Poisson Equation ④ 2 ⑤ =– ⑥ (x)

7.2. Poisson Equation ④ 2 ⑤ =– ⑥ (x)

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 ⑦ ≡ 0.

In what follows, we consider a finite domain ⑧ with a sufficiently smooth boundary ⑨ . Let r s ⑧

and ⑩ s ⑧ , where r = { ❴ , ❛ }, ⑩ ={ ❶ , ❷ }, |r − ⑩ | 2 =( ❴ − ❶ ) 2 +( ❛ − ❷ ) 2 .

7.2.1-1. First boundary value problem. The solution of the first boundary value problem for the Poisson equation

(1) in the domain ⑧ with the nonhomogeneous boundary condition

2 ❢ =− ⑦ (r)

= ✈ (r) for r s ⑨

can be represented as

Here, r

(r, ⑩ ) is the Green’s function of the first boundary value problem, ❼ ❽ is the derivative

of the Green’s function with respect to ❿

, ❷ along the outward normal N to the boundary ⑨ . The integration is performed with respect to ❶ , ❷ , with ✇ ⑧ ❺ = ✇ ❶ ✇ ❷ . The Green’s function ❹ = ❹ (r, ⑩ ) of the first boundary value problem is determined by the following conditions.

1 ❥ . The function ❹ satisfies the Laplace equation in ❴ , ❛ in the domain ⑧ everywhere except for the

point ( ❶ , ❷ ), at which ❹ has a singularity of the form 1 2 1 ➀ ln | r− ➁ | .

2 ❥ . With respect to ❴ , ❛ , the function ❹ satisfies the homogeneous boundary condition of the first ❻

kind at the domain boundary, i.e., the condition ❹ | = 0. The Green’s function can be represented in the form

where the auxiliary function ❜ = ❜ (r, ⑩ ) is determined by solving the first boundary value problem ❻

for the Laplace equation q 2 ❜ = 0 with the boundary condition ❜ ❣ =− 1 2 ➀ ln 1 | r− ➁ | ; in this problem,

is treated as a two-dimensional free parameter. The Green’s function is symmetric with respect to its arguments: ❹ (r, ⑩ )= ❹ ( ⑩ , r).

When using the polar coordinate system, one should set r={ ➋ , ➌ }, ⑩ ={ ❶ , ❷ }, | r− ⑩ | 2 = ➋ 2 + ❶ 2 −2 ➋ ❶ cos( ➌ − ❷ ), ✇ ⑧ ❺ = ❶ ✇ ❶ ✇ ❷

in relations (2) and (3).

7.2.1-2. Second boundary value problem. The second boundary value problem for the Poisson equation (1) is characterized by the boundary

condition

= ✈ (r) for r s ⑨ .

The necessary solvability condition for this problem is ①

(4) The solution of the second boundary value problem, provided that condition (4) is satisfied, can

(r) ✇ ⑧ + ✉ ❻ ✈ (r) ✇ ⑨ = 0.

be represented as

(5) where ③ is an arbitrary constant.

(r) = ✉ ❸ ⑦ ( ⑩ ) ❹ (r, ⑩ ) ✇ ⑧ ❺ + ✉ ❻ ✈ ( ⑩ ) ❹ (r, ⑩ ) ✇ ⑨ ❺ + ③ ,

The Green’s function ❹ = ❹ (r, ⑩ ) of the second boundary value problem is determined by the following conditions:

1 ❥ . The function ❹ satisfies the Laplace equation in ❴ , ❛ in the domain ⑧ everywhere except for the

point ( ❶ , ❷ ), at which ❹ has a singularity of the form 1 2 ➀ ln 1 | r− ➁ | .

2 ❥ . With respect to ❴ , ❛ , the function ❹ satisfies the homogeneous boundary condition of the second kind at the domain boundary:

where ⑨ 0 is the length of the boundary of ⑧ ① . r

The Green’s function is unique up to an additive constant.

The Green’s function cannot be determined by condition 1 ➐ ❥ ❻ and the homogeneous boundary condition ❼ ❽ ➐ ❼ ❾ = 0. The point is that the problem is unsolvable for ❹ in this case, because, on representing ❹ in the form (3), for ➒ 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 ⑧ with

the nonhomogeneous boundary condition

+ ➓ ➔ = ✈ (r) for r s ⑨

is given by formula (5) with r ③ = 0, where ❹ = ❹ (r, ⑩ ) is the Green’s function of the third boundary value problem and is determined by the following conditions:

1 → . The function ❹ satisfies the Laplace equation in ➣ , ↔ in the domain ⑧ everywhere except for the

point ( ❶ , ❷ ), at which ❹ has a singularity of the form 1 2 ➀ ln 1 | r− ➁ | .

2 → . With respect to ➣ , ↔ , the function ❹ satisfies the homogeneous boundary condition of the third kind at the domain boundary, i.e., the condition ↕ ❼ ❽ ❼ ❾ + ➓ ❹ ➙ ❻ = 0. The Green’s function can be represented in the form (3); the auxiliary function ➒ is identified by solving the corresponding third boundary value problem for the Laplace equation ➛ 2 ➒ = 0.

The Green’s function is symmetric with respect to its arguments: ❹ (r, ⑩ )= ❹ ( ⑩ , 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).

7.2.2. Problems in Cartesian Coordinate System

The two-dimensional Poisson equation in the rectangular Cartesian coordinate system has the form

7.2.2-1. Particular solutions of the Poisson equation with a special right-hand side.

1 ➠ → . If ⑦ ( ➡ ➣ , ↔ )=

exp( ➥ ➣ + ➦ ↔ ), the equation has solutions of the form

) sin( ➡ ➦ ➡ ↔ + ➩ ), the equation admits solutions of the form

2 sin( ➥ ➣ + ➨ ) sin( ➦ ↔

7.2.2-2. Domain: − ➫ < ➣ < ➫ ,− ➫ < ↔ < ➫ . Solution:

7.2.2-3. Domain: − ➯

< ➣ < ➫ ,0≤ ↔ < ➫ . First boundary value problem.

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

Reference : A. G. Butkovskiy (1979). ➯

7.2.2-4. Domain: − ➫ < ➣ < ➫ ,0≤ ↔ < ➫ . Second boundary value problem.

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

is an arbitrary constant. ➯

Reference : V. S. Vladimirov (1988).

