Poisson Equation ü 3 ý + þ (x) = 0
8.2. Poisson Equation ü 3 ý + þ (x) = 0
8.2.1. Preliminary Remarks. Solution Structure
Like the three-dimensional Laplace equation, the three-dimensional Poisson equation is often en- countered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. In particular, the Poisson equation describes stationary temperature distribution in the presence of thermal sources or sinks in the domain under consideration.
The Laplace equation is a special case of the Poisson equation with ÿ ≡ 0. Throughout this section, we consider a three-dimensional bounded domain with a sufficiently
smooth boundary ✁ . We assume that r ✂
and ✄ ✂
, where r = { ☎ , ✆ , ✝ } and ✄ ={ ✞ , ✟ , ✠ }.
TABLE 26 Orthogonal coordinates ¯ ☎ ,¯ ✆ ,¯ ✝ that allow separable solutions of the form
= õ (¯ ☎ ) ö (¯ ✆ ) ✡ (¯ ✝ ) for the three-dimensional Laplace equation ☛ 3 ☞ =0 Coordinates
Transformations Particular solutions (or equations for õ , ö , ✡ )
= cos( ✌ 1 ☎ + ✍ 1 ) cos( ✌ 2 ✆ + ✍ 2 ) cosh( ✌ 3 ✝ + ✍ 3 ), Cartesian
1 + ✌ ✆ 2 ✝ , where ✌ 2 2 = ✌ , 2 3 , ;
see also Paragraph 8.1.1-1 ✎ ✎
=[ ✑ ✒ Ô ( ✌ )+ ✓ ✔ Ô ( ✌ )] cos( ✕ ✏ + ✖ ) exp( ✗ ✌ ✝ ), Cylindrical ✎
( ✝ ) are the Bessel functions;
see also Paragraph 8.1.2-1 Parabolic
= ✘ ✙ −1 ✚ 2 ( ✗ ✛ ✞ ) ✘ − ✙ −1 ✚ 2 ( ✗ ✛ ✟ ) cos( ✌ ✝ + ✍ ), cylindrical 2
where ✛ = ó 2 ✌ , ✘ ✙ ( ✝ ) is the parabolic
cylinder function
Ce ( ,− ) ce ( ,− ) cos( + ), cosh cos ,
Elliptic ✍
) se ( ,− ) cos( + ), sinh sin ✢ ,
and Se Ô are the modified Mathieu functions,
ce and se are the Mathieu functions, ✤ = 1 4 ù 2 ✌ 2
(cos ✥ ) cos( ✌ ✎ ✏ + ✍ ),
( ✞ ) are the associated Legendre functions,
= ✜ cos ✥ see also Paragraph 8.1.3-1 Prolate ✜
= ✦ ✧ (cosh ) ✦ ✧ (cos ✢ ) cos( ✌ ✏ spheroidal ✍
= ù sinh ✜ sin ✢ cos ✏ ,
= ù sinh sin ✜ ✢ sin ✏ ,
( ) are the associated Legendre functions =
cosh cos ✢
= ù cosh ✜ sin ✢ cos ✏ , spheroidal ✜
Oblate ✜
= ✦ ✧ (− ★ sinh ) ✦ ✧ (cos ✢ ) cos( ✌ ✏ + ✆ ✍ ✜ ✏ Ô ), = ù cosh sin ✢ sin ,
( ) are the associated Legendre functions sinh cos
= ✩✫✪ ✧ ( ✬ Parabolic ✞ ) ✒ ✪ ✧ ( ✬ ✟ ) cos( ✌ ✏ + ✍ ),
= ù ✞ ✟ cos ✏ ,
sin ,
( ) are the Bessel functions,
( ✜ ✝ ) are the modified Bessel functions
2, − ✮ ), Paraboloidal ✜ ✆ =2 ✭
=2 ✭ cosh ✜ cos ✢ sinh ✏ ,
Ce Ô ( ✜ ,− ✮ ) ce Ô ( ✢ ,− ✮ ) Ce Ô ( ✏ + ★✰✯ ✱
sinh sin ✢ cosh ✜ ✏ , Se Ô ( ,− ✮ ) se Ô ( ✢ ,− ✮ ) Se Ô ( ✏ + ★✰✯ ✱
= 1 2 ✭ (cosh 2
= 1 2 ✭ ✬ ; ce and se
are the Mathieu functions,
Ce
+ cos 2 ✢ − cosh 2 ✏
and Se
are the modified Mathieu functions
+( ✬ 2 + ✬ 1 sn 2 ✞ ) = 0 (Lam´e equation), General
+( 2 + ✬ 1 sn ✟ ) = 0 (Lam´e equation), =
2 1 sn 2 ✠ ) = 0 (Lam´e equation),
+[ ✬ − ✕ ( ✕ + 1) ✌ 2 sn 2 ✞ ] = 0 (Lam´e equation), Conical
+[ − ✲ ✕ ( ✕ + 1) ✌ 2 sn 2 ✟ ] ✡ = 0 (Lam´e equation), , ,
= ✴ ❃ ✭ ✳ where = sn ( ✞ , ✌ ), ✳ = sn ( ✟ , ✌ ), ✌ = ❀ ✭ , and ✡ are expressed in terms of the Lam´e polynomials
TABLE 27
Coordinates ¯ ✄ ☎ ✆ ✄ ,¯ ,¯ that allow ✝ -separated solutions of the form
(¯ ✞ ) for the three-dimensional Laplace equation ☞ 3 =0 New coordinates,
of coordinates
(equations for ✠ , ✡ , ☛ )
= ✏✑✝ −1 sin ✌
cos ,
( ✌ )= ✓ 1 ✔ ✖ ✕ (cos ✌ )+ ✓ 2 ✗ ✕ ✖ (cos ✌
( )= 1 cosh ( ✚ + 1 2 ) ✍ ✛ + ✘ 2 sinh ✙ ( ✚ + 1 2 ) ✍ ✛ , coordinates , , ,
( )= 1 cos( )+ 2 sin( ),
= cosh ✍ −cos ✌
≤ , is any,
= ✏✑✝ −1 sinh ✌ cos ✎ ,
( ✌ )= ✓ 1 ✔ ✖ Toroidal ✕ −1 ✥ 2 (cosh ✌ )+ ✓ 2 ✗ ✕ ✖ (cosh
( ✍ )= ✘ 1 cos( ✚ ✍ )+ ✘ 2 sin( ✚ coordinates ✍ , , , ),
= ✏✑✝ −1 sinh ✌ sin ✎ ,
= ✏✑✝ ✝ −1 = cosh ✍ ✌
−cos ✎ ✍ sin ; ( )= 1 cos( )+ ✜ 2 sin( ✢ ✎ ),
( ✎ )= ✜ 1 cos( ✚ ✎ )+ ✜ 2 sin( ✚ ✎ ),
( ✎ )= ✜ 1 cos( ✚ ✎ )+ ✜ 2 sin( ✚ + ✎ ),
( ✎ )= ✜ 1 cos( ✚ ✎ )+ ✜ 2 sin( ✚ ✎ ),
, ✍ are real numbers
, ✍ are real numbers
8.2.1-1. First boundary value problem. The solution of the first boundary value problem for the Poisson equation
(1) in a domain ✼ with the nonhomogeneous boundary condition
3 ✞ + ✻ (r) = 0
= ✠ (r) for r ✽ ✾
TABLE 28 The volume elements and distances occurring in relations (2) and (5) in some coordinate systems. In all cases, ✿ ={ ❀ , ❁ , ❂ }
Coordinate Volume
Distance, system
Gradient, ∇ ✫✤❄
=| r− ✿ | Cartesian ✄ ☎ ✆
+ −2 cos( − )+( − ) Spherical
−2 cos , where r={ , , }
sin ❁ ❃ ❀ ❃ ❁ ❃ ❂ ❅ ❈ ❉
sin
cos ● = cos ❋ cos ❁ + sin ❋ sin ❁ cos( ✎ − ❂ )
can be represented in the form
(r, ✿ ) is the Green’s function of the first boundary value problem, ❈ ◆ is the derivative
of the Green’s function with respect to P
, ❁ , ❂ along the outward normal N the boundary ✾ of the domain ✼ . Integration is everywhere with respect to ❀ , ❁ , ❂ . The volume elements in solution (2) for basic coordinate systems are presented in Table 28 . In addition, the expressions of the gradients are given, which enable one to find the derivative along the normal in accordance with the formula ❈ ◆ = (N ⋅ ∇ ✫ ❏ ).
= ❏ (r, ✿ ) of the first boundary value problem is determined by the following conditions:
The Green’s function P
1 ✄ ◗ . The function ❏ ☎ ✆ satisfies the Laplace equation with respect to , , in the domain ✼ everywhere except for the point ( ❀ , ❁ , ❂ ), at which it can have a singularity of the form 1 4 1 ❘ |r− ❙ | .
2 ✆ . The function , with respect to , , , satisfies the homogeneous boundary condition of the first kind at the boundary, i.e., the condition ❑
The Green’s function can be represented as
where the auxiliary function ❄ = ❄ (r, ✿ ) is determined by solving the first boundary value problem ❑
for the Laplace equation ☞ 3 ❄ = 0 with the boundary condition ❄ ❚ =− 1 4 1 ❘ |r− ❙ | ; the vector quantity ✿
in this problem is treated as a three-dimensional free parameter. The Green’s function possesses the symmetry property with respect to their arguments: ❏ (r, ✿ )=
( ✿ , r). The construction of Green’s functions is discussed in Paragraphs 8.3.1-4 and 8.3.1-6 through 8.3.1-8 for ❯ ❱✑❲ ❳ ❨❬❩ ✱ = 0. ❭ ❪
For outer first boundary value problems for the Laplace equation, the following condition is usually set at infinity: | ❫ |< ✓ ✭ | r| (|r| ❴ ❵ , ❛ = const).
8.2.1-2. Second boundary value problem. The second boundary value problem for the Poisson equation (1) is characterized by the boundary
condition
= ❜ (r) for r ✽ ✾ .
Necessary condition solvability of the inner problem:
(r) ❃ r+ ❍ ❑ ❜ (r) ❃ ✾ = 0.
The solution of the second boundary value problem can be written as
(5) where ❢ is an arbitrary constant, provided that the solvability condition is met.
(r) = ❍ ■ ✻ ( ✿ ) ❏ (r, ✿ ) ❃ ✼ ✫ + ❍ ❑ ❜ ( ✿ ) ❏ (r, ✿ ) ❃ ✾ ✫ + ❢ ,
The Green’s function ❏ = ❏ (r, ✿ ) of the second boundary value problem is determined by the following conditions:
1 ◗ . The function ❏ satisfies the Laplace equation with respect to ❣ , ❤ , ✐ in the domain ✼ everywhere except for the point ( ❀ , ❁ , ❂ ) at which it has a singularity of the form 1 4 1 ❘ |r− ❙ | .
2 ◗ . The function ❏ , with respect to ❣ , ❤ , ✐ , satisfies the homogeneous condition of the second kind at the boundary, i.e., the condition
where ✾ 0 is the area of the surface ✾ . The Green’s function is unique up to an additive constant. ❯ ❱✑❲ ❳ ❨❬❩ ❦ ❪
The Green’s function cannot be identified with condition 1 ❥ ◗ and the homogeneous boundary condition ❈ ◆ ❥ ❑ ❈ ❖ = 0; this problem for ❏ has no solution, 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 (2) is not met. ❯ ❱✑❲ ❳ ❨❬❩ ❧ ❪
Condition (4) is not extended to the outer second boundary value problem (for infinite domain).
