Problems in Spherical Coordinates ✼
3.1.3. Problems in Spherical Coordinates ✼
The heat equation in the spherical coordinate system has the form
This representation is convenient to describe three-dimensional heat and mass exchange phenomena in domains bounded by coordinate surfaces of the spherical coordinate system.
One-dimensional problems with central symmetry that have solutions of the form ★ = ( ✟ , ✍ ) are discussed in Subsection 1.2.3.
3.1.3-1. Domain: 0 ≤ ✟ ≤ ✸ ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . First boundary value problem.
A spherical domain is considered. The following conditions are prescribed:
= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0
(initial condition),
= ✫ ( ✿ , ✠ , ✍ ) at ✟ = ✸
(boundary condition).
× ❃ ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ),
Here, the ✾ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽
) are the associated Legendre functions ✘
expressed in terms of the Legendre polynomials ❃ ❃●✜ ❈ ( ✽ ✘ ) as follows: ✘
and the ❇ ✻ ❍❏■ are positive roots of the transcendental equation +1 2 ( ❇ ✸ ) = 0.
References : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984). ✼
3.1.3-2. Domain: 0 ≤ ✟ ≤ ✸ ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Second boundary value problem.
A spherical domain is considered. The following conditions are prescribed:
= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0
(initial condition),
= ✫ ( ✿ , ✠ , ✍ ) at ✟ = ✸
(boundary condition).
× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ), for ❉ = 0,
Here, the ✼ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ) are the associated Legendre functions (see Paragraph 3.1.3-1), and the ❇ ✻ ✼ are positive roots of the transcendental equation ✘ ✜ ✘ ✜
3.1.3-3. Domain: 0 ≤ ✟ ≤ ✸ ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Third boundary value problem.
A spherical domain is considered. The following conditions are prescribed:
= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0
(initial condition),
+ ❉ ★ = ✫ ( ✿ , ✠ , ✍ ) at ✟ = ✸ (boundary condition). The solution ★ ( ✟
, ✘ ✿ , ✠ , ✍ ) is determined by the formula in Paragraph 3.1.3-2 where
× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ◆ ( ✠ − ✌ )] exp(− ❇ ✻ ✑ ✍ ),
Here, the ✘ +1 ✼ 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ) are the associated Legendre functions (see Paragraph 3.1.3-1), and the ❇ ✻ ✼ ✘ are positive roots of the transcendental equation ✜ ✘ ✜
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). ✼ ✾ ✼
3.1.3-4. Domain: ✸ 1 ≤ ✟ ≤ ✸ 2 ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . First boundary value problem.
A spherical layer is considered. The following conditions are prescribed:
= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0
(initial condition),
= ✫ 1 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 1 (boundary condition),
= ✫ 2 ( ✿ , ✛ ✠ ✛ , ✍ ) at ✟ = ✸ 2 (boundary condition). Solution:
× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ).
where the ✘ +1 2 (
) are the Bessel functions, the ❃ ❈ ✼ ( ✽ ) are the associated Legendre functions ✘ ✼
expressed in terms of the Legendre polynomials ❃ ❃●✜ ❈
( ) as follows: ✘ ✘
are positive roots of the transcendental equation ❙ ✘ ✜
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
3.1.3-5. Domain: ✸ 1 ≤ ✟ ≤ ✸ 2 ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Second boundary value problem.
A spherical layer is considered. The following conditions are prescribed:
= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0
(initial condition),
= ✫ 1 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 1 (boundary condition),
= 2 (boundary condition). Solution:
= ✛ 2 ( , , ) at
( 2 ☛ , ☞ , ✌ ) ( ✟ , ✿ , ✠ , ☛ , ☞ , ✌ , ✍ ) ☛ sin ☞ ✭ ☛ ✭ ☞ ✭ ✌
1 ( ☞ , ✌ , ✯ ) ( ✟ , ✿ , ✠ , ✸ 1 , ☞ , ✌ , ✍ − ✯ ) sin ☞ ✭ ☞ ✭ ✌ ✭ ✯
2 ( ☞ , ✌ , ✯ ) ( ✟ , ✿ , ✠ , ✸ 2 , ☞ , ✌ , ✍ − ✯ ) sin ☞ ✭ ☞ ✭ ✌ ✭ ✯ ,
× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ). Here,
) and ❚ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ✼ ) are the associated Legendre functions (see Paragraph 3.1.3-4), and the ❙ ✼ ✘ ✜ ❇ ✻ are positive roots of the transcendental equation ❙ ✘ ✜
where the ✘ +1 2 (
can be expressed in terms of the Bessel functions and their derivatives; see Budak, Samarskii, and Tikhonov (1980).
The integrals that determine the coefficients ✾
3.1.3-6. Domain: ✸ 1 ≤ ✟ ≤ ✸ 2 ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Third boundary value problem.
A spherical layer is considered. The following conditions are prescribed:
= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0
(initial condition),
− ★ ❉ 1 = ✫ 1 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 1 (boundary condition),
+ ★ ❉ 2 = ✫ 2 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 2 (boundary condition).
The solution ✘ (
, ✿ , ✠ , ✍ ) is determined by the formula in Paragraph 3.1.3-5 where ▲ ❙ ✘ ✜ ✘ ❙ ✘ ✜ ✘
× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ◆ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ). Here,
where the ✾ +1 2 ( ✟ ) and ❚ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ) are the associated Legendre ✼
functions (see Paragraph 3.1.3-4), and the ❙ ✘ ❇ ✻ ✼ ✜ are positive roots of the transcendental equation ❙ ✘ ✜
can be expressed in terms of the Bessel functions and their derivatives. ❍❏■
The integrals that determine the coefficients ✾
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
3.1.3-7. Domain: 0 ≤ ✟ < ✧ ,0≤ ✿ ≤ ✿ 0 ,0≤ ✠ ≤2 ✏ . First boundary value problem.
A cone is considered. The following conditions are prescribed:
= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0
(initial condition),
= ✫ ( ✟ , ✠ , ✍ ) at ✿ = ✿ 0 (boundary condition). Solution: ✛
− ❆ ✻ (cos ✿ ) ❈ − ✻ (cos ☞ ) cos[ ❲ ( ✠ − ✌
( ✽ ) is the modified Legendre function expressed as ✜
, ❬ , ❭ ; ✽ ) is the Gaussian hypergeometric function and ( ❪ ) is the gamma function. The ✻
summation with respect to ❱ is performed over all roots of the equation ❈ − (cos ✿ 0 ) = 0 that are greater than −1 ❫ 2.
Reference : H. S. Carslaw and J. C. Jaeger (1984).