

and j

e

, ( ) U 1 k ^ ( )n

U

c

( )+j i

( )=k

,

U

e c i

)=h U

,

U

( )=h

−1 (

e

U

( ) are defined by the recurrence relations g

g c

2

( )a =0 g

= g

dfe

= c e

where the functions

e e

l U

T

l U

,

= 0. v

This is a two-dimensional sourceless equation of the heat and mass transfer theory in an inhomogeneous anisotropic medium. The functions ^ = ^ ( ) and z = z (a ) are the principal thermal U diffusivities. 1 W . Particular solutions: X

X e

e

( ,a ) = h

1 k

( ,a ) = h

2 k

( ,a ) = h

3 k

U

X

U U

l U

( )

^

U

U l U ^ ( ) U ^

l U

( ) U

+j

1k

−h

2 k

k z

l

l z

a z a

(a )

a

(a ) l

+V

1,

+j

2,

a

(a )

+j

3,

where the h , j , and V 1 are arbitrary constants. A linear combination of these solutions is also a solution of the original equation.

© 2002 by Chapman & Hall/CRC Page 524

525

7.4. OTHER EQUATIONS

2 W . There are multiplicatively separableX particular solutions of the form ( , a ) = { ( )| (a ), U

(1)

U

where { ( ) and | (a ) are determined by the following second-order linear ordinary differential U equations ( h is an arbitrary constant): ( ^ { `_ ) _` = h { , (z | _ ) _ = − h | , ^

[ [

= ^ ( ), U z = z (a ).

(2)

The sum of solutions of the form (1) corresponding to different values of the parameter h in (2) is also a solution of the original equation (the solutions of some boundary value problems may be obtained by separation of variables). 12.

( )q u q

q r

sOt

r

q r

+ q v

( )q u

q w

w

syx

=}

q w

v

. u

This is a two-dimensional equation of the heat and mass transfer theory with a linear source in an inhomogeneous anisotropic medium. The functions ^ = ^ ( ) and z = z (a ) are the principal thermal U diffusivities. 1 W . There are multiplicatively separableX particular solutions of the form ( , a ) = { ( )| (a ), U

(1)

U

where { ( ) and | (a ) are determined by the following second-order linear ordinary differential U equations ( h is an arbitrary constant): ( ^ { _` ) _` = h { , (z | _ ) _ = ( ~ − h ) | ,

= ^ ( ), U z = z ( a ). ^

[ [

(2)

The sum of solutions of the form (1) corresponding to different values of the parameter h in (2) is also a solution of the original equation; the solutions of some boundary value problems may be obtained by separation of variables. solutions of the form 2 W . There are additively separable particular X ( , a ) = ( ) + (a ), U

i

U

where ( ) and (a ) are determined by the following second-order linear ordinary differential U equations i ( V is an arbitrary constant): ( ^ `_ ) _` − ~ =V , (z i _ ) _ − ~ i = − V , [ [

^

= ^ ( ), U z = z (a ).

In the special case ~ = 0, the solutions of these equations can be represented as ( )=V k i

where h

1, h

2, j

1,

and j

2

U

(a ) = − V

U l U +h 1k ^ ( ) U a a k l +h 2k z (a ) ^

l U

( )

z

U

l

a

(a )

+j +j

1, 2,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 525

526

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.4.4. Other Equations Arising in Applications 1.

2

q w

u

2

+ q

2

q r

u

= 0.

2

q w

Tricomi equation. It is used to describe near-sonic flows of gas. 1 W . Particular solutions: X X

=h

+j a

U

= h (3 where h , j , V , and

2

U

+V a + , U

3

−a )+j (

are arbitrary constants.

3

U

U

2 W . Particular solutions with even powers of : X

(a )e = h { c

and j

where h

,

c

{

)=h

−1 ( a

+j a

e

where h

=|

(a )e = h

and j

− 2 m (2 m − 1) k

[

e

(a − ){

0

( ) , l

a

= c |

(a )

=0

2 +1

U

,

(a ) are e defined bye the recurrence relations e

+j

c

e e

U d e

e

where the functions |

c

,

are arbitrary constants ( m = o , ppp , 1). X

|

2

U

c

3 W . Particular solutions with odd powers of :

e

− a 4 ),

(a ) are e defined bye the recurrence relations e

+j a

(a ) {

=0

={

where the functions { e

de

e

2

U

e e

U

= c

e

− a 3 ) + V (6a

, |

)=h

−1 ( a

a

+j

− 2 m (2 m + 1) k

c

[ 0

e

(a − )|

( ) , l

are arbitrary constants ( m = o , ppp , 1).