8.2.1-3. Third boundary value problem. The solution of the third boundary value problem for the Poisson equation (1) in a bounded domain ✼
with the nonhomogeneous boundary condition
+ ♠ ❫ = ❜ (r) for r ✽ ✾
is given by relation (5) with ▼
= 0, where ❏ = ❏ (r, ✿ ) is the Green’s function of the third boundary value problem; the Green’s function is determined by the following conditions:
1 ◗ . The function ❏ satisfies the Laplace equation with respect to ❣ , ❤ , ✐ in ✼ everywhere except for
the point ( ❀ , ❁ , ❂ ) at which it has a singularity of the form 1 4 1 ❘ |r− ❙ | .
2 ◗ . The function ❏ , with respect to ❣ , ❤ , ✐ , satisfies the homogeneous boundary condition of the third kind at the boundary, i.e., the condition ♥ ❈ ◆ ❈ ❖ + ♠ ❏ ♦ ❑ = 0.
The Green’s function can be represented in the form (3), where the auxiliary function ❄ is determined by solving the corresponding third boundary value problem for the Laplace equation ♣
= 0. The construction of Green’s functions is discussed in Paragraphs 8.3.1-4 and 8.3.1-6 through 8.3.1-8 for r s✉t = 0.
References for Subsection 8.2.1: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970).
8.2.2. Problems in Cartesian Coordinates
The three-dimensional Poisson equation in the rectangular Cartesian system of coordinates has the form
8.2.2-1. Domain: − ❵ < ❣ < ❵ ,− ❵ < ❤ < ❵ ,− ❵ < ✐ < ❵ . Solution:
Reference : R. Courant and D. Hilbert (1989). ② ②
8.2.2-2. Domain: − ❵ < ❣ < ❵ ,− ❵ < ❤ < ❵ ,0≤ ✐ < ❵ . First boundary value problem.
A half-space is considered. A boundary condition is prescribed:
References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.2.2-3. Domain: − ❻ < ❸ < ❻ ,− ❻ < ❹ < ❻ ,0≤ ❺ < ❻ . Third boundary value problem.
A half-space is considered. A boundary condition is prescribed:
− ❾ ⑧ = ❿ ( ❸ , ❹ ) at ❺ = 0.
Solution:
where
exp(− ❾ ➂ ) ⑥ ➂
8.2.2-4. Domain: − ❻ < ❸ < ❻ ,0≤ ❹ < ❻ ,0≤ ❺ < ❻ . First boundary value problem.
A dihedral angle is considered. Boundary conditions are prescribed:
= ⑧ ❿ 1 ( ❸ , ❺ ) at ❹ = 0, = ❿ 2 ( ❸ , ❹ ) at ❺ = 0. Solution:
References : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), A. G. Butkovskiy (1979).
8.2.2-5. Domain: 0 ≤ ❸ < ❻ ,0≤ ❹ < ❻ ,0≤ ❺ < ❻ . First boundary value problem. An octant is considered. Boundary conditions are prescribed:
= 0, ⑧ = ❿ 3 ( ❸ , ❹ ) at ❺ = 0. Solution:
References : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), A. G. Butkovskiy (1979).
8.2.2-6. Domain: − ❻ < ❸ < ❻ ,− ❻ < ❹ < ❻ ,0≤ ❺ ≤ ➈ . First boundary value problem. An infinite layer is considered. Boundary conditions are prescribed:
= ⑧ ❿ 1 ( ❸ , ❹ ) at ❺ = 0, = ❿ 2 ( ❸ , ❹ ) at ❺ = ➈ . Solution:
Green’s function:
References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.2.2-7. Domain: − ❻ < ❸ < ❻ ,− ❻ < ❹ < ❻ ,0≤ ❺ ≤ ➈ . Mixed boundary value problem. An infinite layer is considered. Boundary conditions are prescribed:
= 0, ⑧ = ❿ 2 ( ❸ , ❹ ) at ❺ = ➈ . Solution:
Green’s function:
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.2.2-8. Domain: 0 ≤ ❸ < ❻ ,− ❻ < ❹ < ❻ ,0≤ ❺ ≤ ➈ . First boundary value problem.
A semiinfinite layer is considered. Boundary conditions are prescribed:
, ⑧ ) at ❸ = 0, = ❿ 2 ( ❸ , ❹ ) at ❺ = 0, = ❿ 3 ( ❸ , ❹ ) at ❺ = ➈ . Solution:
Green’s function:
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.2.2-9. Domain: 0 ≤ ❸ ≤ ➈ ,0≤ ❹ ≤ ➒ ,− ❻ < ❺ < ❻ . First boundary value problem. An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions
are prescribed:
= ⑧ ❿ 1 ( ❹ , ❺ ) at ❸ = 0, = ❿ 2 ( ❹ , ❺ ) at ❸ = ➈ ,
) ⑧ at ❹ = 0, = ❿ 4 ( ❸ , ❺ ) at ❹ = ➒ . Solution:
Green’s function:
) sin( ↔ ➈ ④ ➒ ) sin( ➣ ➋ ③ ➔ ) exp(− ➋ ➔ | ❺ − ⑤ |),
sin( ➣ ❸ ) sin( ↔ ➔ ❹
Alternatively, the Green’s function can be represented as
8.2.2-10. Domain: 0 ≤ ❸ ≤ ➈ ,0≤ ❹ ≤ ➒ ,− ❻ < ❺ < ❻ . Third boundary value problem. An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions
are prescribed:
= 0, ⑧ + ❾ 4 = ❿ 4 ( ❸ , ❺ ) at ❹ = ➒ . Solution:
Green’s function:
Here, the ➡ ➋ and ➢ ➔ are positive roots of the transcendental equations
8.2.2-11. Domain: 0 ≤ ❸ ≤ ➈ ,0≤ ❹ ≤ ➒ ,− ❻ < ❺ < ❻ . Mixed boundary value problems.
1 ➤ . An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
1 ( ❹ , ❺ ) at ❸ = 0,
= ❿ ( ❹ , ❺ ) at ❸ = ➈ ,
= ❿ 3 ( ❸ , ❺ ) at ❹ = 0,
= ❿ 4 ( ❸ , ❺ ) at ❹ = ➒ .
Green’s function:
2 ➤ . An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions ➠ ➠
are prescribed:
1 ( ❹ , ❺ ) at ❸ = 0,
= ❿ 2 ( ❹ , ❺ ) at ❸ = ➈ ,
= ❿ 4 ( ❸ , ❺ ) at ❹ = ➒ . Solution: ➠
Green’s function:
sin( ➣ ➋ ❸ ) cos( ↔ ➔ ❹ ) sin( ➣ ➋ ③ ) cos( ↔ ➔ ④ ) exp(− ➋ ➔ | ❺ − ⑤ |),
2 for ≠ 0. Paragraphs 8.2.2-12 through 8.2.2-17 present only Green’s functions; the complete solution is ➧ ↕
constructed with the formulas given in Paragraphs 8.2.1-1 through 8.2.1-3.
8.2.2-12. Domain: 0 ≤ ❸ ≤ ➈ ,0≤ ❹ ≤ ❸ ,− ❻ < ❺ < ❻ . First boundary value problem. An infinite cylindrical domain of triangular cross-section is considered. Boundary conditions are ➠ ➠ ➠
prescribed: = ❿ 1 ( ❹ , ❺ ) at ❸ = 0,
= ❿ 2 ( ❸ , ❺ ) at ❹ = 0,
= ❿ 3 ( ❸ , ❺ ) at ❹ = ❸ .
Green’s function:
sin( ➣ ➋ ❸ ) sin( ➣ ➔ ❹ ) sin( ➣ ➋ ③ ) sin( ➣ ➔ ④ ) exp(− ➋ ➈ ➔ 2
An alternative representation of the Green’s function can be obtained by setting
where the functions ➲ ( ➵ ➳ ➸ ) ➼✉➽ ( ➺ = 1, 2, 3, 4) are specified in Paragraph 8.2.2-9 for ② ② ➻ = ➒ .
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.2.2-13. Domain: 0 ≤ ➾ ≤ ➻ ,0≤ ➚ ≤ ➒ ,0≤ ❺ < ❻ . First boundary value problem.
A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions ➠ ➠
are prescribed:
Green’s function:
sin( ➱ ➵ ➾ ) sin( ✃ ➸ ➚ ) sin( ➱ ➵ ➵ ➹ ➸ ) sin( ✃ ➸ ➻ ➘ ➒ ➷ ➷ ) ❐ ➵ ➸ ( ❒ , ➴ ),
) for > ≥ 0, )= exp(− ➵ ➸ ➴ ) sinh( ➵ ➸ ❒ ) for ➴ > ❒ ≥ 0.
exp(− ➮
) sinh( ➮
An alternative representation of the Green’s function:
where
8.2.2-14. Domain: 0 ≤ ➾ ≤ ➻ ,0≤ ➚ ≤ ➒ ,0≤ ❒ < Ö . Third boundary value problem.
A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed: × Ø ➠
Green’s function:
exp(− ➵ ➸ ❒ ) å ➵ ➸ cosh( ➵ ➸ ➴ )+ ➺ ➮ ➮ 5 sinh( ➵ ➸ ➴ ) æ
Here, the Ü ➵ and Ý ➸ are positive roots of the transcendental equations
tan( ➻ )= 2 ➺ , tan( Ý Ü ➒ ➺ )=
8.2.2-15. Domain: 0 ≤ ➾ ≤ ➻ ,0≤ ➚ ≤ ➒ ,0≤ ❒ < Ö . Mixed boundary value problems.
1 ç . A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed: ➠
Green’s function:
sin( ➱ ➸ ➵ ➾ ) sin( ✃ ➸ ➚ ) sin( ➱ ➵ ➹ ) sin( ✃ ➸ ➘ ) ❐ ➵ ➸ ( ➻ ❒ ➒
> ≥ 0, exp(− ➵ ➸ ➴ ) cosh( ➵ ➸ ❒ ) for ➴ > ❒ ≥ 0.
2 ç . A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed: × Ø ➠
= ➪ 1 ( ➚ , ❒ ) at ➾ = 0,
= ➪ 2 ( ➚ , ❒ ) at ➾ = ➻ ,
= ➪ 3 ( ➾ , ❒ ) at ➚ = 0,
= ➪ 4 ( ➾ , ❒ ) at ➚ = ➒ ,
= ➪ 5 ( ➾ , ➚ ) at ❒ = 0.
Green’s function:
cos( ➱ ➵ ➾ ) cos( ✃ ➸ ➚ ) cos( ➱ ➵ ➻ ➹ ➒ ➷ ➷ ➵ ➸ ) cos( ✃ ➸ ➘ ) ❐ ➵ ➸ ( ❒ , ➴ ),
) for ❒ > ➴ ≥ 0, exp(− ➵ ➸ ➴ ) sinh( ➵ ➸ ❒ ) for ➴ > ❒ ≥ 0.
) sinh( ➵ ➸
8.2.2-16. Domain: 0 ≤ ➾ ≤ ➻ ,0≤ ➚ ≤ ➒ ,0≤ ❒ ≤ é . First boundary value problem.
A rectangular parallelepiped is considered. Boundary conditions are prescribed: ➠ ➠
1 ç . Representation of the Green’s function in the form of a double series:
sin( ➱ ➵ ➾ ) sin( ✃ ➸ ➚ ) sin( ➱ ➵ ➹ ) sin( ✃ ➸ ➘ ) ê ➵ ➸ ( ❒ , ➻ ➴ ➒ ),
This relation can be used to obtain two other representations of the Green’s function by means of the following cyclic permutations:
2 ç . Representation of the Green’s function in the form of a triple series:
sin( ➱ ➵ ➾ ) sin( ✃ ➸ ➚ ) sin( ð ➳ ❒ ) sin( ➱ ➵ ➹ ) sin( ✃ ➸ ➘ ) sin( ð ➳ ➴ ) ( , , , , , )= ➻ ➒ïé
3 ç . An alternative representation of the Green’s function in the form of a triple series:
8.2.2-17. Domain: 0 ≤ ÷ ≤ ➻ ,0≤ ù ≤ ➒ ,0≤ ❒ ≤ é . Third boundary value problem.