7.2.2-5. Domain: − ✄ < ☎ < ✄ ,0≤ ✆ ≤ ✝ . First boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

= ✞ 1 ( ) at = 0, = ✟ 2 ( ☎ ) at ✆ = ✝ . Solution:

sin

cosh[ ( ☎ − ✍ ) ✏ ✝ ] − cos( ✆ ✏ ✝ )

+ ✍ sin

cosh[ ( ☎ − ✍ ) ✏ ✝ ] + cos( ✆ ✏ ✝ ✡ ✡ )

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

7.2.2-6. Domain: − ✄ < ☎ < ✄ ,0≤ ✆ ≤ ✝ . Second boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

) ✞ at ✆ = 0, = ✟ 2 ( ☎ ) at ✆ = ✝ . Solution:

4 ✡ cosh[ ( − ) ] − cos[ ( − ) ] 4 cosh[ (

− ✍ ) ✏ ✝ ] − cos[ ( ✆ + ✓ ) ✏ ✝

where ✚ is an arbitrary constant.

7.2.2-7. Domain: − ✄ < ☎ < ✄ ,0≤ ✆ ≤ ✝ . Third boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

− ✞ 1 = ✟ 1 ( ☎ ) at ✆ = 0, + ✛ 2 = ✟ 2 ( ☎ ) at ✆ = ✝ . The solution ✞ ( ☎ , ✆ ) is determined by the formula in Paragraph 7.2.2-6 where

Here, the ✦ are positive roots of the transcendental equation tan( ✦ ✝ )= ✦ 2 . − ✛ 1 ✛ 2

7.2.2-8. Domain: − ✄ < ☎ < ✄ ,0≤ ✆ ≤ ✝ . Mixed boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

) ✞ at ✆ = 0, = ✟ 2 ( ☎ ) at ✆ = ✝ . Solution:

exp ✧ − ✦ | ☎ − ✍ | ★ sin( ✦ ✆ ) sin( ✦ ✓

7.2.2-9. Domain: 0 ≤ ☎ < ✄ ,0≤ ✆ ≤ ✝ . First boundary value problem.

A semiinfinite strip is considered. Boundary conditions are prescribed:

= 0, ✞ = ✟ 2 ( ☎ ) at ✆ = 0, = ✟ 3 ( ☎ ) at ✆ = ✝ . Solution:

] − cos[ ( + ) ] , ✓ )= ✡ ln

− ✍ ) ✏ ✝ ] − cos[ ( ✆ − ✓ ) ✏ ✝ ] 4 cosh[ ( ☎ + ✍ ) ✏ ✝ ] − cos[ ( ✆ − ✓ ) ✏ ✝

Alternatively, the Green’s function can be represented in the series form

exp ✧ − | ☎ − ✍ | ★ − exp ✧ − | ☎ + ✍ | ★✲✱ sin( ✆ ) sin( ✓

References : N. N. Lebedev, I. P. Skal’skaya, and Ya. S. Uflyand (1955), A. G. Butkovskiy (1979).

7.2.2-10. Domain: 0 ≤ ☎ < ✄ ,0≤ ✆ ≤ ✝ . Third boundary value problem.

A semiinfinite strip is considered. Boundary conditions are prescribed:

) at ✞ = 0, − ✛ 2 = ✟ 2 ( ☎ ) at ✆ = 0, + ✛ 3 = ✟ 3 ( ☎ ) at ✆ = ✝ . Solution:

for ✍ > ☎ .

Here, the are positive roots of the transcendental equation tan( )= ✦ 2 − ✛ 2 ✛

7.2.2-11. Domain: 0 ≤ ☎ < ✄ ,0≤ ✆ ≤ ✝ . Mixed boundary value problems.

1 ✶ . A semiinfinite strip is considered. Boundary conditions are prescribed:

= 0, ✞ = ✟ 3 ( ☎ ) at ✆ = ✝ . Solution:

exp ✧ − | ☎ − ✍ | ★ − exp ✧ − | ☎ + ✍ | ★✲✱ cos( ✆ ) cos( ✓ ),

2 ✶ . A semiinfinite strip is considered. Boundary conditions are prescribed: ✷

= ✞ ✟ 1 ( ✆ ) at ☎ = 0, = ✟ 2 ( ☎ ) at ✆ = 0, = ✟ 3 ( ☎ ) at ✆ = ✝ . Solution:

exp ✧ − | ☎ − ✍ | ★ + exp ✧ − | ☎ + ✍ | ★✲✱ sin( ✆ ) sin( ✓ ), =

7.2.2-12. Domain: 0 ≤ ☎ < ✄ ,0≤ ✆ < ✄ . First boundary value problem.

A quadrant of the plane is considered. Boundary conditions are prescribed:

= ✞ ✟ 1 ( ✆ ) at ☎ = 0, = ✟ 2 ( ☎ ) at ✆ = 0. Solution:

References : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), A. G. Butkovskiy (1979).

7.2.2-13. Domain: 0 ≤ ☎ ≤ ✝ ,0≤ ✆ ≤ ✺ . First boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

= ✞ ✟ 1 ( ✆ ) at ☎ = 0, = ✟ 2 ( ✆ ) at ☎ = ✝ ,

= ✞ ✟ 3 ( ☎ ) at ✆ = 0, = ✟ 4 ( ☎ ) at ✆ = ✺ . Solution:

Two forms of representation of the Green’s function: ✢ ✢ ✢

2 ✢ ✢ sin(

2 ✯ ✽ ✆ ) sin( ✯ ✼ ✍ sin( ) sin( ✽ ✓

sinh( ✼ ✓ ) sinh[ ✼ ( ✺ − ✼ ✆ )] for ✺ ≥ ✆ > ✓ ≥ 0, = ✝ ,

sinh( ✼ ✡ ✆ ✿ ) sinh[ ✼ ( ✺ − ✓ )] for ✺ ≥ ✓ > ✆ ≥ 0,

) sinh[ ( ✝ − ☎ )] for ✝ ≥ ☎ > ✍ ≥ 0, = ✺ ,

) sinh[ ✯ ✽ ( ✝ − ✍ )] for ✝ ≥ ✍ > ☎ ≥ 0.

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

4 ✿ sin(

) sin( ✢ ✆ ) sin( ✼ ✍

Reference : A. G. Butkovskiy (1979).