X 4 W . Separable particular solutions:

( , a ) = yh sinh(3 ] X

U

U

) + j cosh(3 ]

) a yV 1 3 (2 ] a 3 2 ) + 1 3 (2 ] a 3 2 ) , U

) a yV 1 3 (2 ] a 3 2 ) + 1 3 (2 ] a 3 2 ) , where h , j , V , , and ] are arbitrary constants, 1 3 ( F ) and 1 3 ( F ) are the Bessel functions, and 1 3 ( F ) and 1 3 ( F ) are the modified Bessel functions. 5 W . For a > 0, see also equation 7.4.4.2 with o = 1. For a < 0, the change of variable a = − leads to an equation of the form 4.3.3.11 with o = 1. ( , a ) = yh sin(3 ] U

y

2.

U

) + j cos(3 ]

U

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964). 2

w

q

u q r

2

+ q

2

u

2

= 0. X

q w

1 W . Particular solutions:

+V a + , 2h =h 2− a +2 , U (o + 1)(o + 2) c X 6h +2 =h 3− a , U (o + 1)(o + 2) U c X 2h =h a 2− a +3 , U (o + 2)(o + 3) c are arbitrary constants. X

where h , j , V , and

=h

U

a

+j

U

© 2002 by Chapman & Hall/CRC Page 526

527

7.4. OTHER EQUATIONS

2 W . Particular solutions with even powers of : U

X

dfe

= e

e

(a ) = h {

e

where h

e

,

e

)=h

−1 ( a

{

e

+j a

U

X e

de

e

(a ) = h e

and j

(a − ) {

( ) , l

c

2 +1

U

,

(a ) are defined by the recurrence relations e

+j a

(a ) |

=0

=|

where the functions |

where h

e e

=

e [

− 2 m (2 m − 1) k

3 W . Particular solutions with odd powers of :

|

,

are arbitrary constants ( m = , ppp , 1), is any number.

and j

e

2

U

(a ) are defined by the recurrence relations

+j a

e

(a ) {

=0

={

where the functions {

e e

,

e

)=h

−1 ( a

|

a

e

+j

e [

− 2 m (2 m + 1) k

(a − ) | c

( ) , l

are arbitrary constants ( m = , ppp , 1), is any number.

particular solutions: 4 W . Separable X ( , a ) = h sinh( ] X

U

( , a ) = h sin( ] U

U

U

) + j cosh( ] ) + j cos( ]

U

) a V

) a V U

1 2

1 2

(] a ) +

(] a ) +

( ] a ) ,

1 2

1 2

= 12 (o + 2),

( ] a ) ,

where h , j , V , , and ] are arbitrary constants, ( F ) and ( F ) are the Bessel functions, and ( F ) and ( F ) are the modified Bessel functions. 5 W . Fundamental solutions (for a > 0): X

o

1 (

, a , 0 , a 0 ) = m 1 ( 12 )− (~ , ~ , 2~ ; 1 − ),

2 (

, a , 0 , a 0 ) = m 2 ( 12 )− (1 − )1−2 (1 − ~ , 1 − ~ , 2 − 2~ ; 1 − ).

~

=

2(o + 2)

,

=

22

, 12

Here, ( , , ; ) is the hypergeometric function and 1

2

= ( − 0 )2 +

4 a (o + 2)2 c

+2 2

22

= ( − 0 )2 +

4 (o + 2)2 ¤ ¥

+2 2

=

1 4 4¢ o +2¡

2

m 1

=

1 4 4¢ o + 2 ¡

2

m 2

+2

+ a 0c 2 ¡ , +2

− ¤ 0¥ 2 ¡ ,

£ 2 (~

) , (2~ )

£ £

£ 2 (1 − ~

) , (2 − 2~ )

£

where (~ ) is the gamma function; 0 and ¤ 0 are arbitrary constants. The fundamental solutions satisfy the conditions ¦ §

y

§

1 ¨ =0 ¨

= 0,

§

2 ¨ =0 ¨

=0

( and

0

are any, ¤

0

> 0).

The solutions of some boundary value problems can be found in the first book cited below. Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. D. Polyanin (2001a).