A rectangular parallelepiped is considered. Boundary conditions are prescribed: × Ø ý ý × Ø ý ý
Green’s function: ÿ
where the ✟ ó , ✠ ô , and ✡ ✂ are positive roots of the transcendental equations tan( ✟
1 + ü 2 tan( ✠ ➒ )
3 + ü 4 tan( ✡ )
8.2.2-18. Domain: 0 ≤ ÷ ≤ ☛ ,0≤ ù ≤ ➒ ,0≤ ✝ ≤ ☞ . Mixed boundary value problem.
A rectangular parallelepiped is considered. Boundary conditions are prescribed: ý × Ø ý
= ý 1 ( , ) at = 0, = þ 2 ( ù , ✝ ) at ÷ =
) ý at = 0, ✌ = þ 4 ( ÷ , ✝ ) at ù = ➒ ,
= þ 5 ( ÷ , ù ) at ✝ = 0,
= þ 6 ( ÷ , ù ) at ✝ = ☞ .
1 ✦ . A double-series representation of the Green’s function: ★
sin( ✫ ✜ ) sin( ✪ ★ ✬ ✢ ) sin( ✫ ✣ ) sin( ✬ ✪ ✙ ) ê ✪ ( ➒ ✝ ✧ ✧ , ✘ ),
This relation can be used to obtain two other representations of the Green’s function by means of the following cyclic permutations:
2 ✦ . A triple series representation of the Green’s function: ★
) sin( ) sin( ) , , , , , )=
) sin( ) sin(
8.2.3. Problems in Cylindrical Coordinates
The three-dimensional Poisson equation in the cylindrical coordinate system is written as
8.2.3-1. Domain: 0 ≤ ≤ ❄ ,0≤ ❅ ≤2 ,− ❆ < ❇ < ❆ ✲ . First boundary value problem. An infinite circular cylinder is considered. A boundary condition is prescribed: ✾
= ( ❅ , ❇ ) at
Solution: ✚
Green’s function:
2 cos[ ❳ ( ❅ − ❨ )] exp ❩ − ❙
2 =0 P ❖ =1 ◗
( ❚ ) are the Bessel functions; and the ❙ P are positive roots of the transcendental equation ◗ ◗
where ◆ 0 = 1 and = 2 for
≠ 0; the
8.2.3-2. Domain: 0 ≤ ❭ ≤ ❄ ,0≤ ❅ ≤2 ❪ ,− ❆ < ❇ < ❆ . Third boundary value problem. An infinite circular cylinder is considered. A boundary condition is prescribed:
Green’s function:
where 0 = 1 and = 2 for ❳ ≠ 0; the ( ❚ ) are the Bessel functions; and the ❙ P are positive roots of the transcendental equation ❘ ❱
8.2.3-3. Domain: 0 ≤ ❭ ≤ ❄ ,0≤ ❅ ≤2 ❪ ,0≤ ❇ < ❆ . First boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
Green’s function:
( ❇ , ) = exp(− ❙ P
| ❇ − ▲ |) − exp(− ❙ P | ❇ + ▲ |),
for
2 for ❳ ≠ 0, where the ◗
are positive roots of the transcendental equation ( ❙ ❄ ) = 0.
8.2.3-4. Domain: 0 ≤ ❭ ≤ ❄ ,0≤ ❅ ≤2 ❪ ,0≤ ❇ < ❆ . Third boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
Green’s function: ◆
exp(− ❙ P
cosh( ❙ ◆ P ◆ )+ ❜
2 sinh( ❙ P ▲ ◆ ) ❲
cosh( ❙ P ❇ )+ ❜ 2 sinh( ❙ P ❇ ) ❲
for > ,
where the ( ❚ ) are the Bessel functions and the ❘ ❱ ❘ ◆ ❙ P ◆ are positive roots of the transcendental equation
8.2.3-5. Domain: 0 ≤ ❭ ≤ ❄ ,0≤ ❅ ≤2 ❪ ,0≤ ❇ < ❆ . Mixed boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
Green’s function: ◆
( ❇ , ▲ ) = exp(− ❙ P | ❇ − ▲ |) + exp(− ❙ P ◆ | ❇ + ▲ |), = ❤ 1 for
2 for ❳ ≠ 0,
where the ◆ ( ❚ ) are the Bessel functions and the ❙ P
are roots of the transcendental equation ◗ ( ❙ ❄ ) = 0.
Paragraphs 8.2.3-6 through 8.3.3-10 present only Green’s functions; the complete solution is constructed with the formulas given in Subsection 8.2.1. See also Paragraphs 8.3.1-4 and 8.3.1-8 for q = 0.
8.2.3-6. Domain: 0 ≤ ❭ ≤ ❄ ,0≤ ❅ ≤2 ❪ ,0≤ ❇ ≤ r . First boundary value problem.
A circular cylinder of finite length is considered. Boundary conditions are prescribed:
A double series representation of the Green’s function: ✚
cos[ ❄ ❳ ◆ ( − ❨ )] ❣ P ( ❇ , ▲ ),
) 2 ❲ ❙ P sinh( ❙ P r )
=0 P ❖ =1
sinh( ◆
1 for = 0, sinh( ❙ P
) sinh[ ❙ P r
( − )] for ≥ > ≥ 0,
2 for ❳ ≠ 0, where the
≥ ▲ > ◆ ❇ for ≥ 0,
) sinh[ ❙ P ( r − ▲
( ❚ ) are the Bessel functions (the prime denotes the derivative with respect to the ◗
argument) and the ❙ P are positive roots of the transcendental equation ( ❙ ❄ ) = 0.
A triple series representation of the Green’s function: ◆
8.2.3-7. Domain: 0 ≤ ❭ ≤ t ,0≤ ① ≤2 ❪ ,0≤ ✈ ≤ r . Third boundary value problem.
A circular cylinder of finite length is considered. Boundary conditions are prescribed:
Green’s function: ②
)+ q sin( q ✈ ), ❘ ◆ ⑤ ④ ⑤ 2 =
Here, 0 = 1 and = 2 for ❳ ≠ 0; the ( ❚ ) are the Bessel functions; and the ❙ P and q are positive roots of the transcendental equations ❘ ◆ ❱ ❘ ◆ ◗ ◗
tan( )
)+ 1 ( ❙ t ) = 0,
8.2.3-8. Domain: 0 ≤ ❭ ≤ t ,0≤ ① ≤2 ❪ ,0≤ ✈ ≤ r . Mixed boundary value problem.
A circular cylinder of finite length is considered. Boundary conditions are prescribed:
Green’s function: ✚
cos r ✉ ✇ r ✇ , where ◆ 0 = 1 and = 2 for
× cos[ ❘ ◆ ❳ ( ① − ❨ )] cos ✉
( ❘ ❚ ◆ ) are the Bessel functions (the prime denotes the derivative with respect to the argument); and the ❙ P ◗ ◗ are positive roots of the transcendental equation
≠ 0; the
( ❙ t ) = 0.
8.2.3-9. Domain: 0 ≤ ❭ ≤ t ,0≤ ① ≤ ① 0 ,0≤ ✈ ≤ r . First boundary value problem.
A cylindrical sector of finite thickness is considered. Boundary conditions are prescribed:
Green’s function: ✚
where the ⑦⑨⑧
0 ( ❭ ) are the Bessel functions and the ❙ P are positive roots of the transcendental equation
0 ( ❙ t ) = 0.
8.2.3-10. Domain: 0 ≤ ❭ ≤ t ,0≤ ① ≤ ① 0 ,0≤ ✈ ≤ r . Mixed boundary value problem.
A cylindrical sector of finite thickness is considered. Boundary conditions are prescribed:
Green’s function: ✚
0 ( ❭ ) are the Bessel functions; and the ❙ P are positive roots of the transcendental equation ◗ ◗
where ⑦⑨⑧ 0 = 1 and = 2 for
≠ 0; the
0 ( ❙ t ) = 0.
8.2.4. Problems in Spherical Coordinates
The three-dimensional Poisson equation in the spherical coordinate system is written as
Only Green’s functions are presented below; the complete solutions can be constructed with the formulas given in Subsection 8.2.1.
8.2.4-1. Domain: 0 ≤ ❭ ≤ t ,0≤ ⑩ ≤ ❪ ,0≤ ① ≤2 ❪ . First boundary value problem.
A spherical domain is considered. A boundary condition is prescribed:
= ( ① , ⑩ ) at ❭ = t .
Green’s function: ✚
, cos ❻ = cos ⑩ cos ❨ + sin ⑩ sin ❨ cos( ① − ❺ ).
cos
4 ❪ ❷ ❭ 2 ❚ 2 −2 t 2 ❭ ❚ cos ❻ + t 4
An alternative representation of the Green’s function:
where r={ ❸ , ❹ , ✈ },
References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.2.4-2. Domain: 0 ≤ ❭ ≤ t ,0≤ ⑩ ≤ ❪ ,0≤ ① ≤2 ❪ . Second boundary value problem.
A spherical domain is considered. A boundary condition is prescribed:
= ( ① , ⑩ ) at ❭ = t .
Green’s function: ✚
where | r−r 0 |= ❷ ❭ 2 −2 ❭ ❚ cos ❻ + ❚ 2 , | r 0 || r 1 |= ❷ ❭ 2 ❚ 2 −2 t 2 ❭ ❚ cos ❻ + t 4 ,
| r 0 |= ❚ , (r ⋅ r 0 )= ❭ ❚ cos ❻ , cos ❻ = cos ⑩ cos ❨ + sin ⑩ sin ❨ cos( ① − ❺ ). For a solution of the second boundary value problem to exist the solvability condition must be
satisfied (see Paragraph 8.2.1-2). ➀➂➁
Reference : N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970).
8.2.4-3. Domain: 0 ≤ ❭ ≤ t ,0≤ ⑩ ≤ ❪ ,0≤ ① ≤2 ❪ . Third boundary value problem.
A spherical domain is considered. A boundary condition is prescribed:
= ( ⑩ , ① ) at ❭ = t .
Green’s function: ➆
× ➑ ➒ (cos ⑩ ) ➑ ➒ (cos ❨ ) cos[ ➓ ( ➔ − ❺ )],
Here, the ➆ +1
2 ( ➏ ) are the Bessel functions, the ➑ ➒ ( ➝ ) are the associated Legendre functions that ➆
are expressed in terms of the Legendre polynomials ➑
and the ➎ ➈ are positive roots of the transcendental equation ➌ ➆ ➟ ➌ ➆
8.2.4-4. Domain: ↕ ≤ ➏ < ➢ ,0≤ ➤ ≤ ➥ ,0≤ ➔ ≤2 ➥ . First boundary value problem. Three-dimensional space with a spherical cavity is considered. A boundary condition is prescribed:
= ➧ ( ➔ , ➤ ) at ➏ = ↕ .
The Green’s function of the outer first boundary value problem is given by the same relation as that for the inner first boundary value problem (see Paragraph 8.2.4-1), except that ➏ ≥ ↕ and ➐ ≥ ↕ .
8.2.4-5. Domain: ↕ ≤ ➏ < ➢ ,0≤ ➤ ≤ ➥ ,0≤ ➔ ≤2 ➥ . Second boundary value problem. Three-dimensional space with a spherical cavity is considered. A boundary condition is prescribed: ➨ ➩
= ➧ ( ➔ , ➤ ) at ➏ = ↕ .
Green’s function:
1 1 1 (1 − cos )|r| |r 0 , ➐
where | r| = ➏ , | r 0 |= ➐ , | r−r 0 |= ➳ ➏ 2 −2 ➏ ➐ cos ➯ + ➐ 2 , | r 0 || r 1 |= ➳ ➏ 2 ➐ 2 −2 ↕ 2 ➏ ➐ cos ➯ + ↕ 4 ,
(r ⋅ r 0 )= ➏ ➐ cos ➯ , cos ➯ = cos ➤ cos ➵ + sin ➤ sin ➵ cos( ➔ − ➭ ).