7.2.2-14. Domain: 0 ≤ ☎ ≤ ✝ ,0≤ ✆ ≤ ✺ . Third boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

) ✞ at ☎ = 0, + ✛ ✞ 2 = ✟ 2 ( ✆ ) at ☎ = ✝ ,

− ✞ 3 = 3 ( ) at = 0, + ✛ 4 = ✟ 4 ( ☎ ) at ✆ = ✺ . Solution:

Here, ✢

( 1 ) = cos( )+

sin(

( ✆ ) = cos( ✽

sin(

tan( ✦ ✝ )

1 + ✛ 2 tan( ❁ ✺ )

7.2.2-15. Domain: 0 ≤ ☎ ≤ ✝ ,0≤ ✆ ≤ ✺ . Mixed boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

) ✞ at ✆ = 0, = ✟ 4 ( ☎ ) at ✆ = ✺ . Solution:

Two forms of representation of the Green’s function: ✢ ✢

2 ✯ sin( ) sin( ) ✽ ✓

2 sin( ) sin( ) ✽ ☎ ✍

+ 1) sinh( ✼ ✓ ) cosh[ ✼ ( ✺ − ✆ )] for ✺ ≥ ✆ > ✼ ✓ ✆ ✓ ✵ ≥ 0, =

sinh( ✼ ✆ ) cosh[ ✼ ( ✺ − ✓ )] for ✺ ≥ ✓ > ✡ ✆ ✿ ≥ 0,

)] for = ✝ − ≥ ☎ > ✍ ≥ 0,

sinh( ✍ ) cosh[ ( ✝

) cosh[ ✯ ✽ ( ✝ − ✍ )] for ✝ ≥ ✍ > ☎ ≥ 0. The Green’s function can be written in form of a double series: ✢ ✢

) sin( ✢ ✽ ✆ ) sin( ✼ ✍ ) sin( ✯ ✽ ✓

7.2.3. Problems in Polar Coordinate System

The two-dimensional Poisson equation in the polar coordinate system is written as

7.2.3-1. Domain: 0 ≤ ✡ ≤

,0≤ ≤2 . First boundary value problem.

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

= ✟ ( ) at

The magnitude of a vector difference is calculated as | ✝ r− ✺ r 0 | = ✝ 2 2 ✝ ✺ ✍

( ✝ and ✺ are any scalars). Thus, we obtain ❂

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: ✤

Here, the ( ✍ ) are the Bessel functions and the ❇ ❈ ✢ ❋ ❇ ✦ ✽ ❈ ✢ are positive roots of the transcendental equation

7.2.3-3. Domain: ✡

≤ < ✄ ,0≤ ≤2 . First boundary value problem.

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

where the Green’s function ✤

( , , ✍ , ✓ ) is defined by the formula presented in Paragraph 7.2.3-1. ✤ ✔✖✕ ❅

Reference : A. G. Butkovskiy (1979). ✤

1 ≤ ≤ ❃ 2 ,0≤ ≤2 . First boundary value problem. An annular domain is considered. Boundary conditions are prescribed: ✤ ❂ ❂

7.2.3-4. Domain: ✡

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: ❂ ✤ ❂

3 ( ) at = . Solution:

−2 2 cos( − )+ 1 −2 ❂ ✍ cos( + ✓ )+ ❃ 4

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 ≤ ✡ ≤

2. First boundary value problem.

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

= ✟ 3 ( ) at = ✏ 2. Solution: ✤

1 ( ) at

= ✟ 2 ( ) at

See also Example 3 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).

7.2.3-7. Domain: 0 ≤ ≤ ❃ ,0≤ ≤ ▲ . First boundary value problem.

A circular sector is considered. Boundary conditions are prescribed: ❂ ✤ ❂

= ✞ ✟ 1 ( ) at = ❃ , = ✟ 2 ( ) at = 0, = ✟ 3 ( ) at = ▲ . Solution:

. For ▲ = ✏ ✭ , where ✭ is a positive integer, the Green’s function is expressed as ▼

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

. For arbitrary ▲ , the Green’s function is given by

where ◆ = , = ✍ ,¯ = ✍ , and ❱ = −1. ▼ ▼ ▼ ▼

7.2.3-8. Domain: 0 ≤ < ✄ ,0≤ ≤ ▲ . First boundary value problem.

A wedge domain is considered. Boundary conditions are prescribed: ❂ ✤ ❂

Alternatively, the Green’s function can be represented in the complex form ✤ ▼ ▼

( 2 , , , )= ln , = , = , = , = −1.

2 ◆P❖ ❄ ❑ ▼ − ❄ ❑ ▼ ◆

7.2.4. Arbitrary Shape Domain. Conformal Mappings Method

7.2.4-1. Description of the method. Tables of conformal mappings. Any simply connected domain ❬ in the ❭ ❪ -plane with a piecewise smooth boundary can be mapped

in a mutually unique way, with an appropriate conformal mapping, onto the upper half-plane or into a unit circle in a ❫ ❴ -plane. Under a conformal mapping, a Poisson equation in the ✒ ❭ ❪ -plane transforms into a Poisson equation in the ❫ ❴ -plane; what is changed is the function , as well as the function ❵ in the boundary condition. Consequently, a first and a second boundary value problem for the plane domain ❬ can be reduced, respectively, to a first and a second boundary value problem for the upper half-plane or a unit circle. The latter problems are considered above (see Subsections

7.2.2 and 7.2.3).

A large number of conformal mappings (mappings defined by analytic functions) of various domains onto the upper half-plane or a unit circle can be found, for example, in Lavrik and Savenkov (1970), Lavrent’ev and Shabat (1973), and Ivanov and Trubetskov (1994).

Table 22 presents conformal mappings of some domains ❬ in the complex plane onto the upper half-plane Im ✰ ◗ ◗ ❛ ≥ 0 in the complex plane ❛ . In the relations involving square roots, it is assumed

that ❖

= ❜ | | cos ✧ 2 ★ + ❱ sin ✧ 2 ★✲✱ , where = arg (i.e., the first branch of ❜ is taken). Table 23 presents conformal mappings of some domains ❬ ✤ ✤ ✤ in the complex plane onto the unit circle | ❛ | ≤ 1 in the complex plane ❛ .

7.2.4-2. General formula for the Green’s function. Example boundary value problems. Let a function ❛ = ❛ ( ) define a conformal mapping of a domain ❬ in the complex plane onto

the upper half-plane in the complex plane ❖ ❛ . Then the Green’s function of the first boundary value ❖

problem in ❬ for the Poisson (Laplace) equation is expressed as ◗

( )= ❫ ( ❭ , ❪ )+ ❱❞❴ ( ❭ , ❪ ) and ¯ ❛ ( ◆ )= ❖ ❫ ( ❭ , ❪ )− ◆ ❱❞❴ ( ❭ , ❪ ). The solution of the first boundary value problem for the Poisson equation is determined by the ❖ ❖

above Green’s function in accordance with formula (2) specified in Paragraph 7.2.1-1.