© 2002 by Chapman & Hall/CRC Page 527

528

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES 2

3. q

u

2

+ ª

2

+ q

q u

u

= 0.

2

q © © q © q « Elliptic analogue of the Euler–Poisson–Darboux equation.

1 ¬ . For = 1, see Subsections 8.1.2 and 8.2.3 with = ( , F ). For ≠ 1, the transformation = (1 − ) F , ¤ = 1− ® leads to an equation of the form 7.4.4.1: ¦ 2 ¦ 2

2® 1− ®

¤

¦ 2

+ ¦

2

¤

= 0.

2 ¬ . Suppose ® = ® ( , F ) is a solution of the equation in question for a fixed value of the parameter . Then the functions ¯ ® defined by the relations ¦

¯ ®

® = ¦ ,

¯ ®

=

¯ ®

=

¦ F ¦ ® ® ¦ +F ¦ ¦ F ® 2 F ¦ + (F 2

,

¦

® − 2) ¦ + F ® F

are also solutions of this equation. 3 ¬ . Suppose ® = ® ( , F ) is a solution of the equation in question for a fixed value of the parameter . Using this ® , one can construct solutions of the equation with other values of the parameter by the formulas y

4.

−1 ® ¦ ® ¦ ¦ ® F ¦

2− ®

= ®

,

® −2

=

+ ( − 1) ® ,

® −2

=

® −2

= (

® +2

=

® +2

=

® +2

=

2

F 2

−

¦

+ ( − 1) F ® , ¦ ® ¦ F

® − F 2) ¦ + 2 2 F

¦ 1 ® ¦

¦ ® ¦ F

2

¦ ® ¦

−F

,

+ °

2

− ( − 1) F 2 ® ,

¦

® − ¦ , F

2 ¦

¦

®

¦ ® + 2F ¦ F

+ ® .

Reference: A. V. Aksenov (2001). 2

q

u

2

2

+ ( )q

q r

t

r

u

2

= 0.

q w

1 ¬ . Particular solutions:

where ±

=±

=± ¸

=± ¸

= (±

1 ¤ 1¤ 1¤ 1´

2

3

+±

+±

2¤

+±

2 ¤

+±

+±

2´ ¤

+±

+±

2 )¤

1, ± 2, ± 3, ± 4, ± 5,

2

+±

and ±

+±

3 3¤ 3¤

3´ ¤ 6

4,

+±

4

− 2±

+±

4´

− 6±

+±

4¤

+±

1² 1¤ 5´

²

(´ − µ ) ¶ (µ ) · µ + ± ³

³

(´ − µ ) ¶ (µ ) · µ + ±

− 2 ² ³ (´ − µ )( ±

1µ

5, 5,

+±

2)¶

(µ ) · µ + ±

6,

are arbitrary constants, is any number.

© 2002 by Chapman & Hall/CRC Page 528

529

7.4. OTHER EQUATIONS

2 ¬ . Separable particular solution: ¸

§

= (±

+±

1¹ º

2¹

−

§

) » (´ ), º

= » (´ ) is determined by the ordinary

where ± 1 , ± 2 , and ¼ are arbitrary constants, and the function » differential equation » ½J½ + ¼ 2 ¶ (´ ) » = 0. 3 ¬ . Separable particular³ ³ solution: ¸

= [±

1

sin( ¼ ¤ ) + ±

2

cos( ¼ ¤ )] ¾ (´ ),

where ± 1 , ± 2 , and ¼ are arbitrary constants, and the function ¾ = ¾ (´ ) is determined by the ordinary differential equation ¾ ½J½ − ¼ 2 ¶ (´ ) ¾ = 0. ³ even powers of ¤ : 4 ¬ . Particular solutions³ with ¿fÀ

¸ À

,

Ã

À

)=Ä

−1 ( ´

Ã

À

´

+Å

À

1(

q r

ÊOË

r

À

+Å

,

Ã

À

)=Ä

−1 ( ´

Ã

À

´

+Å

− 2 Æ (2 Æ + 1) ² ³ (´ − µ ) ¶ (µ )

É

À

(µ ) · µ , É

are arbitrary constants ( Æ = Ç , ppp , 1), and is any number.

and Å

q

´

À

,

=0 É

É

(´ ) = Ä

2 +1

(´ )Â

(´ ) are defined by the recurrence relations

É É Ã

= Ã

À

=

where the functions

5.