Reference : N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970).
8.2.4-6. Domain: ↕ 1 ≤ ➏ ≤ ↕ 2 ,0≤ ➤ ≤ ➥ ,0≤ ➔ ≤2 ➥ . First boundary value problem.
A spherical layer is considered. Boundary conditions are prescribed:
1 ➦ , ➆ = ➧ 2 ( ➤ , ➔ ) at ➏ = ↕ 2 . Green’s function:
× ➑ (cos ➤ ) ➑ (cos ➵ ) cos[ ➙ ( ➔ − ➭ )],
2 ( ➏ ) are the Bessel functions, the ➑ ( ➝ ) are the associated Legendre functions (see Paragraph 8.2.4-3), and the ➎ ➈ are positive roots of the transcendental equation
8.2.4-7. Domain: 0 ≤ ➏ ≤ ↕ ,0≤ ➤ ≤ ➥ ➚
2, 0 ≤ ➔ ≤2 ➥ . First boundary value problem.
A hemisphere is considered. Boundary conditions are prescribed:
= ➦ ➧ 1 ( ➔ , ➤ ) at ➏ = ↕ , = ➧ 2 ( ➏ , ➔ ) at ➤ = ➥ ➚ 2. Green’s function in the spherical coordinate system:
, ➔ , ➐ , ➵ , ➭ )= s ( ➏ , ➤ , ➔ , ➐ , ➵ , ➭ )− s ( ➏ , ➤ , ➔ , ➐ , ➥ − ➵ , ➭ ), where ➫ s ( ➏ ,
, ➔ ➫ , ➐ , ➵ , ➭ ) is the Green’s functions for a sphere; see Paragraph 8.2.4-1, where must
be replaced by s . Green’s function in the Cartesian coordinate system:
References : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.2.4-8. Domain: 0 ≤ ☎ ≤ ✆ ,0≤ ✝ ≤ ✞ ✟
2, 0 ≤ ✠ ≤ ✞ . First boundary value problem.
A quarter of a sphere is considered. Boundary conditions are prescribed:
, ✡ ✝ ) at ☎ = ✆ , = ☛ 2 ( ☎ , ✠ ) at ✝ = ✞ ✟ 2,
= ✡ ☛ 3 ( ☎ , ✝ ) at ✠ = 0, = ☛ 4 ( ☎ , ✝ ) at ✠ = ✞ . Green’s function in the spherical coordinate system:
, ✝ , ✠ , ✌ , ✍ , ✎ )= s ( ☎ , ✝ , ✠ ☞ , ✌ , ✍ , ✎ )− s ( ☎ , ✝ , ✠ ☞ , ✌ , ✞ − ✍ , ✎ ) + s ( ☎ , ✝ , ✠ , ✌ , ✞ − ✍ ,2 ✞ − ✎ )− s ( ☎ , ✝ , ✠ , ✌ , ✍ ,2 ✞ − ✎ ),
where ☞ s (
, ✝ , ✠ ☞ , ✌ , ✍ , ✎ ) is the Green’s function for a sphere; see Paragraph 8.2.4-1, where must
be replaced by s . Green’s function in the Cartesian coordinate system:
={ ✏ 0 , (−1) ✑ 0 , (−1) ✒ 0 }, r ✗ ✔ ✕ =( ✆ ✟ ☎ 0 ) 2 r ✔ ✕ , where ☎ 0 =| r 0 |; ✙ = 0, 1; ✚ = 0, 1.
}, r ✕ 0 ={ ✏ 0 , ✑ 0 , ✒ 0 }, r ✔
References : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3. Helmholtz Equation ✣ 3 ✤ + ✥ ✤ =– ✦ (x)
A variety of problems related to steady-state oscillations (mechanical, acoustic, thermal, electro- magnetic, etc.) lead to the three-dimensional Helmholtz equation with ✧ > 0. This equation governs mass transfer phenomena with volume chemical reaction of the first order for ✧ < 0. Any elliptic equation with constant coefficients can be reduced to the Helmholtz equation.
8.3.1. General Remarks, Results, and Formulas
8.3.1-1. Some definitions. The Helmholtz equation is called homogeneous if ★ = 0 and nonhomogeneous if ★ ≠ 0. A homo-
geneous boundary value problem is a boundary value problem for a homogeneous equation with homogeneous boundary conditions; ✡ = 0 is a particular solution of a homogeneous boundary value
problem. The values ✧ ✔ of the parameter ✧ for which there are nontrivial solutions (i.e., not identically zero solutions) of a homogeneous boundary value problem are called eigenvalues. The corresponding ✡ ✡
solutions, = ✔ , are called eigenfunctions of this boundary value problem. In what follows, we consider simultaneously the first, second, and third boundary value prob- lems for the three-dimensional Helmholtz equation in a finite three-dimensional domain ✩ with a sufficiently smooth surface ✪ . It is assumed that ✚ > 0 for the third boundary value problem with the boundary condition
+ ✚ ✡ =0 for r ✭ ✪ ,
where ✮ ✯ ✮ ✰ is the derivative along the outward normal to the surface ✪ , and r = { ✏ , ✑ , ✒ }.
8.3.1-2. Properties of eigenvalues and eigenfunctions.
1 ✱ . There are infinitely many eigenvalues { ✧ ✔ }; they form a discrete spectrum of the boundary value problem.
2 ✱ . All eigenvalues are positive, except for one eigenvalue ✧ 0 = 0 of the second boundary value problem (the corresponding eigenfunction is ✡ 0 = const). The eigenvalues are assumed to be ordered
so that ✧ 1 < ✧ 2 < ✧ 3 < ✲✳✲✳✲ .
3 ✱ . The eigenvalues tend to infinity as the number ✙ increases. The following asymptotic estimate holds:
lim ✴ ✵
where ✩ 3 is the volume of the domain under consideration.
4 ✷ ✱ . The eigenfunctions are defined up to a constant multiplier. Any two eigenfunctions, ✔ and , that correspond to different eigenvalues ✧ ✔ ≠ ✧ ✷ ✸ ✹ are orthogonal, that is,
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 this boundary value problem, specifically,
= ✺ ✔ ✔ , where ✔ = ✼
If ☛ is square summable, then the series is convergent in mean.
6 ✱ . The eigenvalues of the first boundary value problem do not increase if the domain is extended. ✽ ✾❀✿ ❁ ❂❄❃ ❅ ❆
In a three-dimensional problem, to each eigenvalue ✧ ✔ ✡ finitely many linearly inde- pendent eigenfunctions (1) ✔ , ❇✳❇✳❇ ✡ , ( ✔ ❈ ) generally correspond. These functions can always be replaced by their linear combinations
= 1, 2, ❇✳❇✳❇ , ● , such that ¯ ✡ (1) ✔ , ❇✳❇✳❇ ✡ ,¯ ( ✔ ❈ ) are now pairwise orthogonal. Therefore, without loss of generality, we
assume that all eigenfunctions are orthogonal. ✛✢✜
Reference : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1984).
8.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions. Three cases are possible.
1 ✱ . If ✧ is not equal to any one of the eigenvalues, then the solution of the problem is given by
, where
2 ✷ ✱ . If ✧ coincides with one of the eigenvalues, ✧ = ✧ , then the condition of the orthogonality of
the function ✷ ★ to the eigenfunction ,
is a necessary condition for a solution of the nonhomogeneous problem to exist. The solution is then given by ✷
is an arbitrary constant and ✡ ✔ 2 = ❏ ✡ 2 ✔ ✺ ✩ .
3 ✡ ✱ . If ✧ = ✧ ✷ ✷ ✺ and ❏ ★ ✩ ≠ 0, then the boundary value problem for the nonhomogeneous equation has no solution. ✽ ✾❀✿ ❁ ❂❄❃ ❑ ❆
mutually orthogonal eigen- functions ( ✔ ❉ ) ( ❋ = 1, ❇✳❇✳❇ , ● ✔ ), then the solution is written as
If to each eigenvalue
there are corresponding
provided that ❍
Reference : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1984).
8.3.1-4. Solution of nonhomogeneous boundary value problems of general form.
1 ✱ . The solution of the first boundary value problem for the Helmholtz equation with the boundary condition
= ☛ (r) for r ✭ ✪
can be represented in the form
. (1) Here, r = { ✏ , ✑ , ✒ }, ❖ ={ ✌ , ✍ , ✎ } (r ✭ ✩ , ❖ ✭ ✩ ); ✮ ✮ ✰ ❘ denotes the derivative along the outward
normal to the surface ✪ with respect to ✌ , ✍ , ✎ . The Green’s function is given by the series
where the ✡ ✔ and ✧ ✔ are the eigenfunctions and eigenvalues of the homogeneous first boundary value problem.
2 ✱ . The solution of the second boundary value problem with the boundary condition ✫
= ☛ (r) for r ✭ ✪
can be represented in the form
(r, ✺ ) (r, ❖ ) ✪ P . (3) Here, the Green’s function is given by the series
and ✡ ✔ are the positive eigenvalues and corresponding eigenfunctions of the homogeneous second boundary
where ✩ 3 is the volume of the three-dimensional domain under consideration, and the ✧ ✔
value problem. For clarity, the term corresponding to the zero eigenvalue ✧
0 ✡ =0( 0 = const) is singled out in (4). It is assumed that ✧ ≠ 0 and ✧ ≠ ✧ ✔ .
3 ✱ . The solution of the third boundary value problem for the Helmholtz equation with the boundary ✫
condition
+ ✚ ✡ = ☛ (r) for r ✭ ✪
is given by relation (3) in which the Green’s function is defined by series (2) with the eigenfunc- ✡
tions ✔ and eigenvalues ✧ ✔ of the homogeneous third boundary value problem.
4 ✱ . Let nonhomogeneous boundary conditions of various types be set on different portions ✷ ✪ ❙ of the surface ❚
[ ✡ ]= ☛ ❙ (r) for r ✭ ✪ ❙ .
Then the solution of the corresponding mixed boundary value problem can be written as
(r, ) if a first-kind boundary condition is set on , (r, ❖ )
(r, ❖ )= ❳
if a second- or third-kind boundary condition is set on ✪ ❭ ❙ . The Green’s function is expressed by series (2) that involves the eigenfunctions ❪ ❫ and eigenval-
ues ✧ ❫ of the homogeneous mixed boundary value problem.
8.3.1-5. Boundary conditions at infinity in the case of an unbounded domain. Below it is assumed that the function ★ is finite or sufficiently rapidly decaying as ❴ ❵ ❛ .
1 ✱ . If ✧ < 0 and the domain is unbounded, the additional condition that the solution must vanish at infinity is set:
0 as ❴ ❵ ❛ .
2 ✱ . If ✧ > 0, the radiation conditions (Sommerfeld conditions) are often used at infinity. In three- dimensional problems, these conditions are expressed as
lim ✵ ❴ ❝ ❞ ❴ + ❡❣❢ ❤ ❪ ✐ = 0, where ❞
lim ✵ ❴ ❪ = const,
2 = −1. The principle of limit absorption and the principle of limit amplitude are also employed to
separate a single solution. ❥✢✜
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.1-6. Green’s function for an infinite cylindrical domain of arbitrary cross-section. Consider the three-dimensional Helmholtz equation
(5) inside an infinite cylindrical domain ♠ = {( ♥ , ♦ ) ♣ q ,− ❛ < r < ❛ } with arbitrary cross-section q .