Example 1. Consider the first boundary value problem for the Poisson equation in the strip − ❡ < ❢ < ❡ ,0≤ ❣ ≤ ❤ . The function that maps this strip onto the upper half-plane has the form ✐ ( ❥ ) = exp( ❦ ❥❊❧♠ ❤ ) (see the second row of Table 22 ). Substituting this expression into relation (1) and performing elementary transformations, we obtain the Green’s function

, ❣ , ♦ , ♣ )= 1 ln cosh[ ( − ) ] − cos[ ( + ) ]

cosh[ ❦ ( ❢ − ♦ ) ❧♠❤ ] − cos[ ❦ ( ❣ − ♣ ) ❧♠❤

Example 2. Consider the first boundary value problem for the Poisson equation in a semicircle of radius q ❤ such that ={ ❢ 2 + ❣ 2 ≤ ❤ 2 , ❣ ≥ 0} . The domain

is conformally mapped onto the upper half-plane by the function ✐ ( ❥ ) = −( ❥❊❧♠❤ + ❤ ❧♠❥ ) (see the sixth row in Table 22 ). Substituting this expression into (1), we arrive at the Green’s function s s

Example 3. Consider the first boundary value problem for the Poisson equation in a quadrant of a circle of radius ❤

, so + ❣ 2 ≤ ❤ 2 , ❢ ≥0 , ❣ ≥ 0} . The conformal mapping of the domain

that ={ ❢ 2

onto the upper half-plane is performed with the function ✐ ( ❥ ) = −( ❥❊❧♠ ❤ ) 2 − ( ❤ ❧♠ ❥ ) 2 (see the seventh row of Table 22 ). Substituting this expression into (1) yields

. Let a function ❛ = ❛ ( ) define a conformal mapping of a domain ❬ in the complex plane onto the unit circle ✇①❛ ✇ ≤ 1 in the complex plane ❖ ❛ . Then the Green’s function of the first boundary value ❖

rtr

problem in ❬ for the Laplace equation is given by ◗

References for Subsection ❖ 7.2.4: N. N. Lebedev, I. P. Skal’skaya, and Ya. S. Uflyand (1955), A. G. Sveshnikov and

A. N. Tikhonov (1974). ◆

TABLE 22 Conformal mapping of some domains ❬ in the -plane onto the upper half-plane Im ❛ ≥ 0 in the ❛ -plane. Notation: = ❖ ❭ + ❱❞❪ and ❛ = ❫ + ❱❞❴

No ❖ Domain

in the -plane

Transformation

First quadrant: ❖

, ✺ are real numbers ❖

Infinite strip of width ⑧ :

= exp( ❝ − ⑧

Semiinfinite strip of width ⑨

= cosh( ❝ 0≤ ⑧

Plane with the cut ⑨

in the real axis

Interior of an infinite sector with angle ❖

= ❄ 0 ≤ arg ❑ ≤ ▲ ,0≤| |< ⑦ (0 < ▲ ≤2 ❝ )

Upper half of a circle of radius ▼

2 Quadrant of a circle of radius ❖ ⑧ : ⑧ 2

Sector of a circle of radius ⑧ with angle ▲ :

2 ❪ 2 ⑧ 2 + ⑩ , 0 ≤ arg ▲ ❛ =− ⑩ ≤ ≤

Upper half-plane with a circular domain ❖

2 ❪ 2 ⑧ 2 removed: = ≥ 0, + ≥ ❖ ⑧ or radius +

Exterior of a parabola: ❖ 1 1

Interior of a parabola: ❖

7.3. Helmholtz Equation ❽ 2 ❾ + ❿ ❾ =– ➀ (x)

Many problems related to steady-state oscillations (mechanical, acoustical, thermal, electromagnetic, etc.) lead to the two-dimensional Helmholtz equation. For ➁ < 0, this equation describes mass transfer processes with volume chemical reactions of the first order. Moreover, any elliptic equation with constant coefficients can be reduced to the Helmholtz equation.

7.3.1. General Remarks, Results, and Formulas

7.3.1-1. Some definitions. The Helmholtz equation is called homogeneous if ➂ = 0 and nonhomogeneous if ➂ ≠ 0. A

homogeneous boundary value problem is a boundary value problem for the homogeneous Helmholtz equation with homogeneous boundary conditions; a particular solution of a homogeneous boundary value problem is ➃ = 0.

The values ➁ ➄ of the parameter ➁ for which there are nontrivial solutions (solutions other

TABLE 23

Conformal mapping of some domains ➉

in the -plane onto the unit circle

| ≤ 1. Notation: = ❻ + , ❼ = + , 0 = ❖ ❻ 0 + 0 , and ¯ 0 = ❻ 0 − 0

No ❖ Domain

in -plane

Transformation

Upper half-plane:

is a real number

A circle of unit radius: 0

is a real number ❖ ❖

Exterior of a circle of radius ➎ :

Infinite strip of width ❖

exp( ❝

) − exp( ❝ 0 ➎ )

− exp( ❝ ❖ ➏ ➎ ) − exp( ❝ ❖ ¯ 0 ➏ ➎ )

Semicircle of radius ➏

2 Sector of a unit circle with angle ❖ :

Exterior of an ellipse with semiaxes ➒t➓

than identical zero) of the homogeneous boundary value problem are called eigenvalues and the corresponding solutions, ➃ = ➃ ➄ , are called eigenfunctions of the boundary value problem.

In what follows, the first, second, and third boundary value problems for the two-dimensional Helmholtz equation in a finite two-dimensional domain ↔ with boundary ↕ are considered. For the third boundary value problem with the boundary condition

+ ➜ ➃ =0 for r ➝ ↕ ,

it is assumed that ➜ > 0. Here, ➞ ➟ is the derivative along the outward normal to the contour ↕ ❺ ➞ ➠ , and r={ ❻ , }.

7.3.1-2. Properties of eigenvalues and eigenfunctions.

1 ➡ . There are infinitely many eigenvalues { ➁ ➄ }; the set of eigenvalues forms a discrete spectrum for the given boundary value problem.

2 ➡ . All eigenvalues are positive, except for the eigenvalue ➁ 0 = 0 existing in the second boundary value problem (the corresponding eigenfunction is ➃ 0 = const). We number the eigenvalues in order

of increasing magnitudes, ➁ 1 < ➁ 2 < ➁ 3 < ➢❊➢❊➢ .

3 ➡ . The eigenvalues tend to infinity as the number ➤ increases. The following asymptotic estimate holds:

lim ➥ ➦ ➁ ➄ =

where ↔ 2 is the area of the two-dimensional domain under study.

, ) are defined up to a constant multiplier. Any two eigenfunctions corresponding to different eigenvalues, ➁ ➄ ≠ ➁ ➧ , are orthogonal:

4 ❺ ➡ . The eigenfunctions ➃ ➄ = ➃ ➄ (

5 ➡ . Any twice continuously differentiable function ➭ = ➭ (r) that satisfies the boundary conditions of

a boundary value problem can be expanded into a uniformly convergent series in the eigenfunctions of the boundary value problem:

where ➭ ➄ = ➲

If ➭ is square summable, then the series converges in mean.