(µ ) · µ , Á

À À

¿À

¸

where Ä

À

− 2 Æ (2 Æ − 1) ² ³ (´ − µ ) ¶ (µ )

Á

5 È . Particular solutions with odd powers of Â :

À

,

are arbitrary constants ( Æ = Ç , ppp , 1), is any number.

and Å

where Ä

À

+Å ´

À

2

(´ ) are defined by the recurrence relations Á

(´ ) = Ä Á Ã

(´ )Â =0 Á

= Á

À

= ¥

À

where the functions

À À

)q Ì q r

+ q Í

2(

q Î

ÊOË

Î

)q Ì

+ Ï ÐÒÑ 1 ( ) + Ñ 2 ( )Ó

q Î

r

Í

Î

Ì

= 0.

This equation is encountered in the theory of vibration of inhomogeneous membranes. Its separable solutions are sought in the form ¸ (´ , Â ) = (´ ) (Â ). The article cited below presents an algorithm Á É for accelerated convergence of solutions to eigenvalue boundary value problems for this equation.

ÔyÕ

Reference: L. D. Akulenko and S. V. Nesterov (1999).

7.4.5. Equations of the Form Ö

2Ù

(× ) Ø Ø

× 2

+ Ø

2Ù

Ø Ú

2

Ù

+ Û (× ) Ø Ø

×

+ Ü ( × )Ù

= – Ý (× , Ú )

7.4.5-1. Statements of boundary value problems. Relations for the Green’s function. Consider two-dimensional boundary value problems for the equation Þ

2¸

(´ ) ß ß

´ 2

2¸

+ ß ß

Â 2

¸

+ à (´ ) ß ß

´

+ á (´ )¸ = − â (´ , Â )

(1)

© 2002 by Chapman & Hall/CRC Page 529

530

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

with general boundary conditions in ´ , ã

¸

1

− Æ 1 ¸ = ¶ 1 (Â ) at + Æ 2 ¸ = ¶ 2 (Â ) at

ã ß ³ ¸ 2 ß ³

= ´ 1, ´ = ´ 2, ´

(2)

and different boundary conditions in Â . We assume that the coefficients of equation (1) and the boundary conditions (2) meet the requirement Þ

(´ ), à (´ ), á (´ ) are continuous functions (´

Þ

≤ ´ ≤ ´ 2 );

1

| ã 1 | + | Æ 1 | > 0,

> 0,

| ã 2 | + | Æ 2 | > 0.

In the general case, the Green’s function can be represented as ä

¿

(´ )

(´ , Â , å , æ ) = ç (å ) è =1 é

Ã

Here,

1 (´ ) = Þ exp (´ ) ç

) í (í ) î

ÊOì

Ã é

à (í Þ

ê

Ã

ê é

ê

, Í

(å )

ê 2

).

Ã

2

ç

2

(í )

1

ì

(3)

Ã

=

Ã é

(Â , æ ; ¼ ë

Ã2

ï

é

(í ) í ,

(4)

î

Ã

and the ð and (í ) are the eigenvalues and eigenfunctions of the homogeneous boundary value ï Ã problem for differential equation Ã the ordinary é

Þ

(í ) ½J½ + à (í ) ½ + [ ð + á (í )] = 0, ã

−Æ

½ é 1 ã ï ï ½ 2é ï

1 é = 0 at + Æ 2 é ï = 0 at

(5)

=é í 1 , í

(6)

= í 2. í

(7)

é The functions ë for various boundaryé conditions in Â are specified in Table 25. ï Equation (5)Ã can be rewritten in self-adjoint form as

[ñ (í ) ½ ] ½ + [ ð ç (í ) − ò (í )] = 0, é where the functions ñ (í ) and ò (í ) are given by

é

ï ï

ñ

(í ) = exp Þ

à (í

) í (í ) î

ÊOì

(8)

, ò (í ) = − Þ

á (í

Í

) exp (í )

Þ ÊOì

à (í

) í (í ) î

, Í

and ç (í ) is defined in (4). The eigenvalue problem (8), (6), (7) possesses the following properties: 1 È . All eigenvalues ð 1 , ð 2 , ppp are real and ð 2 È . The system of eigenfunctions { 1 (í ), weight ç (í ), that is, é

2

ç

3 È . If the conditions

(í )

1

ì

ï

ï

2 (í

é

Ã

) ≥ 0,

ó

.