+ ❤ ❪ =− ❧ (r)
On the surface of this domain, let s = {( ♥ , ♦ ) ♣ t ,− ❛ < r < ❛ }, where t is the boundary of q , the homogeneous boundary condition of general form
=0 for r ♣
and in (6), one can obtain boundary conditions of the first ( = 0, ✇
be set, with ✉ ≥ 0. By appropriately choosing the constants
= 1), second ( ✉ = 1, ✇ = 0), and third ( ✇ ≠ 0) kind. The Green’s function of the first or third boundary value problem can be represented in the
* In Paragraphs 8.3.1-6 through 8.3.1-8, the cross-section ⑨
is assumed to have finite dimensions.
where the ❶ and are the eigenvalues and eigenfunctions of the corresponding two-dimensional boundary value problem in q ⑨ ,
Recall that all ⑨
are positive. In the second boundary value problem, the zero eigenvalue ⑩ ❶ 0 ⑩ = 0 appears, and hence the summation in (7) must start with ➇ = 0. In this case, 0 = 1 and 0 2 = q 2 , where q 2 is the area of the cross-section q .
References ⑨ : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), A. N. Tikhonov and A. A. Samarskii (1990).
8.3.1-7. Green’s function for a semiinfinite cylindrical domain.
1 ➈ . The Green’s function of the three-dimensional first boundary value problem for equation (5) in
a semiinfinite cylindrical domain ♠ = {( ♥ , ♦ ) ♣ q ,0≤ r < ➉ } with arbitrary cross-section q is
given by ⑦ ①
( , )= exp(− ➋ | ➋ r − ④ | ) − exp(− ➋ | r + ④ |) ➍
exp(− ➋ ⑦ r ) sinh( ➋ ⑦ ④ ) for r > ④ ≥ 0,
exp(− ➋ ④ ) sinh( ➋ ➋ r ) for ④ > r ≥ 0,
Relations (9) and (10) involve the eigenfunctions ✉
of the two-dimensional first boundary value problem (8) with = 0 and ✇ = 1. ⑨
and eigenvalues ❶
2 ➈ . The Green’s function of the three-dimensional second boundary value problem for equation (5) in a semiinfinite cylindrical domain ♠ = {( ♥ , ♦ ) ♣ q ,0≤ r < ➉ } with arbitrary cross-section q is
given by ⑦
exp(− ➋ ⑦ | r − ④ | ) + exp(− ➋ | r + ④ ⑦ |) ➍
exp(− ➋ ⑦ r ) cosh( ➋ ⑦ ④ ) for r > ④ ≥ 0,
exp(− ➋ ④ ) cosh( ➋ r ) for ④ > ➋ r ≥ 0,
Relations (11) and (12) involve the eigenfunctions ✉
of the two-dimensional second boundary value problem (8) with = 1 and ✇ ⑨ = 0. Note that in (11) the term corresponding to the zero eigenvalue ❶ 0 = 0 is specially singled out; q 2 is the area of the cross-section q .
and eigenvalues ❶
3 ➈ . The Green’s function of the three-dimensional third boundary value problem for equation (5) with the boundary conditions
− ➅ 1 =0 for r = 0, ✈ + ✇ 2 =0 for r ♣ s
in a semiinfinite cylindrical domain ♠ = {( ♥ , ♦ ) ♣ q ,0≤ r < ➉ } with arbitrary cross-section q
and lateral surface s is given by relation (9) with ⑦ ⑦ ⑦
exp(− ⑦ ➋ r ) ➋ cosh( ➋ ④ )+ ✇ sinh( ➋ ④ ) ➍
for r > ④ ≥ 0, ( r , ④ )= ➏ ➎➏
exp(− ➋ ④ ➋ cosh( ➋ r )+ ✇
1 sinh( ➋ r ) ➍
for > ≥ 0,
(13) Relations (9) and (13) involve the eigenfunctions
and eigenvalues ✉ ❶
of the two-dimensional
third boundary value problem (8) with = 1 and ✇ ⑨ = ✇ 2 .
4 ➈ . The Green’s function of the three-dimensional mixed boundary value problem for equation (5) ⑦ ⑦
with a second-kind boundary condition at the end face and a first-kind boundary condition at the lateral surface is given by relations (9) and (12), where the ❶
and
are the eigenvalues and ✉
eigenfunctions of the two-dimensional first boundary value problem (8) with ⑨ = 0 and ✇ = 1.
The Green’s functions of other mixed boundary value problems can be constructed likewise.
8.3.1-8. Green’s function for a cylindrical domain of finite dimensions.
1 ➈ . The Green’s function of the three-dimensional first boundary value problem for equation (5) in
a cylindrical domain of finite dimensions ♠ = {( ♥ , ♦ ) ♣ q ,0≤ r ≤ ➔ } with arbitrary cross-section q
is given by relation (9) with ⑦
sinh( ➋ ④ ) sinh[ ➋ ( ➔ − r )]
Relations (9) and (14) involve the eigenfunctions ✉
of the two-dimensional first boundary value problem (8) with = 0 and ✇ = 1. ⑨
and eigenvalues ❶
Another representation of the Green’s function: ① ⑦ ⑦
, ♦ ) ( ② , ③ ) sin( ➣ → r ) sin( ➣ → ④ ) ( ♥ , ♦ , r , ② , ③ , ④ )=
It is a consequence of formula (2). ⑨
2 ➈ . The Green’s function of the three-dimensional second boundary value problem for equation (5) in a cylindrical domain of finite dimensions ➙ = {( ➛ , ➜ ) ➝ ➞ ,0≤ ➟ ≤ ➔ } with arbitrary cross-section ➞
is given by relation (11) with ⑦
cosh( ➋ ④ ) cosh[ ➋ ( ➔ − ➟ )]
Relations (11) and (15) involve the eigenfunctions
of the two-dimensional second boundary value problem (8) with ➠ = 1 and ➡ ⑨ = 0.
and eigenvalues ❶
Another representation of the Green’s function: ① ⑦
It is a consequence of formula (4). ⑨
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
3 ➈ . The Green’s function of the three-dimensional third boundary value problem for equation (5) with the boundary conditions
in a cylindrical domain of finite dimensions ➙ = {( ➛ , ➜ ) ➝ ➞ ,0≤ ➟ ≤ ➔ } with arbitrary cross-section ➞
and lateral surface ➧ is given by relation (9) with [ ➋ ➨ cosh( ➋ ➨ ➩ )+ ➡ 1 sinh( ➋ ➨ ➩ )] ➫ ➋ ➨ cosh[ ➋ ➨ ( ➔ − ➟ )]+ ➡ 2 sinh[ ➋ ➨ ( ➔ − ➟ )] ➭
for > ,
2 ) cosh( ➋ ➨ ➔ )+( ➋ 2 + ➡ 1 ➡ 2 ) sinh( ➋ ➔ )]
cosh( ➋ ➨ ➟ )+ ➡ 1 sinh( ➋ ➨ ➟ )] ➫ ➋ ➨ cosh[ ➋ ➨ ( ➔ − ➩ )]+ ➡ 2 sinh[ ➋ ➨ ( ➔ − ➩ )] ➭
for ➟ < ➩ ,
[ ➋ ➨ ( ➡ 1 + ➡ 2 ) cosh( ➋ ➨ ➔ )+( ➋ 2 ➨ + ➡ 1 ➡ 2 ) sinh( ➋ ➨ ➔ )]
Relations (9) and (16) involve the eigenfunctions ➲ ➨ and eigenvalues ➯ ➨ of the two-dimensional
third boundary value problem (8) with ➠ = 1 and ➡ = ➡ 3 .
4 ➈ . The Green’s function of the three-dimensional mixed boundary value problem for equation (5) with second-kind boundary conditions at the end faces and a first-kind boundary condition at the lateral surface is given by relations (9) and (15), where the ➯ ➨ and ➲ ➨ are the eigenvalues and eigenfunctions of the two-dimensional first boundary value problem (8) with ➠ = 0 and ➡ = 1.
The Green’s function of the three-dimensional mixed boundary value problem for equation (5) with the boundary conditions
in a cylindrical domain of finite dimensions ➙ = {( ➛ , ➜ ) ➝ ➞ ,0≤ ➟ ≤ ➔ } with arbitrary cross-section ➞
and lateral surface ➧ is given by relation (9) with sinh( ➋ ➨ ➩ ) cosh[ ➋ ➨ ( ➔ − ➟ )]
Relations (9) and (17) involve the eigenfunctions ➲ ➨ and eigenvalues ➯ ➨ of the two-dimensional first boundary value problem (8) with ➠ = 0 and ➡ = 1.
The Green’s functions of other mixed boundary value problems can be constructed likewise.
8.3.2. Problems in Cartesian Coordinates
The three-dimensional nonhomogeneous Helmholtz equation in the rectangular Cartesian system of coordinates has the form
8.3.2-1. Particular solutions of the homogeneous equation ( ➵ ≡ 0):
=( ➸ 1 cos ➡ ➛ + ➸ 2 sin ➡ ➛ )( ➺ 1 cos ➜ ↕ + ➺ 2 sin ➜ )( ➻ 1 ➟ + ➻ 2 ), ↕ ➓ = ➡ 2 + 2 ↕ ;
=( ➸ 1 cos ➡ ➛ + ➸ 2 sin ➡ ➛ )( ➺ 1 cosh ➜ + ➺ 2 sinh ➜ )( ➻ 1 ↕ ➟ ↕ + ➻ 2 ), ➓ = ➡ 2 − 2 ↕ ;
=( ➸ 1 cos ➡ ➛ + ➸ 2 sin ➡ ➛ )( ➺ 1 cos ➜ ↕ + ➺ 2 sin ➜ ↕ )( ➻ 1 cos ➇ ➟ + ➻ 2 sin ➇ ➟ ), ➓ = ➡ 2 + 2 ↕ + ➇ 2 ;
=( ➸ 1 cosh ➡ ➛ + ➸ 2 sinh ➡ ➛ )( ➺ 1 cos ➜ + ➺ 2 sin ➜ )( ➻ 1 cos ➇ ➟ + ➻ 2 sin ➇ ➟
=( ➸ 1 cosh ➡ ➛ + ➸ 2 sinh ➡ ➛ )( ➺ 1 cosh ➜ + ➺ 2 sinh ➜ )( ➻ 1 cos ➇ ➟ + ➻ 2 sin ➇ ➟
=( ➸ 1 cosh ➡ ➛ + ➸ 2 sinh ➡ ➛ )( ➺ 1 cosh ➜ + ➺ 2 sinh ➜ )( ➻ 1 cosh ➇ ➟ + ➻ 2 sinh ➇ ➟
where ➸ 1 , ➸ 2 , ➺ 1 , ➺ 2 , ➻ 1 , and ➻ 2 are arbitrary constants.
Fundamental solutions: ➼ ➼
8.3.2-2. Domain: − ➉ < ➛ < ➉ ,− ➉ < ➜ < ➉ ,− ➉ < ➟ < ➉ .
1 ➈ . Solution for ➓ =− ➡ 2 < 0:
2 ❮ . Solution for ❰ = ➬ 2 ➹ > 0: ➹
This solution was obtained taking into account the radiation condition at infinity (see Paragraph
Ð✢Ñ 8.3.1-5, Item 2 ).
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.2-3. Domain: − Ò < ➱ < Ò ,− Ò < ✃ < Ò ,0≤ ❐ < Ò . First boundary value problem.
A half-space is considered. A boundary condition is prescribed:
Green’s function for Ø
exp(− ➬ Ü 1 ) exp(− ➬ Ü 2
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.2-4. Domain: − Ò < ➱ < Ò ,− Ò < ✃ < Ò ,0≤ ❐ < Ò . Second boundary value problem.
A half-space is considered. A boundary condition is prescribed:
. Green’s function for ➶
exp(− ➬ Ü 1 ) exp(− ➬ Ü 2 )
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.2-5. Domain: − Ò < ➱ < Ò ,0≤ ✃ < Ò ,0≤ ❐ < Ò . First boundary value problem.