6 ➡ . The eigenvalues of the first boundary value problem do not increase if the domain is extended. ➳ ➵♠➸ ➺ ➻➽➼ ➾ ➚

In a two-dimensional problem, generally correspond to each eigenvalue ➁ ➄ finitely many linearly independent eigenfunctions ➶ ➃ ➄ (1) , ➃ (2) ➄ , ➪❊➪❊➪ , ➃ ( ➄ ) . These functions can always be

replaced by their linear combinations

= 1, 2, ➪❊➪❊➪ , ➷ , so that the new eigenfunctions ¯ ➃ (1) ➄ ,¯ ➃ (2) ➄ , ➪❊➪❊➪ ,¯ ➃ ( ➄ ➶ ) now are pairwise orthogonal. Therefore, without

loss of generality, we assume that all the eigenfunctions are orthogonal. ➬✖➮

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

7.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions. Three cases are possible.

1 ➡ . If the equation parameter ➁ is not equal to any one of the eigenvalues, then there exists the series solution

, where

. If is equal to some eigenvalue, ✃ = ➧ , then the solution of the nonhomogeneous problem ✃

exists only if the function is orthogonal to ❒ ➧ , i.e.,

In this case the system is expressed as

, and ❰ is an arbitrary constant.

≠ 0, then the boundary value problem for the nonhomogeneous equation does not have solutions. ➳ ➵♠➸ ➺ ➻➽➼ Ï ➚

. If = ➧ and

If ➷ mutually orthogonal eigenfunctions ( ➹ ❐ ) ❐ ❐ ❒ ( ➴ = 1, 2, ➪❊➪❊➪ , ➷ ) correspond to each eigenvalue , then, for ✃ ≠

, the solution is written as

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

7.3.1-4. Solution of nonhomogeneous boundary value problem of general form.

1 Ø . The solution of the first boundary value problem for the Helmholtz equation with the boundary condition

= ➭ (r) for r Ù Ú

can be represented in the form ❮

Here, r = { ã

, å } and Ý ={ æ , ç } (r Ù Ñ , Ý Ù Ñ ); è è é ê denotes the derivative along the outward normal to the contour Ú with respect to the variables æ and ç . The Green’s function is given by the series

where the ✃

and are the eigenfunctions and eigenvalues of the homogeneous first boundary ✃ ✃ ✃

value problem. ✃

. The solution of the second boundary value problem with the boundary condition

= á (r) for r Ù Ú

can be written as ã

( Ý ) (r, Ý ) Þ Ñ ß + Û à á ( Ý ) (r, Ý ) Þ Ú ß . (3) Here, the Green’s function is given by the series

2 is the area of the two-dimensional domain under consideration, and the ✃ ✃ ✃ and ❒ are the positive eigenvalues and the corresponding eigenfunctions of the homogeneous second boundary ❐ ✃ ✃

value problem. For clarity, the term corresponding to the zero eigenvalue 0 =0( ❒ 0 = const) is singled out in (4).

3 Ø . The solution of the third boundary value problem for the Helmholtz equation with the boundary condition

+ ì ❒ = á (r) for r Ù Ú

is given by formula (3), where the Green’s function is defined by series (2), which involves the ã eigenfunctions ❐

and eigenvalues of the homogeneous third boundary value problem.

7.3.1-5. Boundary conditions at infinity in the case of an infinite domain. ✃

In what follows, the function is assumed to be finite or sufficiently rapidly decaying as í î ï .

. For < 0, in the case of an infinite domain, the vanishing condition of the solution at infinity is set,

0 as í î ï .

. For > 0, if the domain is unbounded, the radiation conditions (Sommerfeld conditions) at infinity are used. In two-dimensional problems, these conditions are written as

= −1. ò To identify a single solution, the principle of limit absorption and the principle of limit amplitude

are also used. ö✖÷

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

7.3.2. Problems in Cartesian Coordinate System

A two-dimensional nonhomogeneous Helmholtz equation in the rectangular Cartesian system of coordinates has the form

7.3.2-1. Particular solutions and some relations.

1 Ø . Particular solutions of the homogeneous equation ( ≡ 0):

=( ä + ø )( ❰ cos ù å + ú sin ù å

=( ➱ ý ä + ø )( ❰ cosh ù å + ú sinh ù å ), û =− ù 2 ,

=( ý cos ù ä + ø sin ù ä

=( 2 ý cosh ù ä + ø sinh ù ä )( ❰ å + ú ), û =− ù ,

2 =( 2 cos + sin )( cos + sin ), =

=( ý cos ù 1 ä + ø sin ù 1 ä )( ❰ cosh ù 2 å + ú sinh ù 2 å ),

2 =( 2 cosh + sinh )( cos + sin ), =− ù

1 − ù 2 2 , where ý , ø , ❰ , and ú are arbitrary constants.

=( ý cosh ù 1 ä

+ 2 ø sinh ù 1 ä )( ❰ cosh ù 2 å + ú sinh ù 2 å ), û =− ù

2 Ø . Fundamental solutions: þ þ

0 ( ☎ ) and 0 ( ☎ ) 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,

= ä 2 + å 2 , 0 ( ☎ ) is the modified Bessel function of the second kind, (1)

1 ð is given by 1

ln .

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

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

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

7.3.2-2. Domain: − ï < ä < ï ,− ï < å < ï .

1 ✠ . Solution for û =− ✁ 2 < 0:

2 ✠ . Solution for û = ì 2 > 0: ☞ ✌ ë ☞ ✌ ë

The radiation conditions (Sommerfeld conditions) at infinity were used to obtain this solution (see Paragraph 7.3.1-5, Item 2 ö✖÷ ✠ ).

References : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), A. N. Tikhonov and A. A. Samarskii (1990).

7.3.2-3. Domain: − ï < ✒ < ï ,0≤ ✓ < ï . First boundary value problem.

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

1 ✠ . The Green’s function for û =− ✁ 2 < 0:

2 ✠ . The Green’s function for û = ✏ 2 > 0:

The radiation conditions at infinity were used to obtain this relation (see Paragraph 7.3.1-5, Item 2 ✠ ).

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

7.3.2-4. Domain: − ï < ✒ < ï ,0≤ ✓ < ï . Second boundary value problem.

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

1 ✠ . The Green’s function for û =− ✁ 2 < 0:

2 ✠ . The Green’s function for û = ✏ 2 > 0: ( ✂

The radiation conditions at infinity were used to obtain this relation (see Paragraph 7.3.1-5, Item 2 ✠ ).

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

7.3.2-5. Domain: 0 ≤ ✒ < ï ,0≤ ✓ < ï . First boundary value problem.

A quadrant of the plane is considered. Boundary conditions are prescribed:

= ü ✖ 1 ( ✓ ) at ✒ = 0, = ✖ 2 ( ✒ ) at ✓ = 0. Solution:

1 ✠ . The Green’s function for û

2 ✠ . The Green’s function for û = ✏ 2 > 0:

7.3.2-6. Domain: 0 ≤ ✒ < ï ,0≤ ✓ < ï . Second boundary value problem.