ô

ó

ô

), ppp } is orthogonal on the interval í

1

≤í ≤í

2

with

(í ) í = 0 for Ç ≠ ö .

(í ) é

ò (í

as Ç Ã

î

é õ ã

1Æ 1

≥ 0,

ã

2Æ 2

≥0

(9)

are satisfied, there are no negative eigenvalues. If ò ≡ 0 and Æ 1 = Æ 2 = 0, then the least eigenvalue is ð 0 = 0 and the corresponding eigenfunction is 0 = const; in this case, the summation in (3) must start with Ç = 0. In the other cases, if conditions (9) are satisfied, all eigenvalues are positive; for é example, the first inequality in (9) holds if á (í ) ≤ 0.

© 2002 by Chapman & Hall/CRC Page 530

531

7.4. OTHER EQUATIONS

The functions ë Domain −

|ù | < ô

Ã

1

ó

= 0 for ù

Function ë

for Â ô

0≤
æ , cosh(÷ ) for æ >

−û üGÿ

[÷ cosh(÷ æ ) + 3 sinh(÷ æ )] for > æ , [÷ cosh(÷ ) + 3 sinh(÷ )] for æ > sinh(÷ æ ) sinh[÷ ( − )] for > æ , sinh(÷ ) sinh[÷ ( − æ )] for æ > cosh(÷ æ ) cosh[÷ ( − )] for > æ , cosh(÷ ) cosh[÷ ( − æ )] for æ >

1 û ü sinh(û ü )

= 0, =

−û üGþ

(Â , æ ; ð

sinh(÷ æ ) cosh[÷ ( − )] for > æ , sinh(÷ ) cosh[÷ ( − æ )] for æ >

Subsection 1.8.9 presents some relations for estimating the eigenvalues ð and eigenfunctions (í ). é Green’s function of the two-dimensional third boundary value problem (1)–(2) augmented The by the boundary conditions ù ß

− 3 ù = 0 at

= 0,

ù ß

ß

+ 4 ù = 0 at

=

ß

is given by relation (3) with ë

( , æ ; ð ) =

÷

cosh(÷ )+ 3 sinh(÷ ) ÷ cosh[÷ ( − )]+ 4 sinh[÷ ( − )] ÷ ÷ ( 3 + 4) cosh(÷ )+(÷ 2 + 3 4) sinh(÷ )

for

> ,

÷

cosh(÷ )+ 3 sinh(÷ ) ÷ cosh[÷ ( − )]+ 4 sinh[÷ ( − )] ÷ ÷ ( 3 + 4) cosh(÷ )+(÷ 2 + 3 4) sinh(÷ )

for

< .

7.4.5-2. Representation of solutions to boundary value problems using the Green’s function. 1 . The solution of the first boundary value problem for equation (1) with the boundary conditions ù

= 1( ) ù

= 3 (í ) at

at í

= í 1, = 0, ù

ù

= 2( )

at í

= 4 (í ) at

* For unbounded domains, the condition of boundedness of the solution as omitted.

= í 2, =

is set; in Table 25, this condition is

© 2002 by Chapman & Hall/CRC Page 531

532

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

is expressed in terms of the Green’s function as ù

(í , ) = (í 1 ) +!

2

ì

1 ( )

0

( , , , )

=

− ( 1

3( ) ( , " , , ) # −!

1 =0 2 +! ! ( , ) ( , " , , ) . 1 0 $

2

2)

!

2 ( )

0

( , " , , )

4( ) ( , " , , ) #

=

1

=

2

2 . The solution of the second boundary value problem for equation (1) with boundary conditions

= 1 (" )

% &

=

at

= 3 ( ) at %

1,

" = 0,

= 2 (" )

%

%

&

at

=

= 4 ( ) at

" =

2,

is expressed in terms of the Green’s function as % ( , " ) = − (

−!

1) 2

!

0

1 ( ) ( , " ,

1,

) + ( 2 ) !

3 ( ) ( , " , , 0) +! 1 2 +! ! ( , ) ( , " , , ) 1 0 $

2

0

2 ( ) ( , " ,

2,

)

4 ( ) ( , " , , )

1

.

3 . The solution of the third boundary value problem for equation (1) in terms of the Green’s function is represented in the same way as the solution of the second boundary value problem (the Green’s function is now different).

© 2002 by Chapman & Hall/CRC Page 532

Chapter 07: Elliptic Equations with Two Space Variables .fr

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