A dihedral angle is considered. Boundary conditions are prescribed:
= 0, Ó = Ô 2 ( ➱ , ✃ ) at ❐ = 0. Solution:
Green’s function for Ø
exp(− ➬ Ü 2 ) exp(− ➬ Ü 3 ) exp(− ➬ Ü 4 ) ( , , , , , × )=
8.3.2-6. Domain: − Ò < ➱ < Ò ,0≤ ✃ < Ò ,0≤ ❐ < Ò . Second boundary value problem.
A dihedral angle is considered. Boundary conditions are prescribed:
= Ô 2 ( ➱ , ✃ ) at ❐ = 0. Solution:
Green’s function for Ø
2 ) exp(− ➬ Ü 3 ) exp(− ➬ Ü 4 ) ( , , , , , × )=
exp(−
1 ) ➹ exp(−
8.3.2-7. Domain: − Ò < ➱ < Ò ,− Ò < ✃ < Ò ,0≤ ❐ ≤ ä . First boundary value problem. An infinite layer is considered. Boundary conditions are prescribed:
= Ó Ô 1 ( ➱ , ✃ ) at ❐ = 0, = Ô 2 ( ➱ , ✃ ) at ❐ = ä . Solution:
Green’s function for Ø
8.3.2-8. Domain: − Ò < ➱ < Ò ,− Ò < ✃ < Ò ,0≤ ❐ ≤ ä . Second boundary value problem. An infinite layer is considered. Boundary conditions are prescribed:
= Ô 2 ( ➱ , ✃ ) at ❐ = ä . Solution:
Green’s function for ❰ =− ➬ 2 < 0: ➹
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.2-9. Domain: 0 ≤ ➱ ≤ ä ,0≤ ✃ ≤ é ,− Ò < ❐ < Ò . First boundary value problem. An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions
are prescribed:
= Ó Ô 1 ( ✃ , ❐ ) at ➱ = 0, = Ô 2 ( ✃ , ❐ ) at ➱ = ä ,
= Ô 3 ( ➱ , ❐ ) at ✃
= 0, Ó = Ô 4 ( ➱ , ❐ ) at ✃ = é .
Green’s function: Ø
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.2-10. Domain: 0 ≤ ➱ ≤ ä ,0≤ ✃ ≤ é ,− Ò < ❐ < Ò . Second boundary value problem. An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions
are prescribed:
Green’s function: Ø
cos( î ç ➱ ) cos( ï ì ✃ ) cos( î ç ➘ ) cos( ï ì × ➴ ) exp(− ç ì | ❐ − |),
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.2-11. Domain: 0 ≤ ➱ ≤ ä ,0≤ ✃ ≤ é ,− Ò < ❐ < Ò . Third boundary value problem. An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions
are prescribed:
1 Ó ( ✃ , ❐ ) at ➱ = 0, + ➬
2 = Ô 2 ( ✃ , ❐ ) at ➱ = ä ,
− 3 = 3 ( , ) at
+ 4 = 4 ( , ) at = .
The solution Ó ( ➱ , ✃ , ❐ ) is determined by the formula in Paragraph 8.3.2-10 where
( ø ì 2 + ➬ 2 )( ø ì 3 2 + ➬ 2 2 4 ) where the ÷ ç and ø ì are positive roots of the transcendental equations
8.3.2-12. Domain: 0 ≤ ➱ ≤ ä ,0≤ ✃ ≤ é ,− Ò < ❐ < Ò . Mixed boundary value problems.
1 ❮ . An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
Green’s function: Ø
) sin( ï ì ➴ ) exp(− ç ì | ❐ −
2 ❮ . An infinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
1 Ó ( ✃ , ❐ ) at ➱ = 0, = Ô 2 ( ✃ , ❐ ) at ➱ = ä ,
) at ✃ = 0,
= Ô 4 ( ➱ , ❐ ) at ✃ = é .
Green’s function: Ø
sin( î ç ➱ ) cos( ï ì ✃ ) sin( î ç ➘ ) cos( ï ì ➴ ) exp(− ç ì | ❐ −
8.3.2-13. Domain: 0 ≤ ➱ ≤ ä ,0≤ ✃ ≤ é ,0≤ ❐ < Ò . First boundary value problem.
A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
Green’s function: Ø
, sin( î ç ➱ ) sin( ï ì ✃ ) sin( î ç ç ì ➘ ) sin( ï ì ➴ û ç × ì ä é ) ( ❐ , ),
exp(− ç ì
) sinh( ç ì
) for > × ≥ 0,
exp(−
) sinh(
) for × > ❐ ≥ 0.
8.3.2-14. Domain: 0 ≤ ➱ ≤ ä ,0≤ ✃ ≤ é ,0≤ ❐ < Ò . Second boundary value problem.
A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
= Ó Ô 1 ( ✃ , ❐ ) at ➱ = 0, = Ô ✃ ❐
2 ( , ) at
= , = Ô 5 ( á ➱ , Ö ✃ ) at ❐ = 0. Solution:
Green’s function: Ø
cos( î ç ÿ ) cos( ï ì ) cos( î ç ➘ ) cos( ï ì þ ) û ç × ì ä é ç ì ( ✁ , × ),
) cosh( ç ì ) for û ó ç ì ✁ ü í í ✁ × > × ≥ 0,
exp(− í
( , × )= exp(− ç ì
) cosh( ç ì ✁ ) for × > ✁ ≥ 0.
8.3.2-15. Domain: 0 ≤ ÿ ≤ ä ,0≤ ≤ é ,0≤ ✁ < Ò . Third boundary value problem.
A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
+ 4 = 4 ( , ) at = , − ✆ 5 = Ô 5 ( Ö ÿ á ,) at ✁ = 0. The solution Ó ( ÿ ,, ✁ ) is determined by the formula in Paragraph 8.3.2-14 where
( ÿ ,)=( ÷ ç cos ÷ ç ÿ + ✆ 1 sin ÷ ç ÿ )( ø ì cos ø ì + ✆ 3 sin ø ì ),
where the ÷ ç and ø ì are positive roots of the transcendental equations
tan( )= ÷ 2 ✆ ✆ ,
tan( ø é )=
8.3.2-16. Domain: 0 ≤ ÿ ≤ ä ,0≤ ≤ é ,0≤ ✁ < Ò . Mixed boundary value problems.
1 ✡ . A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
Green’s function: Ø
) cosh( ç û ì ç ì ✁ ü í í × ) for ✁ >
2 ✡ . A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions are prescribed:
) at ÿ = 0,
= Ô 2 (, ✁ ) at ÿ = ä ,
) Ó at = 0,
= Ô 4 ( ÿ , ✁ ) at
= Ô 5 ( ÿ ,) at ✁ = 0.
Solution:
Green’s function:
cos( î ç ÿ ) cos( ï ì ) cos( î ç ➘ ) cos( ï ì þ ) û ç ì × ✁ ä é ( , ).
) sinh( ç ì ✁ ) for × > ✁ ≥ 0.
Paragraphs 8.3.2-17 through 8.3.2-23 present only the eigenvalues and eigenfunctions of ho- Û
mogeneous boundary value problems for the homogeneous Helmholtz equation (with Û ☞ ≡ 0). The solutions of the corresponding nonhomogeneous boundary value problems (with
0) can be constructed by the relations specified in Paragraphs 8.3.1-4 and 8.3.1-8.
8.3.2-17. Domain: 0 ≤ ÿ ≤ ä ,0≤ ≤ é ,0≤ ✁ ≤ ✌ . First boundary value problem.
A rectangular parallelepiped is considered. Boundary conditions are prescribed:
1 ✡ . Eigenvalues of the homogeneous problem:
Eigenfunctions and the norm squared:
2 ✡ . A double-series representation of the Green’s function:
sinh( ✗ ✁ ) sinh[ ✥ ✖ ✗ ( ✌ − ✚ )]
for ≥ ✚ > ✁ ≥ 0,
sinh( ✥ ✖ ✗
= ✗ ✘ , ✥ = ✦ ✜ 2 ✖ + ✢ 2 ✗ − ☎ . This relation can be used to obtain two other representations of the Green’s function with the aid of
the cyclic permutations of triples:
A triple series representation of the Green’s function:
) sin( ✬ ✍ ✚ ) , , þ , ✚ )= ✘ ä ✌ ✖ ✫ ✗ ✫
sin( ✖ ÿ
) sin( ✗ ) sin( ✬ ✍ ✁ ) sin( ✜ ✖ ✣
References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.2-18. Domain: 0 ≤ ✘
≤ ä ,0≤ ✴ ≤ ,0≤ ✵ ≤ ✶ . Second boundary value problem.
A rectangular parallelepiped is considered. Boundary conditions are prescribed:
1 ✻ . Eigenvalues of the homogeneous problem:
The square of the norm of an eigenfunction is defined as
1 for = ✑ (1 + 0 )(1 + 0 )(1 + 0 ), 0 = = 0,
8 0 for ✑ ≠ 0.
2 ✻ . A double series representation of the Green’s function:
cosh( ✗ ✚ ) cosh[ ✥ ( ✶ − ✵ )]
cosh( ✗ ✥ ✖ ✗ ✵ ) cosh[ ✥ ( ✶ − ✚ )]
2 for ✑ ≠ 0. This relation can be used to obtain two other representations of the Green’s function with the aid of
the cyclic permutations:
A triple series representation of the Green’s function:
) cos( ✽ ✚ ) ( ÿ , ✴ , ✵ , , ❃ , ✚ )= ❄ ✘
cos( ✜ ÿ ) cos( ✢ ✴ ) cos( ✬ ✵ ) cos( ✜ ) cos( ✢ ✗ ❃
References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.2-19. Domain: 0 ≤ ✘
≤ ,0≤ ✴ ≤ ,0≤ ✵ ≤ ✶ . Third boundary value problem.
A rectangular parallelepiped is considered. Boundary conditions are prescribed:
− ✕ 1 = ✹ 1 ( ✴ , ✵ ) at ÿ = 0, + 2 = ✹ 2 ( ✴ , ✵ ) at ÿ
3 3 ( ÿ , ✵ ) at ✴ = 0,
+ ✿ 4 = ✹ 4 ( ÿ , ✵ ) at ✴ = ,
5 = ✹ 5 ( ÿ , ✴ ) at ✵ = 0,
+ ✿ 6 = ✹ 6 ( ÿ , ✴ ) at ✵ = ✶ .
Eigenvalues of the homogeneous problem:
Here, the ✗ , , and ■ ❋ are positive roots of the transcendental equations
= ✦ ■ 2 ❋ + 2 ✿ 5 . The square of the norm of an eigenfunction is defined as
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.2-20. Domain: 0 ≤ ÿ
≤ ✶ . Mixed boundary value problems.
1 ✻ . A rectangular parallelepiped is considered. Boundary conditions are prescribed:
= ✹ 6 ( ÿ , ✴ ) at ✵ = ✶ . Eigenvalues of the homogeneous problem:
. The square of the norm of an eigenfunction is defined as ❚
2 ✻ . A rectangular parallelepiped is considered. Boundary conditions are prescribed:
= ✹ 6 ( ÿ , ✴ ) at ✵ = ✶ . Eigenvalues of the homogeneous problem:
. The square of the norm of an eigenfunction is defined as ❚
= sin
cos
cos
❀◗P ❀
for ❳
8 0 for ❳ ≠ 0.
8.3.2-21. Domain: 0 ≤ ÿ
≤ ❄ ,0≤ ✴ ≤ ÿ ,0≤ ✵ ≤ ✶ . First boundary value problem.
A right prism whose base is an isosceles right-angled triangle is considered. Boundary conditions are prescribed: P
Eigenvalues of the homogeneous problem:
8.3.2-22. Domain: 0 ≤ ❄ ÿ ≤ ,0≤ ✴ ≤ ÿ ,0≤ ✵ ≤ ✶ . Second boundary value problem.