A quadrant of the plane is considered. Boundary conditions are prescribed:

( ✒ ) at ✓ = 0. Solution:

1 ✠ . The Green’s function for û =− ✁ 2 < 0:

2 ✠ . The Green’s function for û = ✏ 2 > 0:

7.3.2-7. Domain: − ï < ✒ < ï ,0≤ ✓ ≤ ✩ . First boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

= ü ✖ 1 ( ✒ ) at ✓ = 0, = ✖ 2 ( ✒ ) at ✓ = ✩ . Solution:

Green’s function:

exp ✯ − | ✒ − ✍ | ✰ sin( ✱ ✓ ) sin( ✱ ✎ ),

Alternatively, the Green’s function for ✵ =− ✶ 2 < 0 can be represented as

7.3.2-8. Domain: − ï < ✒ < ï ,0≤ ✓ ≤ ✩ . Second boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

= ✖ 2 ( ✒ ) at ✓ = ✩ . Solution:

Green’s function:

exp ✯ − | ✒ − ✍ | ✰ cos( ✱ ✓

= ✽ 1 for ✳ = 0,

≠ 0. Alternatively, the Green’s function for ✵ =− ✶ 2 < 0 can be represented as

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

7.3.2-9. Domain: − ❀ < ❁ < ❀ ,0≤ ❂ ≤ ✩ . Third boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

+ ❃ 2 ✺ = ❄ 2 ( ❁ ) at ❂ = ✩ . The solution ✺ ( ❁ , ❂ ) is determined by the formula in Paragraph 7.3.2-8 where ✬ ✬

Here, the ù are positive roots of the transcendental equation tan( ù ❍ )= ù 2 . − ❃ 1 ❃ 2

7.3.2-10. Domain: − ❀ < ❁ < ❀ ,0≤ ❂ ≤ ❍ . Mixed boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

= ❄ 2 ( ❁ ) at ❂ = ❍ . Solution:

exp − | − | sin(

7.3.2-11. Domain: 0 ≤ ❁ < ❀ ,0≤ ❂ ≤ ❍ . First boundary value problem.

A semiinfinite strip is considered. Boundary conditions are prescribed:

= 3 ( ) at = . Solution:

exp − | − | − exp − | + | sin( ) sin( ),

7.3.2-12. Domain: 0 ≤ ❁ < ❀ ,0≤ ❂ ≤ ❍ . Second boundary value problem.

A semiinfinite strip is considered. Boundary conditions are prescribed:

= ❄ 3 ( ❁ ) at ❂ = ❍ . Solution:

exp − | − | ❪ + exp ❭ − | ❁ + ❇ | ❪❙❘ cos( ❫ ❂ ) cos( ❫ ❈ ),

7.3.2-13. Domain: 0 ≤ ❁ < ❀ ,0≤ ❂ ≤ ❍ . Third boundary value problem.

A semiinfinite strip is considered. Boundary conditions are prescribed:

+ ❃ 3 = ❄ 3 ( ❁ ) at ✺ ❂ ✺ ✺ ✺ = ❍ . The solution ( ❁ , ❂ ) is determined by the formula in Paragraph 7.3.2-12 where ❩ ❩ ✺ ❩ ❩ ❩

)= ù cos( ù ❂ )+ ❃ 2 sin( ù ❂

Here, the ù

are positive roots of the transcendental equation tan( ù ❍

7.3.2-14. Domain: 0 ≤ < ❢ ,0≤ ❂ ≤ ❍ . Mixed boundary value problems.

1 ❣ . A semiinfinite strip is considered. Boundary conditions are prescribed: ❡ ❡ ❡

= ❤ ✐ 1 ( ❂ ) at = 0, ❥ ❑ = ✐ 2 ( ) at ❂ = 0, ❥ ❑ = ✐ 3 ( ) at ❂ = ❦ . Solution: ❡

exp ❭ − | − ♥ | ❪ − exp ❭ − | + ♥ | ❪❙④ cos( ❫ ❂ ) cos( ❫ ♦ ❧ ),

for ❵

2 for

. A semiinfinite strip is considered. Boundary conditions are prescribed:

= ❤ 1 ( ) at = 0, = 2 ( ) at = 0, = ✐ 3 ( ) at ❂ = ❦ . Solution: ❡

where r

exp ❭ − | − ♥ | ❪ + exp ❭ − | + ♥ | ❪❙④ sin( ❫ ❂ ) sin( ❫ ♦ ❧ ❦ ),

7.3.2-15. Domain: 0 ≤ ≤ ❦ ,0≤ ❂ ≤ ⑧ . First boundary value problem.

A rectangle is considered. Boundary conditions are prescribed: ❡

1 ❤ ( ❂ ❡ ) at = 0, = ✐ ❂ ❡ 2 ( ) at

= ❤ ✐ 3 ( ) at ❂ = 0, = ✐ 4 ( ) at ❂ = ⑧ .

1 ❣ . Eigenvalues of the one-dimensional problem (it is convenient to label them with a double subscript):

; ❶ = 1, 2, Eigenfunctions and the norm squared: ❡

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and ① ⑨

A. N. Tikhonov (1980).

2 ❣ . Solution for ❡ ❽ ≠ ❽

Two forms of representation of the Green’s function: ⑨

2 ① ✇✭① sin(

) sin( ➀ ♥ )

sin( ❫ ❾ ) sin( ❫ ❧ )

) sinh[ ( ⑧ − ❾ )] for ⑧ ≥ ❾ > ➇ ≥ 0, =

sinh(

sinh( ❾ ⑨ ) sinh[ ⑨ ( ⑧ − ➇ )] for ⑧ ≥ ➇ > ❾ ≥ 0,

Alternatively, the Green’s function can be written as the double series ⑨

) sin( ❫ ❾ ❾ ) sin( ➀

sin( ➀ ➄

) sin(

7.3.2-16. Domain: 0 ≤ ☎ ≤ ✆ ,0≤ ✝ ≤ ✞ . Second boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

1 ✌ . Eigenvalues of the homogeneous problem:

Eigenfunctions and the norm squared:

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

. Solution for ≠

Two forms of representation of the Green’s function:

cos( ✮ ☎ ) cos( ✮ ✧ ) ✎

cos( ✱ ✝ ) cos( ✱ ★ ) ✏

cosh( ✯ ✎ ) cosh[ ✯ ✎ ( − ✝ )]

cosh( ✝ ) cosh[ ( ✞ − ★ )]

for ★ > ✝ ,

cosh(

) cosh[ ( − )] for > ,

cosh( ✏

) cosh[ ( ✆ − ✧ )] for ✧ > ☎ ,

= ✚ 1 for ✓ = 0,

2 for ✓ ≠ 0. The Green’s function can also be written as the double series ✭

In Paragraphs 7.3.2-17 through 7.3.2-20, only the eigenvalues and eigenfunctions of homoge- neous boundary value problems for the homogeneous Helmholtz equation (with ≡ 0) are given. The solutions of the corresponding nonhomogeneous problems can be constructed using formulas ✦

presented in Paragraphs 7.3.1-3 and 7.3.1-4.

7.3.2-17. Domain: 0 ≤ ☎ ≤ ✆ ,0≤ ✝ ≤ ✞ . Third boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

1 =0 at ☎ = 0,

=0 at ☎ = ✆ ,

+ ✷ 4 =0 at ✝ = ✞ . Eigenvalues:

− ✷ 3 =0 at ✝ = 0,

where the ✏ and

are positive roots of the transcendental equations

cos ✸ ✝ + ✷ 3 sin ✸ ✝ ). The square of the norm of an eigenfunction:

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

7.3.2-18. Domain: 0 ≤ ☎ ≤ ✆ ,0≤ ✝ ≤ ✞ . Mixed boundary value problems.

1 ✌ . A rectangle is considered. Boundary conditions are prescribed:

Eigenfunctions and the norm squared:

2 ✌ . A rectangle is considered. Boundary conditions are prescribed:

= 0, 1, 2, Eigenfunctions and the norm squared:

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

7.3.2-19. First boundary value problem for a triangular domain. The sides of the triangle are defined by the equations

The unknown quantity is zero for these sides. Eigenvalues:

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

7.3.2-20. Second boundary value problem for a triangular domain. The sides of the triangle are defined by the equations

The normal derivative of the unknown quantity for these sides is zero. Eigenvalues:

7.3.3. Problems in Polar Coordinate System

A two-dimensional nonhomogeneous Helmholtz equation in the polar coordinate system is written as

7.3.3-1. Particular solutions of the homogeneous equation ( ✾ ✾ ≡ 0):

, ✲ are arbitrary constants; the ❃ ( ) and ❅ ( ) are the Bessel functions; and the ❈ ( ) and ❉

( ) are the modified Bessel functions.

In Paragraphs 7.3.3-2 through 7.3.3-11, only the eigenvalues and eigenfunctions of homoge- neous boundary value problems for the homogeneous Helmholtz equation (with ≡ 0) are given. The solutions of the corresponding nonhomogeneous problems can be constructed using formulas ✦

presented in Paragraphs 7.3.1-3 and 7.3.1-4.

7.3.3-2. Domain: 0 ≤ ≤ ❊ . First boundary value problem.

A circle is considered. A boundary condition is prescribed:

Here, the ✲ are positive zeros of the Bessel functions, ❃ ( ) = 0. Eigenfunctions:

. The square of the norm of an eigenfunction is given by

Eigenfunctions possessing the axial symmetry property: ● 0 =

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

7.3.3-3. Domain: 0 ≤ ≤ ❊ . Second boundary value problem.

A circle is considered. A boundary condition is prescribed:

=0 at

Eigenvalues:

where the ✲ ◆

are roots of the transcendental equation ❃ ◆ P ( ) = 0. Eigenfunctions:

) sin ❨ . Here, ❨ ✲ = 0, 1, 2, ❩✗❩✗❩ ; for ❨ ≠ 0, the parameter ❬ assumes the values ❬ = 1, 2, 3, ❩✗❩✗❩ ; for ❨ = 0,

) cos ❨ , ❑ (2) ◆ ❏ = ❃ ◆ ( ❁

a root 00 = 0 (the corresponding eigenfunction is ❑ 00 = 1).

= ❃ 0 0 ❏ . The square of the norm of an eigenfunction is given by

Eigenfunctions possessing the axial symmetry property: ●

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

7.3.3-4. Domain: 0 ≤ ≤ ❊ . Third boundary value problem.

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

Here, the ✲ ◆

th root of the transcendental equation ✲ ❃ ◆ P ( )+ ✷ ❊ ❃ ◆ ✲ ( ) = 0. Eigenfunctions:

is the ❬

sin ❨ . The square of the norm of an eigenfunction is given by

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

7.3.3-5. Domain: ❊ 1 ≤ ≤ ❊ 2 . First boundary value problem.

An annular domain is considered. Boundary conditions are prescribed:

Here, the ✲ ◆

are positive roots of the transcendental equation

) ❅ ◆ ( ◆ ❏ ❊ 1 )− ❃ ◆ ( ◆ ❏ ❊ 1 ) ❅ ◆ ( ◆ ❏ )] sin ❨ . The square of the norm of an eigenfunction is given by

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

7.3.3-6. Domain: ❊ 1 ≤ ≤ ❊ 2 . Second boundary value problem.

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

= ❊ 2 . Eigenvalues:

=0 at

=0 at

Here, the ✲ ◆

are roots of the transcendental equation

1 ✲ ) ❅ ◆ P ( ❊ 2 )− ❃ ◆ P ✲ ( ❊ 2 ) ❅ ◆ P ✲ ( ❊ 1 ) = 0. If ❨

00 = 1. Eigenfunctions:

= 0, there is a root ✲ 00 = 0 and the corresponding eigenfunction is ❑ (1)

=[ ❃ ( ❏ ✾ ) ❅ ◆ P ( ❏ ❊ 1 )− ❃ ◆ P ( ❏ ❊ 1 ) ❅ ( ❏ ✾ )] cos ❨ ,

=[ ❃ ◆ ( ◆ ❏ ) ❅ ◆ P ( ◆ ❏ ❊ 1 )− ❃ ◆ P ✲ ( ◆ ❏ ❊ 1 ) ❅ ◆ ✲ ( ◆ ❏ )] sin ❨ . The square of the norm of an eigenfunction is given by

(1) ❧ 2 ♠ t 2 2 ♥♣♦❚q 1 for r s = ,

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

7.3.3-7. Domain: ✐ 1 ≤ ✇ ≤ ✐ 2 . Third boundary value problem.

An annular domain is considered. Boundary conditions are prescribed:

+ ③ ❑ =0 at ✇ = ✐ 2 . Eigenvalues:

− ③ ❑ =0 at ✇ = ✐ 1 ,

= 0, 1, 2, ⑦✗⑦✗⑦ ; ⑧ = 1, 2, 3, ⑦✗⑦✗⑦ ; where the ❡ ❏ are positive roots of the transcendental equation

1 ) ⑩ 2 ( ❡ ✐ 2 )− 2 ( ❡ ✐ 2 ) ⑩ 1 ( ❡ ✐ 1 ) = 0. Here, we use the notation

=[ ⑩ 1 ( ❸ ❏ ✐ 1 ) ❤ ( ❸ ❏ ✇ )− 1 ( ❸ ❏ ✐ 1 ) ❷ ( ❸ ❏ ✇ )] sin ⑥ . The square of the norm of an eigenfunction is given by ( ❤ ❹ = 1, 2)

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

7.3.3-8. Domain: 0 ≤ ✇ ≤ ✐ ,0≤ ➄ ≤ ➅ . First boundary value problem.