A right prism whose base is an isosceles right-angled triangle is considered. Boundary conditions are prescribed: ✷ ✸ P
where P = N⋅∇ =
Eigenvalues of the homogeneous problem:
8.3.2-23. Domain: 0 ≤ ÿ
≤ ❄ ,0≤ ✴ ≤ ÿ ,0≤ ✵ ≤ ✶ . Mixed boundary value problems.
1 ✻ . A right prism whose base is an isosceles right-angled triangle is considered. Boundary conditions are prescribed: P
Eigenvalues of the homogeneous problem:
2 ✻ . A right prism whose base is an isosceles right-angled triangle is considered. Boundary conditions are prescribed: ✷ ✸ P
Eigenvalues of the homogeneous problem:
; ✿ = 1, 2, 3, Eigenfunctions:
= ❱ ❫ cos
( ❲ + ❳ ) ÿ cos ❄
− (−1) cos ❄ ❳
cos ❘ ❄ ( ❲ + ) ✴ ❳ ❙ ❴ sin
8.3.3. Problems in Cylindrical Coordinates
The three-dimensional nonhomogeneous Helmholtz equation in the cylindrical coordinate system is written as
8.3.3-1. Particular solutions of the homogeneous equation ( ❞ ≡ 0): ❜ ✐ ❥ ❦ ❜ ✐ ❥ ❦
)( ♥ cosh s ♣ + ♦ sinh s ♣ ), >− s ❥ 2 ,
)( cosh ♣ + sinh ♣ ), ❥ >− s 2 ,
where r
= 0, 1, 2, ; , , , , 1 , r 2 , 1 , 2 r , and are arbitrary constants; the ( ) and
r ❩✔❩✔❩ t
( ✈ ) are the Bessel functions; and the ✉ ( ✈ ) and ✇ ( ✈ ) are the modified Bessel functions.
,0≤ ≤2 ② ,− ③ < ♣ < ③ . First boundary value problem. An infinite circular cylinder is considered. A boundary condition is prescribed: ❜
8.3.3-2. Domain: 0 ≤ ❝ ≤
cos[ ( − )] exp ❤ − | ♣ − | ,
where the r
( ✈ ) are the Bessel functions and the s are positive roots of the transcendental equation ( s ➉✯➊ ① ) = 0.
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.3-3. Domain: 0 ≤ ❽ ≤ ① ,0≤ ❾ ≤2 ② ,− ③ < ♣ < ③ . Second boundary value problem. An infinite circular cylinder is considered. A boundary condition is prescribed:
= ( , ♣ ) at = ( , ♣ ) at
( ✓ ) are the Bessel functions and the
are positive roots of the transcendental equation
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.3-4. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,− ✭ < ✞ < ✭ . Third boundary value problem. An infinite circular cylinder is considered. A boundary condition is prescribed:
where the ✢
are positive roots of the transcendental equation
( ✓ ) are the Bessel functions and the
8.3.3-5. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,0≤ ✞ < ✭ . First boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
1 ( ✝ , ✞ ) at ✆ = ✟ ,
= ☞ 2 ( ✆ , ✝ ) at ✞ = 0.
( ✞ , ✍ ) = exp(− ✦ ✜ | ✞ − ✍ |) − exp(− ✦ ✜ | ✞ + ✍
2 for ✥ ≠ 0,
where the ✢
( ✓ ) are the Bessel functions and the
are positive roots of the transcendental equation
8.3.3-6. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,0≤ ✞ < ✭ . Second boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
2 ( ✆ , ✝ ) at ✞ = 0. Solution:
1 for ✥ = 0, ( , ) = exp(− ✚
where the ✢
( ✓ ) are the Bessel functions and the
are positive roots of the transcendental equation
8.3.3-7. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,0≤ ✞ < ✭ . Third boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
) ☎ at ✆
2 ( ✆ , ✝ ) at ✞ = 0.
exp(− ✦ ✜ ✞ )[ ✦ ✜ cosh( ✦ ✜ ✍
exp(− ✦ ✜ ✍ )[ ✦ ✜ cosh( ✦ ✜ ✞ )+ ✰ 2 sinh( ✦ ✜ ✞ )] for ✍
where the ✤ ( ✓ ) are the Bessel functions,
are positive roots of the transcendental equation
= ✽ ✚ ✣ ✚ ✜ − ✖ , and the
8.3.3-8. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,0≤ ✞ < ✭ . Mixed boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
= ☞ 2 ( ✆ , ✝ ) at ✞ = 0. Solution:
( ✞ , ✍ ) = exp(− ✦ ✜ | ✞ − ✍ |) + exp(− ✦ ✜ | ✞ + ✍ |), ✦ ✜
where the ✢
( ✓ ) are the Bessel functions and the
are positive roots of the transcendental equation
Paragraphs 8.3.3-9 through 8.3.3-16 present only the eigenvalues and eigenfunctions of homoge- neous boundary value problems for the homogeneous Helmholtz equation (with ✒ ≡ 0). The solutions of the corresponding nonhomogeneous boundary value problems ( ✒ ✿
0) can be constructed by the relations specified in Paragraphs 8.3.1-4 and 8.3.1-8.
8.3.3-9. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,0≤ ✞ ≤ ❀ . First boundary value problem.
A circular cylinder of finite length is considered. Boundary conditions are prescribed:
= ☎ ✟ , =0 at ✞ = 0, =0 at ✞ = ❀ . Eigenvalues:
=0 at ✆
Here, the ✜ are positive zeros of the Bessel functions, ✤ ( ) = 0. Eigenfunctions:
cos( ✥ ✝ ) sin
sin( ✥ ✟ ✝ ❆ ) sin
Eigenfunctions possessing the axial symmetry property: ❅ ✣ ❅
= ✤ 0 0 ✜ ✟ ❆ sin
The square of the norm of an eigenfunction is defined as
References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.3-10. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,0≤ ✞ ≤ ❀ . Second boundary value problem.
A circular cylinder of finite length is considered. Boundary conditions are prescribed:
=0 ☎ at = , =0 at ✞ = 0, =0 at ✞ = ❀ . Eigenvalues:
Here, the ✜ are roots of the transcendental equation ✤ ✩ ( ) = 0. Eigenfunctions:
sin( ) cos
The square of the norm of an eigenfunction is defined as
= ✘ ✟ ✜ ❀ , where ❈ ✪✬✫ 0 is the Kronecker delta.
References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.3-11. Domain: 0 ≤ ✆ ≤ ✟ ,0≤ ✝ ≤2 ✘ ,0≤ ✞ ≤ ❀ . Third boundary value problem.
A circular cylinder of finite length is considered. Boundary conditions are prescribed:
1 ☎ =0 at ✆ = ☎ ✟ , − ✰ 2 =0 at ✞ = 0, + ✰ 3 =0 at ✞ = ❀ .
Eigenvalues: ✚
where the ❋ ❊ and
are positive roots of the transcendental equations ✣ ✚ ✣ ✚ ✣
The square of the norm of an eigenfunction is defined as ✚ ✚ ✣ ✚ ✣ ✚
where ❈ 0 is the Kronecker delta.
8.3.3-12. Domain: ✟ 1 ≤ ✆ ≤ ✟ 2 ,0≤ ✝ ≤2 ✘ ,0≤ ✞ ≤ ❀ . First boundary value problem.
A hollow circular cylinder of finite length is considered. Boundary conditions are prescribed:
Here, the ✜ are positive roots of the transcendental equation ✚ ✣ ✚ ✣ ✚ ✣ ✚ ✣
)] cos( ✥ ✝ ) sin
. The square of the norm of an eigenfunction is defined as ✚ ✣ ✚ ✚ ✣ ✚
1 )− ✤ ( ✜ ✟ 1 ) ■ ( ✜ ✆ )] sin( ✥ ✝ ) sin
References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.3-13. Domain: ✟ 1 ≤ ✆ ≤ ✟ 2 ,0≤ ✝ ≤2 ✘ ,0≤ ✞ ≤ ❀ . Second boundary value problem.
A hollow circular cylinder of finite length is considered. Boundary conditions are prescribed:
Here, the ✜ are roots of the transcendental equation ✚ ✣ ✚ ✣ ✚ ✣
1 ) ■ ( ✜ ✆ )] cos( ✥ ✝ ) cos ,
. To the zero eigenvalue ✖
=[ ✤ ( ✜ ✆ ) ■ ✩ ( ✜ ✟ 1 )− ✤ ✩ ( ✜ ✟ 1 ) ■ ( ✜ ✆ )] sin( ✥ ✝ ) cos
000 = 1. The square of the norm of an eigenfunction is defined as ✚
000 ☎ = 0 there is a corresponding eigenfunction (1)
where ❈ ✪✬✫ 0 is the Kronecker delta.
References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
8.3.3-14. Domain: ❖ 1 ≤ P ≤ ❖ 2 ,0≤ ◗ ≤2 ✘ ,0≤ ❘ ≤ ❀ . Mixed boundary value problems.
1 ❙ . A hollow circular cylinder of finite length is considered. Boundary conditions are prescribed:
Here, the ✜ are roots of the transcendental equation ✚ ✣ ✚ ✣ ✚ ✣
( ✜ ✣ ✚ )] cos( ◗ ) cos ,
. The square of the norm of an eigenfunction is defined as ✚ ✚ ✣ ✚ ✚ ✣ ✚ ✚ ✚
)] sin( ✥ ◗ ) cos
= ❉ 2 for = 0,
1 for ✥ ≠ 0.
2 ❙ . A hollow circular cylinder of finite length is considered. Boundary conditions are prescribed: ✮ ✯
=0 ✯ at
1 ❚ , =0 at P = ❖ 2 ,
=0 ❚ at ❘ = 0, =0 at ❘ = ❀ . Eigenvalues: ✚
Here, the ✣ are roots of the transcendental equation
)] cos( ) sin ❅
(2) ✜ ❁ =[ ✤ ( ✜ P ) ■ ✩ ( ✜ ❖
)] sin( ✥ ◗ ) sin
The square of the norm of an eigenfunction is defined as ✚
, where ❯ is defined in Item 1 ❙ .
8.3.3-15. Domain: 0 ≤ P ≤ ❖ ,0≤ ◗ ≤ ◗ 0 ,0≤ ❘ ≤ ❀ . First boundary value problem.
A cylindrical sector of finite thickness is considered. Boundary conditions are prescribed:
Here, the ✜ are positive roots of the transcendental equation ✣ ✚ ✤
0 ( ) = 0. Eigenfunctions:
. The square of the norm of an eigenfunction is defined as ✚ ✚ ❩ ❬❪❭ ✣ ✚
sin ◗
sin
8.3.3-16. Domain: 0 ≤ P ≤ ❖ ,0≤ ◗ ≤ ◗ 0 ,0≤ ❘ ≤ ❀ . Mixed boundary value problem.
A cylindrical sector of finite thickness is considered. Boundary conditions are prescribed:
Here, the ✜ are positive roots of the transcendental equation ✤ ✣ ✚
0 ( ) = 0. Eigenfunctions:
. The square of the norm of an eigenfunction is defined as ✚ ✚ ❩ ❬❪❭ ✣ ✚
sin ◗
cos
1 for ✰ = 0,
0 for ✰ ≠ 0.