A circular sector is considered. Boundary conditions are prescribed:

=0 at ➄ = ➅ . Eigenvalues:

Here, the ✐

are positive zeros of the Bessel functions, ➇ ( ❸ ) = 0. Eigenfunctions:

sin ❿

The square of the norm of an eigenfunction is given by ❢

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and ❤ ❶

A. N. Tikhonov (1980).

7.3.3-9. Domain: 0 ≤ ✇ ≤ ✐ ,0≤ ➄ ≤ ➅ . Second boundary value problem.

A circular sector is considered. Boundary conditions are prescribed: ① ② ① ➊ ① ➊

=0 at ➄ = ➅ . Eigenvalues:

Here, the ✐

are roots of the transcendental equation

The square of the norm of an eigenfunction is given by ❤ ➆

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

7.3.3-10. Domain: 0 ≤ ✇ ≤ ✐ ,0≤ ➄ ≤ ➅ . Third boundary value problem.

A circular sector is considered. Boundary conditions are prescribed: ① ➊ ① ➊

+ ③ 3 ❑ =0 at ➄ = ➅ . Eigenvalues:

Here, the ⑤

are positive roots of the transcendental equation ❸ ➍➏➎ ( ❸ )+ ③ 1 ✐ ➍➏➎ ( ❸ ) = 0; the ➐

are positive roots of the transcendental equation tan(

Eigenfunctions:

)+ 2 sin( ➐ ❑ ➄ cos( ③ )

The square of the norm of an eigenfunction is given by ⑤

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

7.3.3-11. Domain: ➓ 1 ≤ ✇ ≤ ➓ 2 ,0≤ ➄ ≤ ➅ . First boundary value problem. Boundary conditions are prescribed: ↕

where the ❸ are positive roots of the transcendental equation

sin( ). The square of the norm of an eigenfunction is given by

References ↔ : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

7.3.4. Other Orthogonal Coordinate Systems. Elliptic Domain

In Paragraphs 7.3.4-1 and 7.3.4-2, two other orthogonal systems of coordinates are described in which the homogeneous Helmholtz equation admits separation of variables.

7.3.4-1. Parabolic coordinate system. In the parabolic coordinates that are introduced by the relations

(0 ≤ ➛ < ➞ ,− ➞ < ➜ < ➞ ), the Helmholtz equation has the form ↕

Setting = ➟ ( ➛ ) ➠ ( ➜ ), we arrive at the following linear ordinary differential equations for ➟ = ➟ ( ➛ ) and ➠ = ➠ ( ➜ ):

where ③ is the separation constant. The general solutions of these equations are given by ❶➡❶ ❶➡❶

1 , 2 , and ⑩ 2 are arbitrary constants, and ➢ ➍ ( ➧ ) is the parabolic cylinder function,

, we have ➫

− ( ➥ )=2 2 exp ➨ − 1 ➧ 2 ➩✗➯

2 ➯ −1 2 ➧ , where ( ➧ ) = (−1) exp ➨ ➧ 2 ➩ ➲

exp ➨ − ➧ 2 ➩ .

References : M. Abramowitz and I. Stegun (1964), W. Miller, Jr. (1977).

7.3.4-2. Elliptic coordinate system. In the elliptic coordinates that are introduced by the relations

= ➳ cosh ➵ cos ➸ , ➝ = ➳ sinh ➵ sin ➸ (0 ≤ ➵ < ➞ ,0≤ ➸ <2 ➺ , ➳ > 0), the Helmholtz equation is expressed as ↕ ↕

(cosh ➵ − cos 2 ➸ ) = 0.

Setting = ❽ ( ➵ ) ➻ ( ➸ ), we arrive at the following linear ordinary differential equations for

= ❽ ( ➵ ) and ➻ = ➻ ( ➸ ):

− 2 cos 2 − ➻ = 0, where ➾ is the separation constant. The solutions of these equations periodic in ➸ are given by

4 , where Ce ➶ ( ➵ , ➹ ) and Se ➶ ( ➵ , ➹ ) are the modified Mathieu functions, and ce ➶ ( ➸ , ➹ ) and se ➶ ( ➸ , ➹ ) are

the Mathieu functions; to each value of ➹ there is a corresponding ➾ = ➾ ➶ ( ➹ ➘✢➴ ).

References : M. Abramowitz and I. Stegun (1964), W. Miller, Jr. (1977).

7.3.4-3. Domain: ( ➬ ➳ ) 2 +( ➝ ) 2 ≤ 1. First boundary value problem.

The unknown quantity is zero at the boundary of the elliptic domain: ↕

=0 ➷ if ( ➳ ) 2 +( ➝ ➷ ➬ ) 2 =1 ➬ ( ➳ ≥ ).

The first three eigenvalues and eigenfunctions are given by the approximate relations

2 = ➮ 11 1 3 ➱ (s) ❰ 2 + ➬ 2 ✃ , ❐ 2 ( ❒ , Ï )= ❮ 1 ( ❒ ➮ 11 ) sin Ï 4 , where 10 = 2.4048 and 11 = 3.8317 are the first roots of the Bessel functions ❮ 0 and ➮ ❮ ➮ 1 , i.e.,

0 ( 10 ) = 0 and ❮ 1 ( 11 ) = 0; ➮ ❒ ➮ = Ð ( Ñ ➷ ❰ ) 2 +( Ò ➷ ➬ ) 2 .

The above relations were obtained using the generalized (nonorthogonal) polar coordinates ❒ , Ï

defined by

(0 ≤ ❒ ≤ 1, 0 ≤ Ï ≤2 Ó ) and the variational method.

= ❰ ❒ cos Ï , Ò

= ➬ ❒ sin Ï

For (c) = 1−( ) ≤ 0.9, the above formulas provide an accuracy of 1% for

2 ➽ do not exceed 0.01%, and the maximum

2 ➽ . For Ô ≤ 0.5, the errors in calculating 1 and (c)

error in determining (s)

2 is 0.12%. In the limit case Ô = 0 that corresponds to a circular domain, the above formulas are exact. ➘✢➴

Reference : L. D. Akulenko and S. V. Nesterov (2000).

TABLE 24

Transformations reducing equation 7.4.1.3 to the Helmholtz equation ➬ Õ 2 + Õ 2

No Exponent ➾

Transformation ➬ Factor

= 1 2 ln( Ñ 2 + Ò 2 ), Ú = arctan