8.3.4. Problems in Spherical Coordinates
The three-dimensional homogeneous Helmholtz equation in the spherical coordinate system is written as ❅
8.3.4-1. Particular solutions: ✣
(cos )( cos
+ sin
+1 2 ( ❢ P ) ❣ ❤ (cos ❫ )( ✐ cos ❄ ◗ + ❜ sin ❄ ❬ ◗ ❡ ), ❴ = ❢ ,
+1 ❬ 2 ( ) ❡ (cos )( cos
( ❢ P ) ❣ ❤ (cos ❫ )( ✐ cos ❄ ◗ + ❜ sin ❄ ◗ ), ❴ =− ❢ P 2 ❦ ❡ +1 2 ,
where ❧ , ❄ = 0, 1, 2, ❂❃❂❃❂ ; ✐ and ❜ are arbitrary constants; ❞ ♠ ( ♥ ) and ■ ( ♠ ♥ ) are the Bessel functions;
( ) are the associated Legendre functions that are expressed in terms of the Legendre polynomials ❬ ❡ ❣ ❡ ( ♥ ) as
( ♥ ) and
( ) are the modified Bessel functions; and the
8.3.4-2. Domain: 0 ≤ P ≤ ❖ . First boundary value problem.
1 ❙ . A spherical domain is considered. A homogeneous boundary condition is prescribed,
Here, the ❢ ❁ are positive zeros of the Bessel functions, +1 2 ( ❢ ) = 0. Note that the
+1 2 ( ) can
be expressed in terms of elementary functions, see Bateman and Erd ´elyi (1953, Vol. 2). Eigenfunctions:
(cos ❫ ) cos ❄ ◗ ,
(cos ❫ ) sin ❄ ◗ ,
Here, the ❣ ❤ ❡ ( ♥ ) are the associated Legendre functions. Eigenfunctions possessing central symmetry (i.e., independent of ❫
and ):
(1) 00 P
Eigenfunctions possessing axial symmetry (i.e., independent of ❬ ❅ ◗ ):
The square of the norm of an eigenfunction: ❬
. A spherical domain is considered. A nonhomogeneous boundary condition is prescribed, ✉
( ➂ ❹ , ❺ )= ➃ ➇ ⑩ (cos ❹ ) sin ➂ ❺ for ➂ = 1, 2, ➈❃➈❃➈ , ❯ = ➉ 2 for = 0,
(cos ❹ ) cos ➂ ❺
for ➂ = −1, −2, ➈❃➈❃➈ ,
for
➊✬➋ The solution was written out under the assumption that +1 ❽ ❼ 2 (
) ≠ 0 for = 0, 1, 2,
References : M. M. Smirnov (1975), A. N. Tikhonov and A. A. Samarskii (1990). ⑩
Paragraphs 8.3.4-3 through 8.3.4-6 present only the eigenvalues and eigenfunctions of homoge- neous boundary value problems for the homogeneous Helmholtz equation (with ➍ ≡ 0). The solutions of the corresponding nonhomogeneous boundary value problems ( ➍ ➎
0) can be constructed by the relations specified in Paragraph 8.3.1-4.
8.3.4-3. Domain: 0 ≤ ⑦ ≤ ❶ . Second boundary value problem.
A spherical domain is considered. A boundary condition is prescribed: ➏ ➐➒➑
=0 at ⑦ = ❶ .
Eigenvalues:
= 1, 2, 3, Here, the ➔
are roots of the transcendental equation
(cos ) cos
(cos ) sin
The square of the norm of an eigenfunction: ❸
Reference : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964).
8.3.4-4. Domain: 0 ≤ ⑦ ≤ ❶ . Third boundary value problem.
A spherical domain is considered. A boundary condition is prescribed:
Here, the ➓
are positive roots of the transcendental equation
(cos ) cos
(cos ❹ ) sin ➂ ❺ ,
Here, the ➇ ( ♥ ) are the associated Legendre functions. The square of the norm of an eigenfunction: ⑩ ❸
Reference ⑩ ❸ ➓ : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964). ⑩ ❸ ➓
8.3.4-5. Domain: ❶ ≤ ⑦ < ➟ . First boundary value problem.
A spherical cavity is considered and the dependent variable is prescribed at its surface:
= ➠ ( ❹ , ❺ ) at ➡ = ❶ ,
and the radiation conditions are prescribed at infinity (see Paragraph 8.3.1-5, Item 2 ➢ ). Solution for ⑨ = ♣ 2 > 0:
+1 ➳ 2 ( ➭ ) is the Hankel function of the second kind and the other quantities are defined just ➊✬➋ as in Paragraph 8.3.4-2, Item 2 ⑩
Reference : A. N. Tikhonov and A. A. Samarskii (1990).
8.3.4-6. Domain: ❶ 1 ≤ ➡ ≤ ❶ 2 . First boundary value problem.
A spherical layer is considered. Boundary conditions are prescribed:
Here, the ➓
are positive roots of the transcendental equation
(cos ➩ ) cos ➪ ➫ ,
Here, the ⑩
( ➶ ) are the associated Legendre functions and
The square of the norm of an eigenfunction: ⑩
8.3.5. Other Orthogonal Curvilinear Coordinates
The homogenous three-dimensional Helmholtz equation admits separation of variables in the eleven orthogonal systems of coordinates listed in Table 29 .
For the parabolic cylindrical system of coordinates, the multipliers ➠ and ➴ are expressed in terms of the parabolic cylinder functions as
2 ❐ ( ➸ 2 − ) −1 ➳ 2 , ➱ = ❒ 4( ➸ 2 − ) ❮ 1 ➔ 4 , where ➷ 1 , ❜ 1 , ➷ 2 , and ❜ 2 are arbitrary constants.
For the elliptic cylindrical system of coordinates, the functions ➠ and ➴ are determined by the modified Mathieu equation and Mathieu equation, respectively, so that
where Ce ( ❰ , Ï
) and Se ⑩ ( ❰ , Ï ⑩ ) are the modified Mathieu functions, and ce ( Ð , Ï ) and se ( Ð , Ï ) are the Mathieu functions; to each value of the parameter ⑩ ⑩ Ï there are certain corresponding eigenvalues ⑩ ⑩
= ❐ ( Ï ) [see Abramowitz and Stegun (1964)]. In the prolate and oblate spheroidal systems of coordinates, the equations for ⑩ ➠ and ➴ are different forms of the spheroidal wave equation, whose bounded solutions are given by ➵ ➵
) = Ps Ò | | (cosh ❰ , 2 ),
) = Ps | | (cos Ð , 2 Ñ ➵ )
for prolate spheroid,
2 | ➠ ⑩ ( ❰ ) = Ps (− Ó sinh ❰ Ò ,
), ➴ ( ❰ ) = Ps | (cos Ð ,− 2 Ñ )
for oblate spheroid,
is an integer, ➹ = 0, 1, 2, ⑩ Ô❃Ô❃Ô , − ➹ ≤ ➸ ≤ ➹ ,
where Ps Ò ( Õ ,
) are the spheroidal wave functions; see Bateman and Erd ´elyi (1955, Vol. 3), Arscott (1964), and Meixner and Sch¨afke (1965). The separation of variables for the Helmholtz equation ⑩
in modified prolate and oblate spheroidal systems of coordinates, as well as the spheroidal wave functions, are discussed in Abramowitz and Stegun (1964).
In the parabolic coordinate system, the solutions of the equations for ➠ and ➴ are expressed in terms of the degenerate hypergeometric functions [see Miller, Jr. (1977)] as follows:
exp ➯
( )= exp
TABLE 29
Orthogonal coordinates ¯ ,¯ ,¯ Õ that allow separable solutions of the form ➵
(¯ á ) ➴ (¯ ) ã = á (¯ Õ ) for the three-dimensional Helmholtz equation ä 3 + =0
Coordinates Transformations Particular solutions (or equations for â , ➴ , ã )
1 à ) cos( ➸ 2 + å 2 ) cos( ➵ ➸ 3 Õ + å 3 ), Cartesian
= ß , = cos( ➸ 1 + å
where ➸ 2 1 + ➸ 2 ➸ 2 , , Õ
see also Paragraph 8.3.2-1
+ è ) cos( + å ), Cylindrical
where ➺ ➸ ç + ❐ ç = , see also Paragraph 8.3.3-1
are the Bessel functions) Parabolic
and ➨
) ➴ ( ✃ ) cos( ➸ Õ + å ), cylindrical
= Ñ cosh ❰ cos Ð , = â ➵ ( ❰ ) ➴ ( Ð ) cos( ➸ Õ + å ), cylindrical
(cos æ ➩ ) cos( ➪ ➫ + å ),
(cos ) cos( + ),
= cos
where = 2 ❐ ; see also Paragraph 8.3.4-1
sinh ❰ sin Ð cos ➫ , = â ( ❰ ) ➴ ( ➵ Ð ) cos( ➸ ➫ + å ), spheroidal à = sinh ❰ sin Ð sin ➫ ,
Prolate á =
+ â é coth ❰ + (− + 2 2 2 Ñ 2 ➵ ì ❐ Ñ sinh ❰ − ➸ sinh ❰ ) â = 0,
= Ñ cosh ❰ cos Ð
cosh ❰ sin Ð cos ➫ , = â ( ❰ ) ➴ ➵ ( Ð ) cos( ➸ ➫ + å ), spheroidal
Oblate á =
+ â é tanh ❰ + (− + 2 ➵ cosh = 2 Ñ cosh sin sin , ❐ Ñ ❰ + ➸ 2 ì cosh 2 ❰ ) â = 0,
+ ➴ é cot Ð +( − 2 2 2 ì = 2
sinh cos
cosh ❰ cos Ð sinh ➫ ,
sinh ❰ sin Ð
cosh 2 ➫ − 1 + cos 2 2 Ð − cosh 2 ➫ ) Ñ ❐ 2 Ñ cosh 4 ➫ ) ã =0
General â ( −1 ) ( )[ ( ) ] +( + 1 + 2 ) = 0, ellipsoidal í
where ☛ = sn 2 ( ☎ , ✡ ), ☞ = sn 2 ( ✝ , ✡ ), ✡ = ÷ ý
In the case of the paraboloidal coordinate system, the equations for ✌ , û , and ✆ are reduced to the Whittaker–Hill equation ✍
Denote by gc ✕ ( ✔ ; ✑ , ✒ ) and gs ✕ ( ✔ ; ✑ , ✒ ), respectively, the even and odd 2 ✖ -periodic solutions of the Whittaker–Hill equation, which is a generalization of the Mathieu equation. The subscript
= 0, 1, 2, ✗✘✗✘✗ labels the discrete eigenvalues ☛ = ☛ ✕ . Each of the solutions gc ✕ and gs ✕ can
be represented in the form of an infinite convergent trigonometric series in cos ✠ ✔ and sin ✠ ✔ , respectively; see Urvin and Arscott (1970). The functions ✌ , û , and ✆ can be expressed in terms of the periodic solutions of the Whittaker–Hill equation as follows [Miller, Jr. (1977)]:
where ✜ = ÷ ù and ✡ = ☛ ✕ − 1 2 ý 2 ù .
For the general ellipsoidal coordinates, the functions ✌ , û , and ✆ are expressed in terms of the ellipsoidal wave functions; for details, see Arscott (1964) and Miller, Jr. (1977). For the conical coordinate system, the functions û õ and ✆ õ are determined by the Lam ´e equations that involve the Jacobian elliptic function sn = sn( , ✡ ). The unambiguity conditions for the transformation yield ✠ = 0, 1, 2, ✗✘✗✘✗ It is known that, for any positive integer ✠ , there exist exactly 2 ✠ +1 solutions corresponding to 2 ✠ +1 different eigenvalues ú . These solutions can be represented the form of finite series known as Lam ´e polynomials. For more details about the Lam´e equation and its solutions, see Whittaker and Watson (1963), Arscott (1964), Bateman and Erd´elyi (1955), and Miller, Jr. (1977).
Unlike the Laplace equation, there are no nontrivial transformations for the three-dimensional Helmholtz equation that allow the ✥ ✦★✧ -separation of variables.
References for Subsection 8.3.5: F. M. Morse and H. Feshbach (1953, Vols. 1–2), P. Moon and D. Spencer (1961), A. Makarov, J. Smorodinsky, K. Valiev, and P. Winternitz (1967), W. Miller, Jr. (1977).